Metamath Proof Explorer

Theorem poimirlem27

Description: Lemma for poimir showing that the difference between admissible faces in the whole cube and admissible faces on the back face is even. Equation (7) of Kulpa p. 548. (Contributed by Brendan Leahy, 21-Aug-2020)

Ref Expression
Hypotheses poimir.0 
|- ( ph -> N e. NN )
poimirlem28.1 
|- ( p = ( ( 1st ` s ) oF + ( ( ( ( 2nd ` s ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` s ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) -> B = C )
poimirlem28.2 
|- ( ( ph /\ p : ( 1 ... N ) --> ( 0 ... K ) ) -> B e. ( 0 ... N ) )
poimirlem28.3 
|- ( ( ph /\ ( n e. ( 1 ... N ) /\ p : ( 1 ... N ) --> ( 0 ... K ) /\ ( p ` n ) = 0 ) ) -> B < n )
poimirlem28.4 
|- ( ( ph /\ ( n e. ( 1 ... N ) /\ p : ( 1 ... N ) --> ( 0 ... K ) /\ ( p ` n ) = K ) ) -> B =/= ( n - 1 ) )
Assertion poimirlem27 
|- ( ph -> 2 || ( ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } ) - ( # ` { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } ) ) )

Proof

Step Hyp Ref Expression
1 poimir.0 
 |-  ( ph -> N e. NN )
2 poimirlem28.1 
 |-  ( p = ( ( 1st ` s ) oF + ( ( ( ( 2nd ` s ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` s ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) -> B = C )
3 poimirlem28.2 
 |-  ( ( ph /\ p : ( 1 ... N ) --> ( 0 ... K ) ) -> B e. ( 0 ... N ) )
4 poimirlem28.3 
 |-  ( ( ph /\ ( n e. ( 1 ... N ) /\ p : ( 1 ... N ) --> ( 0 ... K ) /\ ( p ` n ) = 0 ) ) -> B < n )
5 poimirlem28.4 
 |-  ( ( ph /\ ( n e. ( 1 ... N ) /\ p : ( 1 ... N ) --> ( 0 ... K ) /\ ( p ` n ) = K ) ) -> B =/= ( n - 1 ) )
6 fzfi 
 |-  ( 0 ... K ) e. Fin
7 fzfi 
 |-  ( 1 ... N ) e. Fin
8 mapfi 
 |-  ( ( ( 0 ... K ) e. Fin /\ ( 1 ... N ) e. Fin ) -> ( ( 0 ... K ) ^m ( 1 ... N ) ) e. Fin )
9 6 7 8 mp2an 
 |-  ( ( 0 ... K ) ^m ( 1 ... N ) ) e. Fin
10 fzfi 
 |-  ( 0 ... ( N - 1 ) ) e. Fin
11 mapfi 
 |-  ( ( ( ( 0 ... K ) ^m ( 1 ... N ) ) e. Fin /\ ( 0 ... ( N - 1 ) ) e. Fin ) -> ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) e. Fin )
12 9 10 11 mp2an 
 |-  ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) e. Fin
13 12 a1i 
 |-  ( ph -> ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) e. Fin )
14 2z 
 |-  2 e. ZZ
15 14 a1i 
 |-  ( ph -> 2 e. ZZ )
16 fzofi 
 |-  ( 0 ..^ K ) e. Fin
17 mapfi 
 |-  ( ( ( 0 ..^ K ) e. Fin /\ ( 1 ... N ) e. Fin ) -> ( ( 0 ..^ K ) ^m ( 1 ... N ) ) e. Fin )
18 16 7 17 mp2an 
 |-  ( ( 0 ..^ K ) ^m ( 1 ... N ) ) e. Fin
19 mapfi 
 |-  ( ( ( 1 ... N ) e. Fin /\ ( 1 ... N ) e. Fin ) -> ( ( 1 ... N ) ^m ( 1 ... N ) ) e. Fin )
20 7 7 19 mp2an 
 |-  ( ( 1 ... N ) ^m ( 1 ... N ) ) e. Fin
21 f1of 
 |-  ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> f : ( 1 ... N ) --> ( 1 ... N ) )
22 21 ss2abi 
 |-  { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } C_ { f | f : ( 1 ... N ) --> ( 1 ... N ) }
23 ovex 
 |-  ( 1 ... N ) e. _V
24 23 23 mapval 
 |-  ( ( 1 ... N ) ^m ( 1 ... N ) ) = { f | f : ( 1 ... N ) --> ( 1 ... N ) }
25 22 24 sseqtrri 
 |-  { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } C_ ( ( 1 ... N ) ^m ( 1 ... N ) )
26 ssfi 
 |-  ( ( ( ( 1 ... N ) ^m ( 1 ... N ) ) e. Fin /\ { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } C_ ( ( 1 ... N ) ^m ( 1 ... N ) ) ) -> { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } e. Fin )
27 20 25 26 mp2an 
 |-  { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } e. Fin
28 xpfi 
 |-  ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) e. Fin /\ { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } e. Fin ) -> ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) e. Fin )
29 18 27 28 mp2an 
 |-  ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) e. Fin
30 fzfi 
 |-  ( 0 ... N ) e. Fin
31 xpfi 
 |-  ( ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) e. Fin /\ ( 0 ... N ) e. Fin ) -> ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) e. Fin )
32 29 30 31 mp2an 
 |-  ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) e. Fin
33 rabfi 
 |-  ( ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) e. Fin -> { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } e. Fin )
34 32 33 ax-mp 
 |-  { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } e. Fin
35 hashcl 
 |-  ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } e. Fin -> ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ) e. NN0 )
36 35 nn0zd 
 |-  ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } e. Fin -> ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ) e. ZZ )
37 34 36 mp1i 
 |-  ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) -> ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ) e. ZZ )
38 dfrex2 
 |-  ( E. t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) <-> -. A. t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -. ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) )
39 nfv 
 |-  F/ t ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) )
40 nfcv 
 |-  F/_ t 2
41 nfcv 
 |-  F/_ t ||
42 nfcv 
 |-  F/_ t #
43 nfrab1 
 |-  F/_ t { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) }
44 42 43 nffv 
 |-  F/_ t ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } )
45 40 41 44 nfbr 
 |-  F/ t 2 || ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } )
46 neq0 
 |-  ( -. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } = (/) <-> E. s s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } )
47 iddvds 
 |-  ( 2 e. ZZ -> 2 || 2 )
48 14 47 ax-mp 
 |-  2 || 2
49 vex 
 |-  s e. _V
50 hashsng 
 |-  ( s e. _V -> ( # ` { s } ) = 1 )
51 49 50 ax-mp 
 |-  ( # ` { s } ) = 1
52 51 oveq2i 
 |-  ( 1 + ( # ` { s } ) ) = ( 1 + 1 )
53 df-2 
 |-  2 = ( 1 + 1 )
54 52 53 eqtr4i 
 |-  ( 1 + ( # ` { s } ) ) = 2
55 48 54 breqtrri 
 |-  2 || ( 1 + ( # ` { s } ) )
56 rabfi 
 |-  ( ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) e. Fin -> { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } e. Fin )
57 diffi 
 |-  ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } e. Fin -> ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) e. Fin )
58 32 56 57 mp2b 
 |-  ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) e. Fin
59 snfi 
 |-  { s } e. Fin
60 incom 
 |-  ( ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) i^i { s } ) = ( { s } i^i ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) )
61 disjdif 
 |-  ( { s } i^i ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) ) = (/)
62 60 61 eqtri 
 |-  ( ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) i^i { s } ) = (/)
63 hashun 
 |-  ( ( ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) e. Fin /\ { s } e. Fin /\ ( ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) i^i { s } ) = (/) ) -> ( # ` ( ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) u. { s } ) ) = ( ( # ` ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) ) + ( # ` { s } ) ) )
64 58 59 62 63 mp3an 
 |-  ( # ` ( ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) u. { s } ) ) = ( ( # ` ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) ) + ( # ` { s } ) )
65 difsnid 
 |-  ( s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } -> ( ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) u. { s } ) = { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } )
66 65 fveq2d 
 |-  ( s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } -> ( # ` ( ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) u. { s } ) ) = ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) )
67 64 66 eqtr3id 
 |-  ( s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } -> ( ( # ` ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) ) + ( # ` { s } ) ) = ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) )
68 67 adantl 
 |-  ( ( ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) /\ s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) -> ( ( # ` ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) ) + ( # ` { s } ) ) = ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) )
69 1 ad3antrrr 
 |-  ( ( ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) /\ s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) -> N e. NN )
70 fveq2 
 |-  ( t = u -> ( 2nd ` t ) = ( 2nd ` u ) )
71 70 breq2d 
 |-  ( t = u -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` u ) ) )
72 71 ifbid 
 |-  ( t = u -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` u ) , y , ( y + 1 ) ) )
73 72 csbeq1d 
 |-  ( t = u -> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = [_ if ( y < ( 2nd ` u ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) )
74 2fveq3 
 |-  ( t = u -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` u ) ) )
75 2fveq3 
 |-  ( t = u -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` u ) ) )
76 75 imaeq1d 
 |-  ( t = u -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) )
77 76 xpeq1d 
 |-  ( t = u -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) X. { 1 } ) )
78 75 imaeq1d 
 |-  ( t = u -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) )
79 78 xpeq1d 
 |-  ( t = u -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) )
80 77 79 uneq12d 
 |-  ( t = u -> ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) = ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) )
81 74 80 oveq12d 
 |-  ( t = u -> ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` u ) ) oF + ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) )
82 81 csbeq2dv 
 |-  ( t = u -> [_ if ( y < ( 2nd ` u ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = [_ if ( y < ( 2nd ` u ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` u ) ) oF + ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) )
83 73 82 eqtrd 
 |-  ( t = u -> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = [_ if ( y < ( 2nd ` u ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` u ) ) oF + ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) )
84 83 mpteq2dv 
 |-  ( t = u -> ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` u ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` u ) ) oF + ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) )
85 breq1 
 |-  ( y = w -> ( y < ( 2nd ` u ) <-> w < ( 2nd ` u ) ) )
86 id 
 |-  ( y = w -> y = w )
87 oveq1 
 |-  ( y = w -> ( y + 1 ) = ( w + 1 ) )
88 85 86 87 ifbieq12d 
 |-  ( y = w -> if ( y < ( 2nd ` u ) , y , ( y + 1 ) ) = if ( w < ( 2nd ` u ) , w , ( w + 1 ) ) )
89 88 csbeq1d 
 |-  ( y = w -> [_ if ( y < ( 2nd ` u ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` u ) ) oF + ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = [_ if ( w < ( 2nd ` u ) , w , ( w + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` u ) ) oF + ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) )
90 oveq2 
 |-  ( j = i -> ( 1 ... j ) = ( 1 ... i ) )
91 90 imaeq2d 
 |-  ( j = i -> ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... i ) ) )
92 91 xpeq1d 
 |-  ( j = i -> ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... i ) ) X. { 1 } ) )
93 oveq1 
 |-  ( j = i -> ( j + 1 ) = ( i + 1 ) )
94 93 oveq1d 
 |-  ( j = i -> ( ( j + 1 ) ... N ) = ( ( i + 1 ) ... N ) )
95 94 imaeq2d 
 |-  ( j = i -> ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` u ) ) " ( ( i + 1 ) ... N ) ) )
96 95 xpeq1d 
 |-  ( j = i -> ( ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` u ) ) " ( ( i + 1 ) ... N ) ) X. { 0 } ) )
97 92 96 uneq12d 
 |-  ( j = i -> ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) = ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... i ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( i + 1 ) ... N ) ) X. { 0 } ) ) )
98 97 oveq2d 
 |-  ( j = i -> ( ( 1st ` ( 1st ` u ) ) oF + ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` u ) ) oF + ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... i ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( i + 1 ) ... N ) ) X. { 0 } ) ) ) )
99 98 cbvcsbv 
 |-  [_ if ( w < ( 2nd ` u ) , w , ( w + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` u ) ) oF + ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = [_ if ( w < ( 2nd ` u ) , w , ( w + 1 ) ) / i ]_ ( ( 1st ` ( 1st ` u ) ) oF + ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... i ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( i + 1 ) ... N ) ) X. { 0 } ) ) )
100 89 99 eqtrdi 
 |-  ( y = w -> [_ if ( y < ( 2nd ` u ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` u ) ) oF + ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = [_ if ( w < ( 2nd ` u ) , w , ( w + 1 ) ) / i ]_ ( ( 1st ` ( 1st ` u ) ) oF + ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... i ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( i + 1 ) ... N ) ) X. { 0 } ) ) ) )
101 100 cbvmptv 
 |-  ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` u ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` u ) ) oF + ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) = ( w e. ( 0 ... ( N - 1 ) ) |-> [_ if ( w < ( 2nd ` u ) , w , ( w + 1 ) ) / i ]_ ( ( 1st ` ( 1st ` u ) ) oF + ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... i ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( i + 1 ) ... N ) ) X. { 0 } ) ) ) )
102 84 101 eqtrdi 
 |-  ( t = u -> ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) = ( w e. ( 0 ... ( N - 1 ) ) |-> [_ if ( w < ( 2nd ` u ) , w , ( w + 1 ) ) / i ]_ ( ( 1st ` ( 1st ` u ) ) oF + ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... i ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( i + 1 ) ... N ) ) X. { 0 } ) ) ) ) )
103 102 eqeq2d 
 |-  ( t = u -> ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) <-> x = ( w e. ( 0 ... ( N - 1 ) ) |-> [_ if ( w < ( 2nd ` u ) , w , ( w + 1 ) ) / i ]_ ( ( 1st ` ( 1st ` u ) ) oF + ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... i ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( i + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) )
104 103 cbvrabv 
 |-  { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } = { u e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( w e. ( 0 ... ( N - 1 ) ) |-> [_ if ( w < ( 2nd ` u ) , w , ( w + 1 ) ) / i ]_ ( ( 1st ` ( 1st ` u ) ) oF + ( ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... i ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` u ) ) " ( ( i + 1 ) ... N ) ) X. { 0 } ) ) ) ) }
105 elmapi 
 |-  ( x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) -> x : ( 0 ... ( N - 1 ) ) --> ( ( 0 ... K ) ^m ( 1 ... N ) ) )
106 105 ad3antlr 
 |-  ( ( ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) /\ s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) -> x : ( 0 ... ( N - 1 ) ) --> ( ( 0 ... K ) ^m ( 1 ... N ) ) )
107 simpr 
 |-  ( ( ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) /\ s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) -> s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } )
108 simpl 
 |-  ( ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) -> E. p e. ran x ( p ` n ) =/= 0 )
109 108 ralimi 
 |-  ( A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) -> A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= 0 )
110 109 ad2antlr 
 |-  ( ( ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) /\ s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) -> A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= 0 )
111 fveq2 
 |-  ( n = m -> ( p ` n ) = ( p ` m ) )
112 111 neeq1d 
 |-  ( n = m -> ( ( p ` n ) =/= 0 <-> ( p ` m ) =/= 0 ) )
113 112 rexbidv 
 |-  ( n = m -> ( E. p e. ran x ( p ` n ) =/= 0 <-> E. p e. ran x ( p ` m ) =/= 0 ) )
114 fveq1 
 |-  ( p = q -> ( p ` m ) = ( q ` m ) )
115 114 neeq1d 
 |-  ( p = q -> ( ( p ` m ) =/= 0 <-> ( q ` m ) =/= 0 ) )
116 115 cbvrexvw 
 |-  ( E. p e. ran x ( p ` m ) =/= 0 <-> E. q e. ran x ( q ` m ) =/= 0 )
117 113 116 bitrdi 
 |-  ( n = m -> ( E. p e. ran x ( p ` n ) =/= 0 <-> E. q e. ran x ( q ` m ) =/= 0 ) )
118 117 rspccva 
 |-  ( ( A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= 0 /\ m e. ( 1 ... N ) ) -> E. q e. ran x ( q ` m ) =/= 0 )
119 110 118 sylan 
 |-  ( ( ( ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) /\ s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) /\ m e. ( 1 ... N ) ) -> E. q e. ran x ( q ` m ) =/= 0 )
120 simpr 
 |-  ( ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) -> E. p e. ran x ( p ` n ) =/= K )
121 120 ralimi 
 |-  ( A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) -> A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= K )
122 121 ad2antlr 
 |-  ( ( ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) /\ s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) -> A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= K )
123 111 neeq1d 
 |-  ( n = m -> ( ( p ` n ) =/= K <-> ( p ` m ) =/= K ) )
124 123 rexbidv 
 |-  ( n = m -> ( E. p e. ran x ( p ` n ) =/= K <-> E. p e. ran x ( p ` m ) =/= K ) )
125 114 neeq1d 
 |-  ( p = q -> ( ( p ` m ) =/= K <-> ( q ` m ) =/= K ) )
126 125 cbvrexvw 
 |-  ( E. p e. ran x ( p ` m ) =/= K <-> E. q e. ran x ( q ` m ) =/= K )
127 124 126 bitrdi 
 |-  ( n = m -> ( E. p e. ran x ( p ` n ) =/= K <-> E. q e. ran x ( q ` m ) =/= K ) )
128 127 rspccva 
 |-  ( ( A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= K /\ m e. ( 1 ... N ) ) -> E. q e. ran x ( q ` m ) =/= K )
129 122 128 sylan 
 |-  ( ( ( ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) /\ s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) /\ m e. ( 1 ... N ) ) -> E. q e. ran x ( q ` m ) =/= K )
130 69 104 106 107 119 129 poimirlem22 
 |-  ( ( ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) /\ s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) -> E! z e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } z =/= s )
131 eldifsn 
 |-  ( z e. ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) <-> ( z e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } /\ z =/= s ) )
132 131 eubii 
 |-  ( E! z z e. ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) <-> E! z ( z e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } /\ z =/= s ) )
133 58 elexi 
 |-  ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) e. _V
134 euhash1 
 |-  ( ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) e. _V -> ( ( # ` ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) ) = 1 <-> E! z z e. ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) ) )
135 133 134 ax-mp 
 |-  ( ( # ` ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) ) = 1 <-> E! z z e. ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) )
136 df-reu 
 |-  ( E! z e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } z =/= s <-> E! z ( z e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } /\ z =/= s ) )
137 132 135 136 3bitr4ri 
 |-  ( E! z e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } z =/= s <-> ( # ` ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) ) = 1 )
138 130 137 sylib 
 |-  ( ( ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) /\ s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) -> ( # ` ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) ) = 1 )
139 138 oveq1d 
 |-  ( ( ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) /\ s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) -> ( ( # ` ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } \ { s } ) ) + ( # ` { s } ) ) = ( 1 + ( # ` { s } ) ) )
140 68 139 eqtr3d 
 |-  ( ( ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) /\ s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) -> ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) = ( 1 + ( # ` { s } ) ) )
141 55 140 breqtrrid 
 |-  ( ( ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) /\ s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) -> 2 || ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) )
142 141 ex 
 |-  ( ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) -> ( s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } -> 2 || ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) ) )
143 142 exlimdv 
 |-  ( ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) -> ( E. s s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } -> 2 || ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) ) )
144 46 143 syl5bi 
 |-  ( ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) -> ( -. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } = (/) -> 2 || ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) ) )
145 dvds0 
 |-  ( 2 e. ZZ -> 2 || 0 )
146 14 145 ax-mp 
 |-  2 || 0
147 hash0 
 |-  ( # ` (/) ) = 0
148 146 147 breqtrri 
 |-  2 || ( # ` (/) )
149 fveq2 
 |-  ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } = (/) -> ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) = ( # ` (/) ) )
150 148 149 breqtrrid 
 |-  ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } = (/) -> 2 || ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) )
151 144 150 pm2.61d2 
 |-  ( ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) -> 2 || ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) )
152 151 ex 
 |-  ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) -> ( A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) -> 2 || ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) ) )
153 152 adantld 
 |-  ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) -> ( ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) -> 2 || ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) ) )
154 iba 
 |-  ( ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) -> ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) <-> ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) ) )
155 154 rabbidv 
 |-  ( ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) -> { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } = { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } )
156 155 fveq2d 
 |-  ( ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) -> ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) = ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ) )
157 156 breq2d 
 |-  ( ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) -> ( 2 || ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } ) <-> 2 || ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ) ) )
158 153 157 mpbidi 
 |-  ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) -> ( ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) -> 2 || ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ) ) )
159 158 a1d 
 |-  ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) -> ( t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) -> 2 || ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ) ) ) )
160 39 45 159 rexlimd 
 |-  ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) -> ( E. t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) -> 2 || ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ) ) )
161 38 160 syl5bir 
 |-  ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) -> ( -. A. t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -. ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) -> 2 || ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ) ) )
162 simpr 
 |-  ( ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) -> ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) )
163 162 con3i 
 |-  ( -. ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) -> -. ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) )
164 163 ralimi 
 |-  ( A. t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -. ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) -> A. t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -. ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) )
165 rabeq0 
 |-  ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } = (/) <-> A. t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -. ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) )
166 164 165 sylibr 
 |-  ( A. t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -. ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) -> { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } = (/) )
167 166 fveq2d 
 |-  ( A. t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -. ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) -> ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ) = ( # ` (/) ) )
168 148 167 breqtrrid 
 |-  ( A. t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -. ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) -> 2 || ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ) )
169 161 168 pm2.61d2 
 |-  ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) -> 2 || ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ) )
170 13 15 37 169 fsumdvds 
 |-  ( ph -> 2 || sum_ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ) )
171 rabfi 
 |-  ( ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) e. Fin -> { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } e. Fin )
172 32 171 ax-mp 
 |-  { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } e. Fin
173 simp1 
 |-  ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C )
174 sneq 
 |-  ( ( 2nd ` t ) = N -> { ( 2nd ` t ) } = { N } )
175 174 difeq2d 
 |-  ( ( 2nd ` t ) = N -> ( ( 0 ... N ) \ { ( 2nd ` t ) } ) = ( ( 0 ... N ) \ { N } ) )
176 difun2 
 |-  ( ( ( 0 ... ( N - 1 ) ) u. { N } ) \ { N } ) = ( ( 0 ... ( N - 1 ) ) \ { N } )
177 1 nnnn0d 
 |-  ( ph -> N e. NN0 )
178 nn0uz 
 |-  NN0 = ( ZZ>= ` 0 )
179 177 178 eleqtrdi 
 |-  ( ph -> N e. ( ZZ>= ` 0 ) )
180 fzm1 
 |-  ( N e. ( ZZ>= ` 0 ) -> ( n e. ( 0 ... N ) <-> ( n e. ( 0 ... ( N - 1 ) ) \/ n = N ) ) )
181 179 180 syl 
 |-  ( ph -> ( n e. ( 0 ... N ) <-> ( n e. ( 0 ... ( N - 1 ) ) \/ n = N ) ) )
182 elun 
 |-  ( n e. ( ( 0 ... ( N - 1 ) ) u. { N } ) <-> ( n e. ( 0 ... ( N - 1 ) ) \/ n e. { N } ) )
183 velsn 
 |-  ( n e. { N } <-> n = N )
184 183 orbi2i 
 |-  ( ( n e. ( 0 ... ( N - 1 ) ) \/ n e. { N } ) <-> ( n e. ( 0 ... ( N - 1 ) ) \/ n = N ) )
185 182 184 bitri 
 |-  ( n e. ( ( 0 ... ( N - 1 ) ) u. { N } ) <-> ( n e. ( 0 ... ( N - 1 ) ) \/ n = N ) )
186 181 185 bitr4di 
 |-  ( ph -> ( n e. ( 0 ... N ) <-> n e. ( ( 0 ... ( N - 1 ) ) u. { N } ) ) )
187 186 eqrdv 
 |-  ( ph -> ( 0 ... N ) = ( ( 0 ... ( N - 1 ) ) u. { N } ) )
188 187 difeq1d 
 |-  ( ph -> ( ( 0 ... N ) \ { N } ) = ( ( ( 0 ... ( N - 1 ) ) u. { N } ) \ { N } ) )
189 1 nnzd 
 |-  ( ph -> N e. ZZ )
190 uzid 
 |-  ( N e. ZZ -> N e. ( ZZ>= ` N ) )
191 uznfz 
 |-  ( N e. ( ZZ>= ` N ) -> -. N e. ( 0 ... ( N - 1 ) ) )
192 189 190 191 3syl 
 |-  ( ph -> -. N e. ( 0 ... ( N - 1 ) ) )
193 disjsn 
 |-  ( ( ( 0 ... ( N - 1 ) ) i^i { N } ) = (/) <-> -. N e. ( 0 ... ( N - 1 ) ) )
194 disj3 
 |-  ( ( ( 0 ... ( N - 1 ) ) i^i { N } ) = (/) <-> ( 0 ... ( N - 1 ) ) = ( ( 0 ... ( N - 1 ) ) \ { N } ) )
195 193 194 bitr3i 
 |-  ( -. N e. ( 0 ... ( N - 1 ) ) <-> ( 0 ... ( N - 1 ) ) = ( ( 0 ... ( N - 1 ) ) \ { N } ) )
196 192 195 sylib 
 |-  ( ph -> ( 0 ... ( N - 1 ) ) = ( ( 0 ... ( N - 1 ) ) \ { N } ) )
197 176 188 196 3eqtr4a 
 |-  ( ph -> ( ( 0 ... N ) \ { N } ) = ( 0 ... ( N - 1 ) ) )
198 175 197 sylan9eqr 
 |-  ( ( ph /\ ( 2nd ` t ) = N ) -> ( ( 0 ... N ) \ { ( 2nd ` t ) } ) = ( 0 ... ( N - 1 ) ) )
199 198 rexeqdv 
 |-  ( ( ph /\ ( 2nd ` t ) = N ) -> ( E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C <-> E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C ) )
200 199 biimprd 
 |-  ( ( ph /\ ( 2nd ` t ) = N ) -> ( E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C -> E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) )
201 200 ralimdv 
 |-  ( ( ph /\ ( 2nd ` t ) = N ) -> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) )
202 201 expimpd 
 |-  ( ph -> ( ( ( 2nd ` t ) = N /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C ) -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) )
203 173 202 sylan2i 
 |-  ( ph -> ( ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) )
204 203 adantr 
 |-  ( ( ph /\ t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) )
205 204 ss2rabdv 
 |-  ( ph -> { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } C_ { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } )
206 hashssdif 
 |-  ( ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } e. Fin /\ { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } C_ { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } ) -> ( # ` ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } \ { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ) = ( ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } ) - ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ) )
207 172 205 206 sylancr 
 |-  ( ph -> ( # ` ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } \ { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ) = ( ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } ) - ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ) )
208 1 adantr 
 |-  ( ( ph /\ t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> N e. NN )
209 3 adantlr 
 |-  ( ( ( ph /\ t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) /\ p : ( 1 ... N ) --> ( 0 ... K ) ) -> B e. ( 0 ... N ) )
210 xp1st 
 |-  ( t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( 1st ` t ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
211 xp1st 
 |-  ( ( 1st ` t ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( 1st ` ( 1st ` t ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) )
212 elmapi 
 |-  ( ( 1st ` ( 1st ` t ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` t ) ) : ( 1 ... N ) --> ( 0 ..^ K ) )
213 210 211 212 3syl 
 |-  ( t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( 1st ` ( 1st ` t ) ) : ( 1 ... N ) --> ( 0 ..^ K ) )
214 213 adantl 
 |-  ( ( ph /\ t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( 1st ` ( 1st ` t ) ) : ( 1 ... N ) --> ( 0 ..^ K ) )
215 xp2nd 
 |-  ( ( 1st ` t ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( 2nd ` ( 1st ` t ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
216 fvex 
 |-  ( 2nd ` ( 1st ` t ) ) e. _V
217 f1oeq1 
 |-  ( f = ( 2nd ` ( 1st ` t ) ) -> ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( 2nd ` ( 1st ` t ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) )
218 216 217 elab 
 |-  ( ( 2nd ` ( 1st ` t ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } <-> ( 2nd ` ( 1st ` t ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
219 215 218 sylib 
 |-  ( ( 1st ` t ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( 2nd ` ( 1st ` t ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
220 210 219 syl 
 |-  ( t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( 2nd ` ( 1st ` t ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
221 220 adantl 
 |-  ( ( ph /\ t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( 2nd ` ( 1st ` t ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
222 xp2nd 
 |-  ( t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( 2nd ` t ) e. ( 0 ... N ) )
223 222 adantl 
 |-  ( ( ph /\ t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( 2nd ` t ) e. ( 0 ... N ) )
224 208 2 209 214 221 223 poimirlem24 
 |-  ( ( ph /\ t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( E. x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ E. p e. ran x ( p ` N ) =/= 0 ) ) <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ <. ( 1st ` ( 1st ` t ) ) , ( 2nd ` ( 1st ` t ) ) >. / s ]_ C /\ -. ( ( 2nd ` t ) = N /\ ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) ) )
225 210 adantl 
 |-  ( ( ph /\ t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( 1st ` t ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
226 1st2nd2 
 |-  ( ( 1st ` t ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( 1st ` t ) = <. ( 1st ` ( 1st ` t ) ) , ( 2nd ` ( 1st ` t ) ) >. )
227 226 csbeq1d 
 |-  ( ( 1st ` t ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> [_ ( 1st ` t ) / s ]_ C = [_ <. ( 1st ` ( 1st ` t ) ) , ( 2nd ` ( 1st ` t ) ) >. / s ]_ C )
228 227 eqeq2d 
 |-  ( ( 1st ` t ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( i = [_ ( 1st ` t ) / s ]_ C <-> i = [_ <. ( 1st ` ( 1st ` t ) ) , ( 2nd ` ( 1st ` t ) ) >. / s ]_ C ) )
229 228 rexbidv 
 |-  ( ( 1st ` t ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C <-> E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ <. ( 1st ` ( 1st ` t ) ) , ( 2nd ` ( 1st ` t ) ) >. / s ]_ C ) )
230 229 ralbidv 
 |-  ( ( 1st ` t ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C <-> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ <. ( 1st ` ( 1st ` t ) ) , ( 2nd ` ( 1st ` t ) ) >. / s ]_ C ) )
231 230 anbi1d 
 |-  ( ( 1st ` t ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ -. ( ( 2nd ` t ) = N /\ ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ <. ( 1st ` ( 1st ` t ) ) , ( 2nd ` ( 1st ` t ) ) >. / s ]_ C /\ -. ( ( 2nd ` t ) = N /\ ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) ) )
232 225 231 syl 
 |-  ( ( ph /\ t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ -. ( ( 2nd ` t ) = N /\ ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ <. ( 1st ` ( 1st ` t ) ) , ( 2nd ` ( 1st ` t ) ) >. / s ]_ C /\ -. ( ( 2nd ` t ) = N /\ ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) ) )
233 224 232 bitr4d 
 |-  ( ( ph /\ t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( E. x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ E. p e. ran x ( p ` N ) =/= 0 ) ) <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ -. ( ( 2nd ` t ) = N /\ ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) ) )
234 105 frnd 
 |-  ( x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) -> ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) )
235 234 anim2i 
 |-  ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) -> ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) )
236 dfss3 
 |-  ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) <-> A. n e. ( 0 ... ( N - 1 ) ) n e. ran ( p e. ran x |-> B ) )
237 vex 
 |-  n e. _V
238 eqid 
 |-  ( p e. ran x |-> B ) = ( p e. ran x |-> B )
239 238 elrnmpt 
 |-  ( n e. _V -> ( n e. ran ( p e. ran x |-> B ) <-> E. p e. ran x n = B ) )
240 237 239 ax-mp 
 |-  ( n e. ran ( p e. ran x |-> B ) <-> E. p e. ran x n = B )
241 240 ralbii 
 |-  ( A. n e. ( 0 ... ( N - 1 ) ) n e. ran ( p e. ran x |-> B ) <-> A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B )
242 236 241 sylbb 
 |-  ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) -> A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B )
243 1eluzge0 
 |-  1 e. ( ZZ>= ` 0 )
244 fzss1 
 |-  ( 1 e. ( ZZ>= ` 0 ) -> ( 1 ... ( N - 1 ) ) C_ ( 0 ... ( N - 1 ) ) )
245 ssralv 
 |-  ( ( 1 ... ( N - 1 ) ) C_ ( 0 ... ( N - 1 ) ) -> ( A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B -> A. n e. ( 1 ... ( N - 1 ) ) E. p e. ran x n = B ) )
246 243 244 245 mp2b 
 |-  ( A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B -> A. n e. ( 1 ... ( N - 1 ) ) E. p e. ran x n = B )
247 1 nncnd 
 |-  ( ph -> N e. CC )
248 npcan1 
 |-  ( N e. CC -> ( ( N - 1 ) + 1 ) = N )
249 247 248 syl 
 |-  ( ph -> ( ( N - 1 ) + 1 ) = N )
250 peano2zm 
 |-  ( N e. ZZ -> ( N - 1 ) e. ZZ )
251 189 250 syl 
 |-  ( ph -> ( N - 1 ) e. ZZ )
252 uzid 
 |-  ( ( N - 1 ) e. ZZ -> ( N - 1 ) e. ( ZZ>= ` ( N - 1 ) ) )
253 peano2uz 
 |-  ( ( N - 1 ) e. ( ZZ>= ` ( N - 1 ) ) -> ( ( N - 1 ) + 1 ) e. ( ZZ>= ` ( N - 1 ) ) )
254 251 252 253 3syl 
 |-  ( ph -> ( ( N - 1 ) + 1 ) e. ( ZZ>= ` ( N - 1 ) ) )
255 249 254 eqeltrrd 
 |-  ( ph -> N e. ( ZZ>= ` ( N - 1 ) ) )
256 fzss2 
 |-  ( N e. ( ZZ>= ` ( N - 1 ) ) -> ( 1 ... ( N - 1 ) ) C_ ( 1 ... N ) )
257 255 256 syl 
 |-  ( ph -> ( 1 ... ( N - 1 ) ) C_ ( 1 ... N ) )
258 257 sselda 
 |-  ( ( ph /\ n e. ( 1 ... ( N - 1 ) ) ) -> n e. ( 1 ... N ) )
259 258 adantlr 
 |-  ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ n e. ( 1 ... ( N - 1 ) ) ) -> n e. ( 1 ... N ) )
260 simplr 
 |-  ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ n e. ( 1 ... N ) ) -> ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) )
261 ssel2 
 |-  ( ( ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) /\ p e. ran x ) -> p e. ( ( 0 ... K ) ^m ( 1 ... N ) ) )
262 elmapi 
 |-  ( p e. ( ( 0 ... K ) ^m ( 1 ... N ) ) -> p : ( 1 ... N ) --> ( 0 ... K ) )
263 261 262 syl 
 |-  ( ( ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) /\ p e. ran x ) -> p : ( 1 ... N ) --> ( 0 ... K ) )
264 260 263 sylan 
 |-  ( ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ n e. ( 1 ... N ) ) /\ p e. ran x ) -> p : ( 1 ... N ) --> ( 0 ... K ) )
265 elfzelz 
 |-  ( n e. ( 1 ... N ) -> n e. ZZ )
266 265 zred 
 |-  ( n e. ( 1 ... N ) -> n e. RR )
267 266 ltnrd 
 |-  ( n e. ( 1 ... N ) -> -. n < n )
268 breq1 
 |-  ( n = B -> ( n < n <-> B < n ) )
269 268 notbid 
 |-  ( n = B -> ( -. n < n <-> -. B < n ) )
270 267 269 syl5ibcom 
 |-  ( n e. ( 1 ... N ) -> ( n = B -> -. B < n ) )
271 270 necon2ad 
 |-  ( n e. ( 1 ... N ) -> ( B < n -> n =/= B ) )
272 271 3ad2ant1 
 |-  ( ( n e. ( 1 ... N ) /\ p : ( 1 ... N ) --> ( 0 ... K ) /\ ( p ` n ) = 0 ) -> ( B < n -> n =/= B ) )
273 272 adantl 
 |-  ( ( ph /\ ( n e. ( 1 ... N ) /\ p : ( 1 ... N ) --> ( 0 ... K ) /\ ( p ` n ) = 0 ) ) -> ( B < n -> n =/= B ) )
274 4 273 mpd 
 |-  ( ( ph /\ ( n e. ( 1 ... N ) /\ p : ( 1 ... N ) --> ( 0 ... K ) /\ ( p ` n ) = 0 ) ) -> n =/= B )
275 274 3exp2 
 |-  ( ph -> ( n e. ( 1 ... N ) -> ( p : ( 1 ... N ) --> ( 0 ... K ) -> ( ( p ` n ) = 0 -> n =/= B ) ) ) )
276 275 imp31 
 |-  ( ( ( ph /\ n e. ( 1 ... N ) ) /\ p : ( 1 ... N ) --> ( 0 ... K ) ) -> ( ( p ` n ) = 0 -> n =/= B ) )
277 276 necon2d 
 |-  ( ( ( ph /\ n e. ( 1 ... N ) ) /\ p : ( 1 ... N ) --> ( 0 ... K ) ) -> ( n = B -> ( p ` n ) =/= 0 ) )
278 277 adantllr 
 |-  ( ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ n e. ( 1 ... N ) ) /\ p : ( 1 ... N ) --> ( 0 ... K ) ) -> ( n = B -> ( p ` n ) =/= 0 ) )
279 264 278 syldan 
 |-  ( ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ n e. ( 1 ... N ) ) /\ p e. ran x ) -> ( n = B -> ( p ` n ) =/= 0 ) )
280 279 reximdva 
 |-  ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ n e. ( 1 ... N ) ) -> ( E. p e. ran x n = B -> E. p e. ran x ( p ` n ) =/= 0 ) )
281 259 280 syldan 
 |-  ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ n e. ( 1 ... ( N - 1 ) ) ) -> ( E. p e. ran x n = B -> E. p e. ran x ( p ` n ) =/= 0 ) )
282 281 ralimdva 
 |-  ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) -> ( A. n e. ( 1 ... ( N - 1 ) ) E. p e. ran x n = B -> A. n e. ( 1 ... ( N - 1 ) ) E. p e. ran x ( p ` n ) =/= 0 ) )
283 282 imp 
 |-  ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ A. n e. ( 1 ... ( N - 1 ) ) E. p e. ran x n = B ) -> A. n e. ( 1 ... ( N - 1 ) ) E. p e. ran x ( p ` n ) =/= 0 )
284 246 283 sylan2 
 |-  ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B ) -> A. n e. ( 1 ... ( N - 1 ) ) E. p e. ran x ( p ` n ) =/= 0 )
285 284 biantrurd 
 |-  ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B ) -> ( E. p e. ran x ( p ` N ) =/= 0 <-> ( A. n e. ( 1 ... ( N - 1 ) ) E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` N ) =/= 0 ) ) )
286 nnuz 
 |-  NN = ( ZZ>= ` 1 )
287 1 286 eleqtrdi 
 |-  ( ph -> N e. ( ZZ>= ` 1 ) )
288 fzm1 
 |-  ( N e. ( ZZ>= ` 1 ) -> ( n e. ( 1 ... N ) <-> ( n e. ( 1 ... ( N - 1 ) ) \/ n = N ) ) )
289 287 288 syl 
 |-  ( ph -> ( n e. ( 1 ... N ) <-> ( n e. ( 1 ... ( N - 1 ) ) \/ n = N ) ) )
290 elun 
 |-  ( n e. ( ( 1 ... ( N - 1 ) ) u. { N } ) <-> ( n e. ( 1 ... ( N - 1 ) ) \/ n e. { N } ) )
291 183 orbi2i 
 |-  ( ( n e. ( 1 ... ( N - 1 ) ) \/ n e. { N } ) <-> ( n e. ( 1 ... ( N - 1 ) ) \/ n = N ) )
292 290 291 bitri 
 |-  ( n e. ( ( 1 ... ( N - 1 ) ) u. { N } ) <-> ( n e. ( 1 ... ( N - 1 ) ) \/ n = N ) )
293 289 292 bitr4di 
 |-  ( ph -> ( n e. ( 1 ... N ) <-> n e. ( ( 1 ... ( N - 1 ) ) u. { N } ) ) )
294 293 eqrdv 
 |-  ( ph -> ( 1 ... N ) = ( ( 1 ... ( N - 1 ) ) u. { N } ) )
295 294 raleqdv 
 |-  ( ph -> ( A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= 0 <-> A. n e. ( ( 1 ... ( N - 1 ) ) u. { N } ) E. p e. ran x ( p ` n ) =/= 0 ) )
296 ralunb 
 |-  ( A. n e. ( ( 1 ... ( N - 1 ) ) u. { N } ) E. p e. ran x ( p ` n ) =/= 0 <-> ( A. n e. ( 1 ... ( N - 1 ) ) E. p e. ran x ( p ` n ) =/= 0 /\ A. n e. { N } E. p e. ran x ( p ` n ) =/= 0 ) )
297 295 296 bitrdi 
 |-  ( ph -> ( A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= 0 <-> ( A. n e. ( 1 ... ( N - 1 ) ) E. p e. ran x ( p ` n ) =/= 0 /\ A. n e. { N } E. p e. ran x ( p ` n ) =/= 0 ) ) )
298 fveq2 
 |-  ( n = N -> ( p ` n ) = ( p ` N ) )
299 298 neeq1d 
 |-  ( n = N -> ( ( p ` n ) =/= 0 <-> ( p ` N ) =/= 0 ) )
300 299 rexbidv 
 |-  ( n = N -> ( E. p e. ran x ( p ` n ) =/= 0 <-> E. p e. ran x ( p ` N ) =/= 0 ) )
301 300 ralsng 
 |-  ( N e. NN -> ( A. n e. { N } E. p e. ran x ( p ` n ) =/= 0 <-> E. p e. ran x ( p ` N ) =/= 0 ) )
302 1 301 syl 
 |-  ( ph -> ( A. n e. { N } E. p e. ran x ( p ` n ) =/= 0 <-> E. p e. ran x ( p ` N ) =/= 0 ) )
303 302 anbi2d 
 |-  ( ph -> ( ( A. n e. ( 1 ... ( N - 1 ) ) E. p e. ran x ( p ` n ) =/= 0 /\ A. n e. { N } E. p e. ran x ( p ` n ) =/= 0 ) <-> ( A. n e. ( 1 ... ( N - 1 ) ) E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` N ) =/= 0 ) ) )
304 297 303 bitrd 
 |-  ( ph -> ( A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= 0 <-> ( A. n e. ( 1 ... ( N - 1 ) ) E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` N ) =/= 0 ) ) )
305 304 ad2antrr 
 |-  ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B ) -> ( A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= 0 <-> ( A. n e. ( 1 ... ( N - 1 ) ) E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` N ) =/= 0 ) ) )
306 0z 
 |-  0 e. ZZ
307 1z 
 |-  1 e. ZZ
308 fzshftral 
 |-  ( ( 0 e. ZZ /\ ( N - 1 ) e. ZZ /\ 1 e. ZZ ) -> ( A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B <-> A. m e. ( ( 0 + 1 ) ... ( ( N - 1 ) + 1 ) ) [. ( m - 1 ) / n ]. E. p e. ran x n = B ) )
309 306 307 308 mp3an13 
 |-  ( ( N - 1 ) e. ZZ -> ( A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B <-> A. m e. ( ( 0 + 1 ) ... ( ( N - 1 ) + 1 ) ) [. ( m - 1 ) / n ]. E. p e. ran x n = B ) )
310 189 250 309 3syl 
 |-  ( ph -> ( A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B <-> A. m e. ( ( 0 + 1 ) ... ( ( N - 1 ) + 1 ) ) [. ( m - 1 ) / n ]. E. p e. ran x n = B ) )
311 0p1e1 
 |-  ( 0 + 1 ) = 1
312 311 a1i 
 |-  ( ph -> ( 0 + 1 ) = 1 )
313 312 249 oveq12d 
 |-  ( ph -> ( ( 0 + 1 ) ... ( ( N - 1 ) + 1 ) ) = ( 1 ... N ) )
314 313 raleqdv 
 |-  ( ph -> ( A. m e. ( ( 0 + 1 ) ... ( ( N - 1 ) + 1 ) ) [. ( m - 1 ) / n ]. E. p e. ran x n = B <-> A. m e. ( 1 ... N ) [. ( m - 1 ) / n ]. E. p e. ran x n = B ) )
315 310 314 bitrd 
 |-  ( ph -> ( A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B <-> A. m e. ( 1 ... N ) [. ( m - 1 ) / n ]. E. p e. ran x n = B ) )
316 ovex 
 |-  ( m - 1 ) e. _V
317 eqeq1 
 |-  ( n = ( m - 1 ) -> ( n = B <-> ( m - 1 ) = B ) )
318 317 rexbidv 
 |-  ( n = ( m - 1 ) -> ( E. p e. ran x n = B <-> E. p e. ran x ( m - 1 ) = B ) )
319 316 318 sbcie 
 |-  ( [. ( m - 1 ) / n ]. E. p e. ran x n = B <-> E. p e. ran x ( m - 1 ) = B )
320 319 ralbii 
 |-  ( A. m e. ( 1 ... N ) [. ( m - 1 ) / n ]. E. p e. ran x n = B <-> A. m e. ( 1 ... N ) E. p e. ran x ( m - 1 ) = B )
321 oveq1 
 |-  ( m = n -> ( m - 1 ) = ( n - 1 ) )
322 321 eqeq1d 
 |-  ( m = n -> ( ( m - 1 ) = B <-> ( n - 1 ) = B ) )
323 322 rexbidv 
 |-  ( m = n -> ( E. p e. ran x ( m - 1 ) = B <-> E. p e. ran x ( n - 1 ) = B ) )
324 323 cbvralvw 
 |-  ( A. m e. ( 1 ... N ) E. p e. ran x ( m - 1 ) = B <-> A. n e. ( 1 ... N ) E. p e. ran x ( n - 1 ) = B )
325 320 324 bitri 
 |-  ( A. m e. ( 1 ... N ) [. ( m - 1 ) / n ]. E. p e. ran x n = B <-> A. n e. ( 1 ... N ) E. p e. ran x ( n - 1 ) = B )
326 315 325 bitrdi 
 |-  ( ph -> ( A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B <-> A. n e. ( 1 ... N ) E. p e. ran x ( n - 1 ) = B ) )
327 326 biimpa 
 |-  ( ( ph /\ A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B ) -> A. n e. ( 1 ... N ) E. p e. ran x ( n - 1 ) = B )
328 327 adantlr 
 |-  ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B ) -> A. n e. ( 1 ... N ) E. p e. ran x ( n - 1 ) = B )
329 5 necomd 
 |-  ( ( ph /\ ( n e. ( 1 ... N ) /\ p : ( 1 ... N ) --> ( 0 ... K ) /\ ( p ` n ) = K ) ) -> ( n - 1 ) =/= B )
330 329 3exp2 
 |-  ( ph -> ( n e. ( 1 ... N ) -> ( p : ( 1 ... N ) --> ( 0 ... K ) -> ( ( p ` n ) = K -> ( n - 1 ) =/= B ) ) ) )
331 330 imp31 
 |-  ( ( ( ph /\ n e. ( 1 ... N ) ) /\ p : ( 1 ... N ) --> ( 0 ... K ) ) -> ( ( p ` n ) = K -> ( n - 1 ) =/= B ) )
332 331 necon2d 
 |-  ( ( ( ph /\ n e. ( 1 ... N ) ) /\ p : ( 1 ... N ) --> ( 0 ... K ) ) -> ( ( n - 1 ) = B -> ( p ` n ) =/= K ) )
333 332 adantllr 
 |-  ( ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ n e. ( 1 ... N ) ) /\ p : ( 1 ... N ) --> ( 0 ... K ) ) -> ( ( n - 1 ) = B -> ( p ` n ) =/= K ) )
334 264 333 syldan 
 |-  ( ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ n e. ( 1 ... N ) ) /\ p e. ran x ) -> ( ( n - 1 ) = B -> ( p ` n ) =/= K ) )
335 334 reximdva 
 |-  ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ n e. ( 1 ... N ) ) -> ( E. p e. ran x ( n - 1 ) = B -> E. p e. ran x ( p ` n ) =/= K ) )
336 335 ralimdva 
 |-  ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) -> ( A. n e. ( 1 ... N ) E. p e. ran x ( n - 1 ) = B -> A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= K ) )
337 336 imp 
 |-  ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ A. n e. ( 1 ... N ) E. p e. ran x ( n - 1 ) = B ) -> A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= K )
338 328 337 syldan 
 |-  ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B ) -> A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= K )
339 338 biantrud 
 |-  ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B ) -> ( A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= 0 <-> ( A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= 0 /\ A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= K ) ) )
340 r19.26 
 |-  ( A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) <-> ( A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= 0 /\ A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= K ) )
341 339 340 bitr4di 
 |-  ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B ) -> ( A. n e. ( 1 ... N ) E. p e. ran x ( p ` n ) =/= 0 <-> A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) )
342 285 305 341 3bitr2d 
 |-  ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B ) -> ( E. p e. ran x ( p ` N ) =/= 0 <-> A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) )
343 235 242 342 syl2an 
 |-  ( ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) /\ ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) ) -> ( E. p e. ran x ( p ` N ) =/= 0 <-> A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) )
344 343 pm5.32da 
 |-  ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) -> ( ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ E. p e. ran x ( p ` N ) =/= 0 ) <-> ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) )
345 344 anbi2d 
 |-  ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) -> ( ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ E. p e. ran x ( p ` N ) =/= 0 ) ) <-> ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) ) )
346 345 rexbidva 
 |-  ( ph -> ( E. x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ E. p e. ran x ( p ` N ) =/= 0 ) ) <-> E. x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) ) )
347 346 adantr 
 |-  ( ( ph /\ t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( E. x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ E. p e. ran x ( p ` N ) =/= 0 ) ) <-> E. x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) ) )
348 197 rexeqdv 
 |-  ( ph -> ( E. j e. ( ( 0 ... N ) \ { N } ) i = [_ ( 1st ` t ) / s ]_ C <-> E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C ) )
349 348 biimpd 
 |-  ( ph -> ( E. j e. ( ( 0 ... N ) \ { N } ) i = [_ ( 1st ` t ) / s ]_ C -> E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C ) )
350 349 ralimdv 
 |-  ( ph -> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { N } ) i = [_ ( 1st ` t ) / s ]_ C -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C ) )
351 175 rexeqdv 
 |-  ( ( 2nd ` t ) = N -> ( E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C <-> E. j e. ( ( 0 ... N ) \ { N } ) i = [_ ( 1st ` t ) / s ]_ C ) )
352 351 ralbidv 
 |-  ( ( 2nd ` t ) = N -> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C <-> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { N } ) i = [_ ( 1st ` t ) / s ]_ C ) )
353 352 imbi1d 
 |-  ( ( 2nd ` t ) = N -> ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C ) <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { N } ) i = [_ ( 1st ` t ) / s ]_ C -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C ) ) )
354 350 353 syl5ibrcom 
 |-  ( ph -> ( ( 2nd ` t ) = N -> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C ) ) )
355 354 com23 
 |-  ( ph -> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C -> ( ( 2nd ` t ) = N -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C ) ) )
356 355 imp 
 |-  ( ( ph /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> ( ( 2nd ` t ) = N -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C ) )
357 356 adantrd 
 |-  ( ( ph /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> ( ( ( 2nd ` t ) = N /\ ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) -> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C ) )
358 357 pm4.71rd 
 |-  ( ( ph /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> ( ( ( 2nd ` t ) = N /\ ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 2nd ` t ) = N /\ ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) ) )
359 an12 
 |-  ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 2nd ` t ) = N /\ ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) <-> ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) )
360 3anass 
 |-  ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) )
361 360 anbi2i 
 |-  ( ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) <-> ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) )
362 359 361 bitr4i 
 |-  ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 2nd ` t ) = N /\ ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) <-> ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) )
363 358 362 bitrdi 
 |-  ( ( ph /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> ( ( ( 2nd ` t ) = N /\ ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) <-> ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) )
364 363 notbid 
 |-  ( ( ph /\ A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C ) -> ( -. ( ( 2nd ` t ) = N /\ ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) <-> -. ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) )
365 364 pm5.32da 
 |-  ( ph -> ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ -. ( ( 2nd ` t ) = N /\ ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ -. ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) ) )
366 365 adantr 
 |-  ( ( ph /\ t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ -. ( ( 2nd ` t ) = N /\ ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ -. ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) ) )
367 233 347 366 3bitr3d 
 |-  ( ( ph /\ t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( E. x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ -. ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) ) )
368 367 rabbidva 
 |-  ( ph -> { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | E. x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } = { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ -. ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) } )
369 iunrab 
 |-  U_ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } = { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | E. x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) }
370 difrab 
 |-  ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } \ { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) = { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C /\ -. ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) ) }
371 368 369 370 3eqtr4g 
 |-  ( ph -> U_ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } = ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } \ { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) )
372 371 fveq2d 
 |-  ( ph -> ( # ` U_ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ) = ( # ` ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } \ { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ) )
373 32 33 mp1i 
 |-  ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) -> { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } e. Fin )
374 simpl 
 |-  ( ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) -> x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) )
375 374 a1i 
 |-  ( t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) -> x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) )
376 375 ss2rabi 
 |-  { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } C_ { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) }
377 376 sseli 
 |-  ( s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } -> s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } )
378 fveq2 
 |-  ( t = s -> ( 2nd ` t ) = ( 2nd ` s ) )
379 378 breq2d 
 |-  ( t = s -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` s ) ) )
380 379 ifbid 
 |-  ( t = s -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` s ) , y , ( y + 1 ) ) )
381 380 csbeq1d 
 |-  ( t = s -> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = [_ if ( y < ( 2nd ` s ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) )
382 2fveq3 
 |-  ( t = s -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` s ) ) )
383 2fveq3 
 |-  ( t = s -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` s ) ) )
384 383 imaeq1d 
 |-  ( t = s -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) )
385 384 xpeq1d 
 |-  ( t = s -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) X. { 1 } ) )
386 383 imaeq1d 
 |-  ( t = s -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) )
387 386 xpeq1d 
 |-  ( t = s -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) )
388 385 387 uneq12d 
 |-  ( t = s -> ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) = ( ( ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) )
389 382 388 oveq12d 
 |-  ( t = s -> ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` s ) ) oF + ( ( ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) )
390 389 csbeq2dv 
 |-  ( t = s -> [_ if ( y < ( 2nd ` s ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = [_ if ( y < ( 2nd ` s ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` s ) ) oF + ( ( ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) )
391 381 390 eqtrd 
 |-  ( t = s -> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = [_ if ( y < ( 2nd ` s ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` s ) ) oF + ( ( ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) )
392 391 mpteq2dv 
 |-  ( t = s -> ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` s ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` s ) ) oF + ( ( ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) )
393 392 eqeq2d 
 |-  ( t = s -> ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) <-> x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` s ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` s ) ) oF + ( ( ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) )
394 eqcom 
 |-  ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` s ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` s ) ) oF + ( ( ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) <-> ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` s ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` s ) ) oF + ( ( ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) = x )
395 393 394 bitrdi 
 |-  ( t = s -> ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) <-> ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` s ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` s ) ) oF + ( ( ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) = x ) )
396 395 elrab 
 |-  ( s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } <-> ( s e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` s ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` s ) ) oF + ( ( ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) = x ) )
397 396 simprbi 
 |-  ( s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } -> ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` s ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` s ) ) oF + ( ( ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) = x )
398 377 397 syl 
 |-  ( s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } -> ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` s ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` s ) ) oF + ( ( ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) = x )
399 398 rgen 
 |-  A. s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` s ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` s ) ) oF + ( ( ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) = x
400 399 rgenw 
 |-  A. x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) A. s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` s ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` s ) ) oF + ( ( ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) = x
401 invdisj 
 |-  ( A. x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) A. s e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` s ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` s ) ) oF + ( ( ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) = x -> Disj_ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } )
402 400 401 mp1i 
 |-  ( ph -> Disj_ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } )
403 13 373 402 hashiun 
 |-  ( ph -> ( # ` U_ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ) = sum_ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ) )
404 372 403 eqtr3d 
 |-  ( ph -> ( # ` ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } \ { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ) = sum_ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ) )
405 fo1st 
 |-  1st : _V -onto-> _V
406 fofun 
 |-  ( 1st : _V -onto-> _V -> Fun 1st )
407 405 406 ax-mp 
 |-  Fun 1st
408 ssv 
 |-  { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } C_ _V
409 fof 
 |-  ( 1st : _V -onto-> _V -> 1st : _V --> _V )
410 405 409 ax-mp 
 |-  1st : _V --> _V
411 410 fdmi 
 |-  dom 1st = _V
412 408 411 sseqtrri 
 |-  { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } C_ dom 1st
413 fores 
 |-  ( ( Fun 1st /\ { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } C_ dom 1st ) -> ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -onto-> ( 1st " { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) )
414 407 412 413 mp2an 
 |-  ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -onto-> ( 1st " { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } )
415 fveqeq2 
 |-  ( t = x -> ( ( 2nd ` t ) = N <-> ( 2nd ` x ) = N ) )
416 fveq2 
 |-  ( t = x -> ( 1st ` t ) = ( 1st ` x ) )
417 416 csbeq1d 
 |-  ( t = x -> [_ ( 1st ` t ) / s ]_ C = [_ ( 1st ` x ) / s ]_ C )
418 417 eqeq2d 
 |-  ( t = x -> ( i = [_ ( 1st ` t ) / s ]_ C <-> i = [_ ( 1st ` x ) / s ]_ C ) )
419 418 rexbidv 
 |-  ( t = x -> ( E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C <-> E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C ) )
420 419 ralbidv 
 |-  ( t = x -> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C <-> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C ) )
421 2fveq3 
 |-  ( t = x -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` x ) ) )
422 421 fveq1d 
 |-  ( t = x -> ( ( 1st ` ( 1st ` t ) ) ` N ) = ( ( 1st ` ( 1st ` x ) ) ` N ) )
423 422 eqeq1d 
 |-  ( t = x -> ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 <-> ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 ) )
424 2fveq3 
 |-  ( t = x -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` x ) ) )
425 424 fveq1d 
 |-  ( t = x -> ( ( 2nd ` ( 1st ` t ) ) ` N ) = ( ( 2nd ` ( 1st ` x ) ) ` N ) )
426 425 eqeq1d 
 |-  ( t = x -> ( ( ( 2nd ` ( 1st ` t ) ) ` N ) = N <-> ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) )
427 420 423 426 3anbi123d 
 |-  ( t = x -> ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) ) )
428 415 427 anbi12d 
 |-  ( t = x -> ( ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) <-> ( ( 2nd ` x ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) ) ) )
429 428 rexrab 
 |-  ( E. x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ( 1st ` x ) = s <-> E. x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ( ( ( 2nd ` x ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) ) /\ ( 1st ` x ) = s ) )
430 xp1st 
 |-  ( x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( 1st ` x ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
431 430 anim1i 
 |-  ( ( x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) ) -> ( ( 1st ` x ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) ) )
432 eleq1 
 |-  ( ( 1st ` x ) = s -> ( ( 1st ` x ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) <-> s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) )
433 csbeq1a 
 |-  ( s = ( 1st ` x ) -> C = [_ ( 1st ` x ) / s ]_ C )
434 433 eqcoms 
 |-  ( ( 1st ` x ) = s -> C = [_ ( 1st ` x ) / s ]_ C )
435 434 eqcomd 
 |-  ( ( 1st ` x ) = s -> [_ ( 1st ` x ) / s ]_ C = C )
436 435 eqeq2d 
 |-  ( ( 1st ` x ) = s -> ( i = [_ ( 1st ` x ) / s ]_ C <-> i = C ) )
437 436 rexbidv 
 |-  ( ( 1st ` x ) = s -> ( E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C <-> E. j e. ( 0 ... ( N - 1 ) ) i = C ) )
438 437 ralbidv 
 |-  ( ( 1st ` x ) = s -> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C <-> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C ) )
439 fveq2 
 |-  ( ( 1st ` x ) = s -> ( 1st ` ( 1st ` x ) ) = ( 1st ` s ) )
440 439 fveq1d 
 |-  ( ( 1st ` x ) = s -> ( ( 1st ` ( 1st ` x ) ) ` N ) = ( ( 1st ` s ) ` N ) )
441 440 eqeq1d 
 |-  ( ( 1st ` x ) = s -> ( ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 <-> ( ( 1st ` s ) ` N ) = 0 ) )
442 fveq2 
 |-  ( ( 1st ` x ) = s -> ( 2nd ` ( 1st ` x ) ) = ( 2nd ` s ) )
443 442 fveq1d 
 |-  ( ( 1st ` x ) = s -> ( ( 2nd ` ( 1st ` x ) ) ` N ) = ( ( 2nd ` s ) ` N ) )
444 443 eqeq1d 
 |-  ( ( 1st ` x ) = s -> ( ( ( 2nd ` ( 1st ` x ) ) ` N ) = N <-> ( ( 2nd ` s ) ` N ) = N ) )
445 438 441 444 3anbi123d 
 |-  ( ( 1st ` x ) = s -> ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) ) )
446 432 445 anbi12d 
 |-  ( ( 1st ` x ) = s -> ( ( ( 1st ` x ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) ) <-> ( s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) ) ) )
447 431 446 syl5ibcom 
 |-  ( ( x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) ) -> ( ( 1st ` x ) = s -> ( s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) ) ) )
448 447 adantrl 
 |-  ( ( x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ ( ( 2nd ` x ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) ) ) -> ( ( 1st ` x ) = s -> ( s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) ) ) )
449 448 expimpd 
 |-  ( x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( ( ( ( 2nd ` x ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) ) /\ ( 1st ` x ) = s ) -> ( s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) ) ) )
450 449 rexlimiv 
 |-  ( E. x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ( ( ( 2nd ` x ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) ) /\ ( 1st ` x ) = s ) -> ( s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) ) )
451 nn0fz0 
 |-  ( N e. NN0 <-> N e. ( 0 ... N ) )
452 177 451 sylib 
 |-  ( ph -> N e. ( 0 ... N ) )
453 opelxpi 
 |-  ( ( s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ N e. ( 0 ... N ) ) -> <. s , N >. e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) )
454 452 453 sylan2 
 |-  ( ( s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ ph ) -> <. s , N >. e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) )
455 454 ancoms 
 |-  ( ( ph /\ s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) -> <. s , N >. e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) )
456 opelxp2 
 |-  ( <. s , N >. e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> N e. ( 0 ... N ) )
457 op2ndg 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( 2nd ` <. s , N >. ) = N )
458 457 biantrurd 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` <. s , N >. ) / s ]_ C /\ ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) = N ) <-> ( ( 2nd ` <. s , N >. ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` <. s , N >. ) / s ]_ C /\ ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) = N ) ) ) )
459 op1stg 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( 1st ` <. s , N >. ) = s )
460 csbeq1a 
 |-  ( s = ( 1st ` <. s , N >. ) -> C = [_ ( 1st ` <. s , N >. ) / s ]_ C )
461 460 eqcoms 
 |-  ( ( 1st ` <. s , N >. ) = s -> C = [_ ( 1st ` <. s , N >. ) / s ]_ C )
462 461 eqcomd 
 |-  ( ( 1st ` <. s , N >. ) = s -> [_ ( 1st ` <. s , N >. ) / s ]_ C = C )
463 459 462 syl 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> [_ ( 1st ` <. s , N >. ) / s ]_ C = C )
464 463 eqeq2d 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( i = [_ ( 1st ` <. s , N >. ) / s ]_ C <-> i = C ) )
465 464 rexbidv 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` <. s , N >. ) / s ]_ C <-> E. j e. ( 0 ... ( N - 1 ) ) i = C ) )
466 465 ralbidv 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` <. s , N >. ) / s ]_ C <-> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C ) )
467 459 fveq2d 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( 1st ` ( 1st ` <. s , N >. ) ) = ( 1st ` s ) )
468 467 fveq1d 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) = ( ( 1st ` s ) ` N ) )
469 468 eqeq1d 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) = 0 <-> ( ( 1st ` s ) ` N ) = 0 ) )
470 459 fveq2d 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( 2nd ` ( 1st ` <. s , N >. ) ) = ( 2nd ` s ) )
471 470 fveq1d 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) = ( ( 2nd ` s ) ` N ) )
472 471 eqeq1d 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) = N <-> ( ( 2nd ` s ) ` N ) = N ) )
473 466 469 472 3anbi123d 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` <. s , N >. ) / s ]_ C /\ ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) = N ) <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) ) )
474 459 biantrud 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( ( ( 2nd ` <. s , N >. ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` <. s , N >. ) / s ]_ C /\ ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) = N ) ) <-> ( ( ( 2nd ` <. s , N >. ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` <. s , N >. ) / s ]_ C /\ ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) = N ) ) /\ ( 1st ` <. s , N >. ) = s ) ) )
475 458 473 474 3bitr3d 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) <-> ( ( ( 2nd ` <. s , N >. ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` <. s , N >. ) / s ]_ C /\ ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) = N ) ) /\ ( 1st ` <. s , N >. ) = s ) ) )
476 49 456 475 sylancr 
 |-  ( <. s , N >. e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) <-> ( ( ( 2nd ` <. s , N >. ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` <. s , N >. ) / s ]_ C /\ ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) = N ) ) /\ ( 1st ` <. s , N >. ) = s ) ) )
477 476 biimpa 
 |-  ( ( <. s , N >. e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) ) -> ( ( ( 2nd ` <. s , N >. ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` <. s , N >. ) / s ]_ C /\ ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) = N ) ) /\ ( 1st ` <. s , N >. ) = s ) )
478 fveqeq2 
 |-  ( x = <. s , N >. -> ( ( 2nd ` x ) = N <-> ( 2nd ` <. s , N >. ) = N ) )
479 fveq2 
 |-  ( x = <. s , N >. -> ( 1st ` x ) = ( 1st ` <. s , N >. ) )
480 479 csbeq1d 
 |-  ( x = <. s , N >. -> [_ ( 1st ` x ) / s ]_ C = [_ ( 1st ` <. s , N >. ) / s ]_ C )
481 480 eqeq2d 
 |-  ( x = <. s , N >. -> ( i = [_ ( 1st ` x ) / s ]_ C <-> i = [_ ( 1st ` <. s , N >. ) / s ]_ C ) )
482 481 rexbidv 
 |-  ( x = <. s , N >. -> ( E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C <-> E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` <. s , N >. ) / s ]_ C ) )
483 482 ralbidv 
 |-  ( x = <. s , N >. -> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C <-> A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` <. s , N >. ) / s ]_ C ) )
484 2fveq3 
 |-  ( x = <. s , N >. -> ( 1st ` ( 1st ` x ) ) = ( 1st ` ( 1st ` <. s , N >. ) ) )
485 484 fveq1d 
 |-  ( x = <. s , N >. -> ( ( 1st ` ( 1st ` x ) ) ` N ) = ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) )
486 485 eqeq1d 
 |-  ( x = <. s , N >. -> ( ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 <-> ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) = 0 ) )
487 2fveq3 
 |-  ( x = <. s , N >. -> ( 2nd ` ( 1st ` x ) ) = ( 2nd ` ( 1st ` <. s , N >. ) ) )
488 487 fveq1d 
 |-  ( x = <. s , N >. -> ( ( 2nd ` ( 1st ` x ) ) ` N ) = ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) )
489 488 eqeq1d 
 |-  ( x = <. s , N >. -> ( ( ( 2nd ` ( 1st ` x ) ) ` N ) = N <-> ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) = N ) )
490 483 486 489 3anbi123d 
 |-  ( x = <. s , N >. -> ( ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) <-> ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` <. s , N >. ) / s ]_ C /\ ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) = N ) ) )
491 478 490 anbi12d 
 |-  ( x = <. s , N >. -> ( ( ( 2nd ` x ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) ) <-> ( ( 2nd ` <. s , N >. ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` <. s , N >. ) / s ]_ C /\ ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) = N ) ) ) )
492 fveqeq2 
 |-  ( x = <. s , N >. -> ( ( 1st ` x ) = s <-> ( 1st ` <. s , N >. ) = s ) )
493 491 492 anbi12d 
 |-  ( x = <. s , N >. -> ( ( ( ( 2nd ` x ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) ) /\ ( 1st ` x ) = s ) <-> ( ( ( 2nd ` <. s , N >. ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` <. s , N >. ) / s ]_ C /\ ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) = N ) ) /\ ( 1st ` <. s , N >. ) = s ) ) )
494 493 rspcev 
 |-  ( ( <. s , N >. e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ ( ( ( 2nd ` <. s , N >. ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` <. s , N >. ) / s ]_ C /\ ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) = N ) ) /\ ( 1st ` <. s , N >. ) = s ) ) -> E. x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ( ( ( 2nd ` x ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) ) /\ ( 1st ` x ) = s ) )
495 477 494 syldan 
 |-  ( ( <. s , N >. e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) ) -> E. x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ( ( ( 2nd ` x ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) ) /\ ( 1st ` x ) = s ) )
496 455 495 sylan 
 |-  ( ( ( ph /\ s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) ) -> E. x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ( ( ( 2nd ` x ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) ) /\ ( 1st ` x ) = s ) )
497 496 expl 
 |-  ( ph -> ( ( s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) ) -> E. x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ( ( ( 2nd ` x ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) ) /\ ( 1st ` x ) = s ) ) )
498 450 497 impbid2 
 |-  ( ph -> ( E. x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ( ( ( 2nd ` x ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C /\ ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) ) /\ ( 1st ` x ) = s ) <-> ( s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) ) ) )
499 429 498 syl5bb 
 |-  ( ph -> ( E. x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ( 1st ` x ) = s <-> ( s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) ) ) )
500 499 abbidv 
 |-  ( ph -> { s | E. x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ( 1st ` x ) = s } = { s | ( s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) ) } )
501 dfimafn 
 |-  ( ( Fun 1st /\ { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } C_ dom 1st ) -> ( 1st " { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) = { y | E. x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ( 1st ` x ) = y } )
502 407 412 501 mp2an 
 |-  ( 1st " { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) = { y | E. x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ( 1st ` x ) = y }
503 nfv 
 |-  F/ s ( 2nd ` t ) = N
504 nfcv 
 |-  F/_ s ( 0 ... ( N - 1 ) )
505 nfcsb1v 
 |-  F/_ s [_ ( 1st ` t ) / s ]_ C
506 505 nfeq2 
 |-  F/ s i = [_ ( 1st ` t ) / s ]_ C
507 504 506 nfrex 
 |-  F/ s E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C
508 504 507 nfralw 
 |-  F/ s A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C
509 nfv 
 |-  F/ s ( ( 1st ` ( 1st ` t ) ) ` N ) = 0
510 nfv 
 |-  F/ s ( ( 2nd ` ( 1st ` t ) ) ` N ) = N
511 508 509 510 nf3an 
 |-  F/ s ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N )
512 503 511 nfan 
 |-  F/ s ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) )
513 nfcv 
 |-  F/_ s ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) )
514 512 513 nfrabw 
 |-  F/_ s { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) }
515 nfv 
 |-  F/ s ( 1st ` x ) = y
516 514 515 nfrex 
 |-  F/ s E. x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ( 1st ` x ) = y
517 nfv 
 |-  F/ y E. x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ( 1st ` x ) = s
518 eqeq2 
 |-  ( y = s -> ( ( 1st ` x ) = y <-> ( 1st ` x ) = s ) )
519 518 rexbidv 
 |-  ( y = s -> ( E. x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ( 1st ` x ) = y <-> E. x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ( 1st ` x ) = s ) )
520 516 517 519 cbvabw 
 |-  { y | E. x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ( 1st ` x ) = y } = { s | E. x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ( 1st ` x ) = s }
521 502 520 eqtri 
 |-  ( 1st " { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) = { s | E. x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ( 1st ` x ) = s }
522 df-rab 
 |-  { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } = { s | ( s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) ) }
523 500 521 522 3eqtr4g 
 |-  ( ph -> ( 1st " { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) = { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } )
524 foeq3 
 |-  ( ( 1st " { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) = { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } -> ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -onto-> ( 1st " { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) <-> ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -onto-> { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } ) )
525 523 524 syl 
 |-  ( ph -> ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -onto-> ( 1st " { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) <-> ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -onto-> { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } ) )
526 414 525 mpbii 
 |-  ( ph -> ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -onto-> { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } )
527 fof 
 |-  ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -onto-> { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } -> ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } --> { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } )
528 526 527 syl 
 |-  ( ph -> ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } --> { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } )
529 fvres 
 |-  ( x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -> ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ` x ) = ( 1st ` x ) )
530 fvres 
 |-  ( y e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -> ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ` y ) = ( 1st ` y ) )
531 529 530 eqeqan12d 
 |-  ( ( x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } /\ y e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) -> ( ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ` x ) = ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ` y ) <-> ( 1st ` x ) = ( 1st ` y ) ) )
532 simpl 
 |-  ( ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) -> ( 2nd ` t ) = N )
533 532 a1i 
 |-  ( t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) -> ( 2nd ` t ) = N ) )
534 533 ss2rabi 
 |-  { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } C_ { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( 2nd ` t ) = N }
535 534 sseli 
 |-  ( x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -> x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( 2nd ` t ) = N } )
536 415 elrab 
 |-  ( x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( 2nd ` t ) = N } <-> ( x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ ( 2nd ` x ) = N ) )
537 535 536 sylib 
 |-  ( x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -> ( x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ ( 2nd ` x ) = N ) )
538 534 sseli 
 |-  ( y e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -> y e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( 2nd ` t ) = N } )
539 fveqeq2 
 |-  ( t = y -> ( ( 2nd ` t ) = N <-> ( 2nd ` y ) = N ) )
540 539 elrab 
 |-  ( y e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( 2nd ` t ) = N } <-> ( y e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ ( 2nd ` y ) = N ) )
541 538 540 sylib 
 |-  ( y e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -> ( y e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ ( 2nd ` y ) = N ) )
542 eqtr3 
 |-  ( ( ( 2nd ` x ) = N /\ ( 2nd ` y ) = N ) -> ( 2nd ` x ) = ( 2nd ` y ) )
543 xpopth 
 |-  ( ( x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ y e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) = ( 2nd ` y ) ) <-> x = y ) )
544 543 biimpd 
 |-  ( ( x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ y e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) = ( 2nd ` y ) ) -> x = y ) )
545 544 ancomsd 
 |-  ( ( x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ y e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) -> ( ( ( 2nd ` x ) = ( 2nd ` y ) /\ ( 1st ` x ) = ( 1st ` y ) ) -> x = y ) )
546 545 expdimp 
 |-  ( ( ( x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ y e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) /\ ( 2nd ` x ) = ( 2nd ` y ) ) -> ( ( 1st ` x ) = ( 1st ` y ) -> x = y ) )
547 542 546 sylan2 
 |-  ( ( ( x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ y e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) /\ ( ( 2nd ` x ) = N /\ ( 2nd ` y ) = N ) ) -> ( ( 1st ` x ) = ( 1st ` y ) -> x = y ) )
548 547 an4s 
 |-  ( ( ( x e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ ( 2nd ` x ) = N ) /\ ( y e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ ( 2nd ` y ) = N ) ) -> ( ( 1st ` x ) = ( 1st ` y ) -> x = y ) )
549 537 541 548 syl2an 
 |-  ( ( x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } /\ y e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) -> ( ( 1st ` x ) = ( 1st ` y ) -> x = y ) )
550 531 549 sylbid 
 |-  ( ( x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } /\ y e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) -> ( ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ` x ) = ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ` y ) -> x = y ) )
551 550 rgen2 
 |-  A. x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } A. y e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ( ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ` x ) = ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ` y ) -> x = y )
552 528 551 jctir 
 |-  ( ph -> ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } --> { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } /\ A. x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } A. y e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ( ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ` x ) = ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ` y ) -> x = y ) ) )
553 dff13 
 |-  ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -1-1-> { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } <-> ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } --> { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } /\ A. x e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } A. y e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ( ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ` x ) = ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ` y ) -> x = y ) ) )
554 552 553 sylibr 
 |-  ( ph -> ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -1-1-> { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } )
555 df-f1o 
 |-  ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -1-1-onto-> { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } <-> ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -1-1-> { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } /\ ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -onto-> { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } ) )
556 554 526 555 sylanbrc 
 |-  ( ph -> ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -1-1-onto-> { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } )
557 rabfi 
 |-  ( ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) e. Fin -> { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } e. Fin )
558 32 557 ax-mp 
 |-  { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } e. Fin
559 558 elexi 
 |-  { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } e. _V
560 559 f1oen 
 |-  ( ( 1st |` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) : { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } -1-1-onto-> { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } -> { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ~~ { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } )
561 556 560 syl 
 |-  ( ph -> { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ~~ { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } )
562 rabfi 
 |-  ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) e. Fin -> { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } e. Fin )
563 29 562 ax-mp 
 |-  { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } e. Fin
564 hashen 
 |-  ( ( { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } e. Fin /\ { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } e. Fin ) -> ( ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) = ( # ` { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } ) <-> { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ~~ { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } ) )
565 558 563 564 mp2an 
 |-  ( ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) = ( # ` { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } ) <-> { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ~~ { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } )
566 561 565 sylibr 
 |-  ( ph -> ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) = ( # ` { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } ) )
567 566 oveq2d 
 |-  ( ph -> ( ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } ) - ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( ( 2nd ` t ) = N /\ ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C /\ ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 /\ ( ( 2nd ` ( 1st ` t ) ) ` N ) = N ) ) } ) ) = ( ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } ) - ( # ` { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } ) ) )
568 207 404 567 3eqtr3d 
 |-  ( ph -> sum_ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | ( x = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) /\ ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) /\ A. n e. ( 1 ... N ) ( E. p e. ran x ( p ` n ) =/= 0 /\ E. p e. ran x ( p ` n ) =/= K ) ) ) } ) = ( ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } ) - ( # ` { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } ) ) )
569 170 568 breqtrd 
 |-  ( ph -> 2 || ( ( # ` { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( ( 0 ... N ) \ { ( 2nd ` t ) } ) i = [_ ( 1st ` t ) / s ]_ C } ) - ( # ` { s e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) | ( A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = C /\ ( ( 1st ` s ) ` N ) = 0 /\ ( ( 2nd ` s ) ` N ) = N ) } ) ) )