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 disjdifr 
 |-  ( ( { 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 } ) = (/)
61 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 } ) ) )
62 58 59 60 61 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 } ) )
63 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 } ) ) ) ) } )
64 63 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 } ) ) ) ) } ) )
65 62 64 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 } ) ) ) ) } ) )
66 65 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 } ) ) ) ) } ) )
67 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 )
68 fveq2 
 |-  ( t = u -> ( 2nd ` t ) = ( 2nd ` u ) )
69 68 breq2d 
 |-  ( t = u -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` u ) ) )
70 69 ifbid 
 |-  ( t = u -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` u ) , y , ( y + 1 ) ) )
71 70 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 } ) ) ) )
72 2fveq3 
 |-  ( t = u -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` u ) ) )
73 2fveq3 
 |-  ( t = u -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` u ) ) )
74 73 imaeq1d 
 |-  ( t = u -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) )
75 74 xpeq1d 
 |-  ( t = u -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) X. { 1 } ) )
76 73 imaeq1d 
 |-  ( t = u -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) )
77 76 xpeq1d 
 |-  ( t = u -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) )
78 75 77 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 } ) ) )
79 72 78 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 } ) ) ) )
80 79 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 } ) ) ) )
81 71 80 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 } ) ) ) )
82 81 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 } ) ) ) ) )
83 breq1 
 |-  ( y = w -> ( y < ( 2nd ` u ) <-> w < ( 2nd ` u ) ) )
84 id 
 |-  ( y = w -> y = w )
85 oveq1 
 |-  ( y = w -> ( y + 1 ) = ( w + 1 ) )
86 83 84 85 ifbieq12d 
 |-  ( y = w -> if ( y < ( 2nd ` u ) , y , ( y + 1 ) ) = if ( w < ( 2nd ` u ) , w , ( w + 1 ) ) )
87 86 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 } ) ) ) )
88 oveq2 
 |-  ( j = i -> ( 1 ... j ) = ( 1 ... i ) )
89 88 imaeq2d 
 |-  ( j = i -> ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... i ) ) )
90 89 xpeq1d 
 |-  ( j = i -> ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` u ) ) " ( 1 ... i ) ) X. { 1 } ) )
91 oveq1 
 |-  ( j = i -> ( j + 1 ) = ( i + 1 ) )
92 91 oveq1d 
 |-  ( j = i -> ( ( j + 1 ) ... N ) = ( ( i + 1 ) ... N ) )
93 92 imaeq2d 
 |-  ( j = i -> ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` u ) ) " ( ( i + 1 ) ... N ) ) )
94 93 xpeq1d 
 |-  ( j = i -> ( ( ( 2nd ` ( 1st ` u ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` u ) ) " ( ( i + 1 ) ... N ) ) X. { 0 } ) )
95 90 94 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 } ) ) )
96 95 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 } ) ) ) )
97 96 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 } ) ) )
98 87 97 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 } ) ) ) )
99 98 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 } ) ) ) )
100 82 99 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 } ) ) ) ) )
101 100 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 } ) ) ) ) ) )
102 101 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 } ) ) ) ) }
103 elmapi 
 |-  ( x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) -> x : ( 0 ... ( N - 1 ) ) --> ( ( 0 ... K ) ^m ( 1 ... N ) ) )
104 103 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 ) ) )
105 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 } ) ) ) ) } )
106 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 )
107 106 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 )
108 107 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 )
109 fveq2 
 |-  ( n = m -> ( p ` n ) = ( p ` m ) )
110 109 neeq1d 
 |-  ( n = m -> ( ( p ` n ) =/= 0 <-> ( p ` m ) =/= 0 ) )
111 110 rexbidv 
 |-  ( n = m -> ( E. p e. ran x ( p ` n ) =/= 0 <-> E. p e. ran x ( p ` m ) =/= 0 ) )
112 fveq1 
 |-  ( p = q -> ( p ` m ) = ( q ` m ) )
113 112 neeq1d 
 |-  ( p = q -> ( ( p ` m ) =/= 0 <-> ( q ` m ) =/= 0 ) )
114 113 cbvrexvw 
 |-  ( E. p e. ran x ( p ` m ) =/= 0 <-> E. q e. ran x ( q ` m ) =/= 0 )
115 111 114 bitrdi 
 |-  ( n = m -> ( E. p e. ran x ( p ` n ) =/= 0 <-> E. q e. ran x ( q ` m ) =/= 0 ) )
116 115 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 )
117 108 116 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 )
118 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 )
119 118 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 )
120 119 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 )
121 109 neeq1d 
 |-  ( n = m -> ( ( p ` n ) =/= K <-> ( p ` m ) =/= K ) )
122 121 rexbidv 
 |-  ( n = m -> ( E. p e. ran x ( p ` n ) =/= K <-> E. p e. ran x ( p ` m ) =/= K ) )
123 112 neeq1d 
 |-  ( p = q -> ( ( p ` m ) =/= K <-> ( q ` m ) =/= K ) )
124 123 cbvrexvw 
 |-  ( E. p e. ran x ( p ` m ) =/= K <-> E. q e. ran x ( q ` m ) =/= K )
125 122 124 bitrdi 
 |-  ( n = m -> ( E. p e. ran x ( p ` n ) =/= K <-> E. q e. ran x ( q ` m ) =/= K ) )
126 125 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 )
127 120 126 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 )
128 67 102 104 105 117 127 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 )
129 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 ) )
130 129 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 ) )
131 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
132 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 } ) ) )
133 131 132 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 } ) )
134 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 ) )
135 130 133 134 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 )
136 128 135 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 )
137 136 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 } ) ) )
138 66 137 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 } ) ) )
139 55 138 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 } ) ) ) ) } ) )
140 139 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 } ) ) ) ) } ) ) )
141 140 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 } ) ) ) ) } ) ) )
142 46 141 biimtrid 
 |-  ( ( ( 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 } ) ) ) ) } ) ) )
143 dvds0 
 |-  ( 2 e. ZZ -> 2 || 0 )
144 14 143 ax-mp 
 |-  2 || 0
145 hash0 
 |-  ( # ` (/) ) = 0
146 144 145 breqtrri 
 |-  2 || ( # ` (/) )
147 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 } ) ) ) ) } ) = ( # ` (/) ) )
148 146 147 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 } ) ) ) ) } ) )
149 142 148 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 } ) ) ) ) } ) )
150 149 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 } ) ) ) ) } ) ) )
151 150 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 } ) ) ) ) } ) ) )
152 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 ) ) ) ) )
153 152 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 ) ) ) } )
154 153 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 ) ) ) } ) )
155 154 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 ) ) ) } ) ) )
156 151 155 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 ) ) ) } ) ) )
157 156 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 ) ) ) } ) ) ) )
158 39 45 157 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 ) ) ) } ) ) )
159 38 158 biimtrrid 
 |-  ( ( 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 ) ) ) } ) ) )
160 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 ) ) )
161 160 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 ) ) ) )
162 161 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 ) ) ) )
163 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 ) ) ) )
164 162 163 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 ) ) ) } = (/) )
165 164 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 ) ) ) } ) = ( # ` (/) ) )
166 146 165 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 ) ) ) } ) )
167 159 166 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 ) ) ) } ) )
168 13 15 37 167 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 ) ) ) } ) )
169 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 )
170 32 169 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
171 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 )
172 sneq 
 |-  ( ( 2nd ` t ) = N -> { ( 2nd ` t ) } = { N } )
173 172 difeq2d 
 |-  ( ( 2nd ` t ) = N -> ( ( 0 ... N ) \ { ( 2nd ` t ) } ) = ( ( 0 ... N ) \ { N } ) )
174 difun2 
 |-  ( ( ( 0 ... ( N - 1 ) ) u. { N } ) \ { N } ) = ( ( 0 ... ( N - 1 ) ) \ { N } )
175 1 nnnn0d 
 |-  ( ph -> N e. NN0 )
176 nn0uz 
 |-  NN0 = ( ZZ>= ` 0 )
177 175 176 eleqtrdi 
 |-  ( ph -> N e. ( ZZ>= ` 0 ) )
178 fzm1 
 |-  ( N e. ( ZZ>= ` 0 ) -> ( n e. ( 0 ... N ) <-> ( n e. ( 0 ... ( N - 1 ) ) \/ n = N ) ) )
179 177 178 syl 
 |-  ( ph -> ( n e. ( 0 ... N ) <-> ( n e. ( 0 ... ( N - 1 ) ) \/ n = N ) ) )
180 elun 
 |-  ( n e. ( ( 0 ... ( N - 1 ) ) u. { N } ) <-> ( n e. ( 0 ... ( N - 1 ) ) \/ n e. { N } ) )
181 velsn 
 |-  ( n e. { N } <-> n = N )
182 181 orbi2i 
 |-  ( ( n e. ( 0 ... ( N - 1 ) ) \/ n e. { N } ) <-> ( n e. ( 0 ... ( N - 1 ) ) \/ n = N ) )
183 180 182 bitri 
 |-  ( n e. ( ( 0 ... ( N - 1 ) ) u. { N } ) <-> ( n e. ( 0 ... ( N - 1 ) ) \/ n = N ) )
184 179 183 bitr4di 
 |-  ( ph -> ( n e. ( 0 ... N ) <-> n e. ( ( 0 ... ( N - 1 ) ) u. { N } ) ) )
185 184 eqrdv 
 |-  ( ph -> ( 0 ... N ) = ( ( 0 ... ( N - 1 ) ) u. { N } ) )
186 185 difeq1d 
 |-  ( ph -> ( ( 0 ... N ) \ { N } ) = ( ( ( 0 ... ( N - 1 ) ) u. { N } ) \ { N } ) )
187 1 nnzd 
 |-  ( ph -> N e. ZZ )
188 uzid 
 |-  ( N e. ZZ -> N e. ( ZZ>= ` N ) )
189 uznfz 
 |-  ( N e. ( ZZ>= ` N ) -> -. N e. ( 0 ... ( N - 1 ) ) )
190 187 188 189 3syl 
 |-  ( ph -> -. N e. ( 0 ... ( N - 1 ) ) )
191 disjsn 
 |-  ( ( ( 0 ... ( N - 1 ) ) i^i { N } ) = (/) <-> -. N e. ( 0 ... ( N - 1 ) ) )
192 disj3 
 |-  ( ( ( 0 ... ( N - 1 ) ) i^i { N } ) = (/) <-> ( 0 ... ( N - 1 ) ) = ( ( 0 ... ( N - 1 ) ) \ { N } ) )
193 191 192 bitr3i 
 |-  ( -. N e. ( 0 ... ( N - 1 ) ) <-> ( 0 ... ( N - 1 ) ) = ( ( 0 ... ( N - 1 ) ) \ { N } ) )
194 190 193 sylib 
 |-  ( ph -> ( 0 ... ( N - 1 ) ) = ( ( 0 ... ( N - 1 ) ) \ { N } ) )
195 174 186 194 3eqtr4a 
 |-  ( ph -> ( ( 0 ... N ) \ { N } ) = ( 0 ... ( N - 1 ) ) )
196 173 195 sylan9eqr 
 |-  ( ( ph /\ ( 2nd ` t ) = N ) -> ( ( 0 ... N ) \ { ( 2nd ` t ) } ) = ( 0 ... ( N - 1 ) ) )
197 196 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 ) )
198 197 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 ) )
199 198 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 ) )
200 199 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 ) )
201 171 200 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 ) )
202 201 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 ) )
203 202 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 } )
204 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 ) ) } ) ) )
205 170 203 204 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 ) ) } ) ) )
206 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 )
207 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 ) )
208 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 ) } ) )
209 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 ) ) )
210 elmapi 
 |-  ( ( 1st ` ( 1st ` t ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` t ) ) : ( 1 ... N ) --> ( 0 ..^ K ) )
211 208 209 210 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 ) )
212 211 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 ) )
213 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 ) } )
214 fvex 
 |-  ( 2nd ` ( 1st ` t ) ) e. _V
215 f1oeq1 
 |-  ( f = ( 2nd ` ( 1st ` t ) ) -> ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( 2nd ` ( 1st ` t ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) )
216 214 215 elab 
 |-  ( ( 2nd ` ( 1st ` t ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } <-> ( 2nd ` ( 1st ` t ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
217 213 216 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 ) )
218 208 217 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 ) )
219 218 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 ) )
220 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 ) )
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 ` t ) e. ( 0 ... N ) )
222 206 2 207 212 219 221 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 ) ) ) ) )
223 208 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 ) } ) )
224 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 ) ) >. )
225 224 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 )
226 225 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 ) )
227 226 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 ) )
228 227 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 ) )
229 228 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 ) ) ) ) )
230 223 229 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 ) ) ) ) )
231 222 230 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 ) ) ) ) )
232 103 frnd 
 |-  ( x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) -> ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) )
233 232 anim2i 
 |-  ( ( ph /\ x e. ( ( ( 0 ... K ) ^m ( 1 ... N ) ) ^m ( 0 ... ( N - 1 ) ) ) ) -> ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) )
234 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 ) )
235 vex 
 |-  n e. _V
236 eqid 
 |-  ( p e. ran x |-> B ) = ( p e. ran x |-> B )
237 236 elrnmpt 
 |-  ( n e. _V -> ( n e. ran ( p e. ran x |-> B ) <-> E. p e. ran x n = B ) )
238 235 237 ax-mp 
 |-  ( n e. ran ( p e. ran x |-> B ) <-> E. p e. ran x n = B )
239 238 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 )
240 234 239 sylbb 
 |-  ( ( 0 ... ( N - 1 ) ) C_ ran ( p e. ran x |-> B ) -> A. n e. ( 0 ... ( N - 1 ) ) E. p e. ran x n = B )
241 1eluzge0 
 |-  1 e. ( ZZ>= ` 0 )
242 fzss1 
 |-  ( 1 e. ( ZZ>= ` 0 ) -> ( 1 ... ( N - 1 ) ) C_ ( 0 ... ( N - 1 ) ) )
243 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 ) )
244 241 242 243 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 )
245 1 nncnd 
 |-  ( ph -> N e. CC )
246 npcan1 
 |-  ( N e. CC -> ( ( N - 1 ) + 1 ) = N )
247 245 246 syl 
 |-  ( ph -> ( ( N - 1 ) + 1 ) = N )
248 peano2zm 
 |-  ( N e. ZZ -> ( N - 1 ) e. ZZ )
249 uzid 
 |-  ( ( N - 1 ) e. ZZ -> ( N - 1 ) e. ( ZZ>= ` ( N - 1 ) ) )
250 peano2uz 
 |-  ( ( N - 1 ) e. ( ZZ>= ` ( N - 1 ) ) -> ( ( N - 1 ) + 1 ) e. ( ZZ>= ` ( N - 1 ) ) )
251 187 248 249 250 4syl 
 |-  ( ph -> ( ( N - 1 ) + 1 ) e. ( ZZ>= ` ( N - 1 ) ) )
252 247 251 eqeltrrd 
 |-  ( ph -> N e. ( ZZ>= ` ( N - 1 ) ) )
253 fzss2 
 |-  ( N e. ( ZZ>= ` ( N - 1 ) ) -> ( 1 ... ( N - 1 ) ) C_ ( 1 ... N ) )
254 252 253 syl 
 |-  ( ph -> ( 1 ... ( N - 1 ) ) C_ ( 1 ... N ) )
255 254 sselda 
 |-  ( ( ph /\ n e. ( 1 ... ( N - 1 ) ) ) -> n e. ( 1 ... N ) )
256 255 adantlr 
 |-  ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ n e. ( 1 ... ( N - 1 ) ) ) -> n e. ( 1 ... N ) )
257 simplr 
 |-  ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ n e. ( 1 ... N ) ) -> ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) )
258 ssel2 
 |-  ( ( ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) /\ p e. ran x ) -> p e. ( ( 0 ... K ) ^m ( 1 ... N ) ) )
259 elmapi 
 |-  ( p e. ( ( 0 ... K ) ^m ( 1 ... N ) ) -> p : ( 1 ... N ) --> ( 0 ... K ) )
260 258 259 syl 
 |-  ( ( ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) /\ p e. ran x ) -> p : ( 1 ... N ) --> ( 0 ... K ) )
261 257 260 sylan 
 |-  ( ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ n e. ( 1 ... N ) ) /\ p e. ran x ) -> p : ( 1 ... N ) --> ( 0 ... K ) )
262 elfzelz 
 |-  ( n e. ( 1 ... N ) -> n e. ZZ )
263 262 zred 
 |-  ( n e. ( 1 ... N ) -> n e. RR )
264 263 ltnrd 
 |-  ( n e. ( 1 ... N ) -> -. n < n )
265 breq1 
 |-  ( n = B -> ( n < n <-> B < n ) )
266 265 notbid 
 |-  ( n = B -> ( -. n < n <-> -. B < n ) )
267 264 266 syl5ibcom 
 |-  ( n e. ( 1 ... N ) -> ( n = B -> -. B < n ) )
268 267 necon2ad 
 |-  ( n e. ( 1 ... N ) -> ( B < n -> n =/= B ) )
269 268 3ad2ant1 
 |-  ( ( n e. ( 1 ... N ) /\ p : ( 1 ... N ) --> ( 0 ... K ) /\ ( p ` n ) = 0 ) -> ( B < n -> n =/= B ) )
270 269 adantl 
 |-  ( ( ph /\ ( n e. ( 1 ... N ) /\ p : ( 1 ... N ) --> ( 0 ... K ) /\ ( p ` n ) = 0 ) ) -> ( B < n -> n =/= B ) )
271 4 270 mpd 
 |-  ( ( ph /\ ( n e. ( 1 ... N ) /\ p : ( 1 ... N ) --> ( 0 ... K ) /\ ( p ` n ) = 0 ) ) -> n =/= B )
272 271 3exp2 
 |-  ( ph -> ( n e. ( 1 ... N ) -> ( p : ( 1 ... N ) --> ( 0 ... K ) -> ( ( p ` n ) = 0 -> n =/= B ) ) ) )
273 272 imp31 
 |-  ( ( ( ph /\ n e. ( 1 ... N ) ) /\ p : ( 1 ... N ) --> ( 0 ... K ) ) -> ( ( p ` n ) = 0 -> n =/= B ) )
274 273 necon2d 
 |-  ( ( ( ph /\ n e. ( 1 ... N ) ) /\ p : ( 1 ... N ) --> ( 0 ... K ) ) -> ( n = B -> ( p ` n ) =/= 0 ) )
275 274 adantllr 
 |-  ( ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ n e. ( 1 ... N ) ) /\ p : ( 1 ... N ) --> ( 0 ... K ) ) -> ( n = B -> ( p ` n ) =/= 0 ) )
276 261 275 syldan 
 |-  ( ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ n e. ( 1 ... N ) ) /\ p e. ran x ) -> ( n = B -> ( p ` n ) =/= 0 ) )
277 276 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 ) )
278 256 277 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 ) )
279 278 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 ) )
280 279 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 )
281 244 280 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 )
282 281 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 ) ) )
283 nnuz 
 |-  NN = ( ZZ>= ` 1 )
284 1 283 eleqtrdi 
 |-  ( ph -> N e. ( ZZ>= ` 1 ) )
285 fzm1 
 |-  ( N e. ( ZZ>= ` 1 ) -> ( n e. ( 1 ... N ) <-> ( n e. ( 1 ... ( N - 1 ) ) \/ n = N ) ) )
286 284 285 syl 
 |-  ( ph -> ( n e. ( 1 ... N ) <-> ( n e. ( 1 ... ( N - 1 ) ) \/ n = N ) ) )
287 elun 
 |-  ( n e. ( ( 1 ... ( N - 1 ) ) u. { N } ) <-> ( n e. ( 1 ... ( N - 1 ) ) \/ n e. { N } ) )
288 181 orbi2i 
 |-  ( ( n e. ( 1 ... ( N - 1 ) ) \/ n e. { N } ) <-> ( n e. ( 1 ... ( N - 1 ) ) \/ n = N ) )
289 287 288 bitri 
 |-  ( n e. ( ( 1 ... ( N - 1 ) ) u. { N } ) <-> ( n e. ( 1 ... ( N - 1 ) ) \/ n = N ) )
290 286 289 bitr4di 
 |-  ( ph -> ( n e. ( 1 ... N ) <-> n e. ( ( 1 ... ( N - 1 ) ) u. { N } ) ) )
291 290 eqrdv 
 |-  ( ph -> ( 1 ... N ) = ( ( 1 ... ( N - 1 ) ) u. { N } ) )
292 291 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 ) )
293 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 ) )
294 292 293 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 ) ) )
295 fveq2 
 |-  ( n = N -> ( p ` n ) = ( p ` N ) )
296 295 neeq1d 
 |-  ( n = N -> ( ( p ` n ) =/= 0 <-> ( p ` N ) =/= 0 ) )
297 296 rexbidv 
 |-  ( n = N -> ( E. p e. ran x ( p ` n ) =/= 0 <-> E. p e. ran x ( p ` N ) =/= 0 ) )
298 297 ralsng 
 |-  ( N e. NN -> ( A. n e. { N } E. p e. ran x ( p ` n ) =/= 0 <-> E. p e. ran x ( p ` N ) =/= 0 ) )
299 1 298 syl 
 |-  ( ph -> ( A. n e. { N } E. p e. ran x ( p ` n ) =/= 0 <-> E. p e. ran x ( p ` N ) =/= 0 ) )
300 299 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 ) ) )
301 294 300 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 ) ) )
302 301 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 ) ) )
303 0z 
 |-  0 e. ZZ
304 1z 
 |-  1 e. ZZ
305 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 ) )
306 303 304 305 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 ) )
307 187 248 306 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 ) )
308 0p1e1 
 |-  ( 0 + 1 ) = 1
309 308 a1i 
 |-  ( ph -> ( 0 + 1 ) = 1 )
310 309 247 oveq12d 
 |-  ( ph -> ( ( 0 + 1 ) ... ( ( N - 1 ) + 1 ) ) = ( 1 ... N ) )
311 310 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 ) )
312 307 311 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 ) )
313 ovex 
 |-  ( m - 1 ) e. _V
314 eqeq1 
 |-  ( n = ( m - 1 ) -> ( n = B <-> ( m - 1 ) = B ) )
315 314 rexbidv 
 |-  ( n = ( m - 1 ) -> ( E. p e. ran x n = B <-> E. p e. ran x ( m - 1 ) = B ) )
316 313 315 sbcie 
 |-  ( [. ( m - 1 ) / n ]. E. p e. ran x n = B <-> E. p e. ran x ( m - 1 ) = B )
317 316 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 )
318 oveq1 
 |-  ( m = n -> ( m - 1 ) = ( n - 1 ) )
319 318 eqeq1d 
 |-  ( m = n -> ( ( m - 1 ) = B <-> ( n - 1 ) = B ) )
320 319 rexbidv 
 |-  ( m = n -> ( E. p e. ran x ( m - 1 ) = B <-> E. p e. ran x ( n - 1 ) = B ) )
321 320 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 )
322 317 321 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 )
323 312 322 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 ) )
324 323 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 )
325 324 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 )
326 5 necomd 
 |-  ( ( ph /\ ( n e. ( 1 ... N ) /\ p : ( 1 ... N ) --> ( 0 ... K ) /\ ( p ` n ) = K ) ) -> ( n - 1 ) =/= B )
327 326 3exp2 
 |-  ( ph -> ( n e. ( 1 ... N ) -> ( p : ( 1 ... N ) --> ( 0 ... K ) -> ( ( p ` n ) = K -> ( n - 1 ) =/= B ) ) ) )
328 327 imp31 
 |-  ( ( ( ph /\ n e. ( 1 ... N ) ) /\ p : ( 1 ... N ) --> ( 0 ... K ) ) -> ( ( p ` n ) = K -> ( n - 1 ) =/= B ) )
329 328 necon2d 
 |-  ( ( ( ph /\ n e. ( 1 ... N ) ) /\ p : ( 1 ... N ) --> ( 0 ... K ) ) -> ( ( n - 1 ) = B -> ( p ` n ) =/= K ) )
330 329 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 ) )
331 261 330 syldan 
 |-  ( ( ( ( ph /\ ran x C_ ( ( 0 ... K ) ^m ( 1 ... N ) ) ) /\ n e. ( 1 ... N ) ) /\ p e. ran x ) -> ( ( n - 1 ) = B -> ( p ` n ) =/= K ) )
332 331 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 ) )
333 332 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 ) )
334 333 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 )
335 325 334 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 )
336 335 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 ) ) )
337 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 ) )
338 336 337 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 ) ) )
339 282 302 338 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 ) ) )
340 233 240 339 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 ) ) )
341 340 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 ) ) ) )
342 341 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 ) ) ) ) )
343 342 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 ) ) ) ) )
344 343 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 ) ) ) ) )
345 195 rexeqdv 
 |-  ( ph -> ( E. j e. ( ( 0 ... N ) \ { N } ) i = [_ ( 1st ` t ) / s ]_ C <-> E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C ) )
346 345 biimpd 
 |-  ( ph -> ( E. j e. ( ( 0 ... N ) \ { N } ) i = [_ ( 1st ` t ) / s ]_ C -> E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C ) )
347 346 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 ) )
348 173 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 ) )
349 348 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 ) )
350 349 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 ) ) )
351 347 350 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 ) ) )
352 351 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 ) ) )
353 352 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 ) )
354 353 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 ) )
355 354 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 ) ) ) ) )
356 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 ) ) ) )
357 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 ) ) )
358 357 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 ) ) ) )
359 356 358 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 ) ) )
360 355 359 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 ) ) ) )
361 360 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 ) ) ) )
362 361 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 ) ) ) ) )
363 362 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 ) ) ) ) )
364 231 344 363 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 ) ) ) ) )
365 364 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 ) ) ) } )
366 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 ) ) ) }
367 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 ) ) ) }
368 365 366 367 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 ) ) } ) )
369 368 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 ) ) } ) ) )
370 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 )
371 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 } ) ) ) ) )
372 371 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 } ) ) ) ) ) )
373 372 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 } ) ) ) ) }
374 373 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 } ) ) ) ) } )
375 fveq2 
 |-  ( t = s -> ( 2nd ` t ) = ( 2nd ` s ) )
376 375 breq2d 
 |-  ( t = s -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` s ) ) )
377 376 ifbid 
 |-  ( t = s -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` s ) , y , ( y + 1 ) ) )
378 377 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 } ) ) ) )
379 2fveq3 
 |-  ( t = s -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` s ) ) )
380 2fveq3 
 |-  ( t = s -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` s ) ) )
381 380 imaeq1d 
 |-  ( t = s -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) )
382 381 xpeq1d 
 |-  ( t = s -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` s ) ) " ( 1 ... j ) ) X. { 1 } ) )
383 380 imaeq1d 
 |-  ( t = s -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) )
384 383 xpeq1d 
 |-  ( t = s -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` s ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) )
385 382 384 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 } ) ) )
386 379 385 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 } ) ) ) )
387 386 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 } ) ) ) )
388 378 387 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 } ) ) ) )
389 388 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 } ) ) ) ) )
390 389 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 } ) ) ) ) ) )
391 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 )
392 390 391 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 ) )
393 392 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 ) )
394 393 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 )
395 374 394 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 )
396 395 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
397 396 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
398 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 ) ) ) } )
399 397 398 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 ) ) ) } )
400 13 370 399 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 ) ) ) } ) )
401 369 400 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 ) ) ) } ) )
402 fo1st 
 |-  1st : _V -onto-> _V
403 fofun 
 |-  ( 1st : _V -onto-> _V -> Fun 1st )
404 402 403 ax-mp 
 |-  Fun 1st
405 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
406 fof 
 |-  ( 1st : _V -onto-> _V -> 1st : _V --> _V )
407 402 406 ax-mp 
 |-  1st : _V --> _V
408 407 fdmi 
 |-  dom 1st = _V
409 405 408 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
410 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 ) ) } ) )
411 404 409 410 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 ) ) } )
412 fveqeq2 
 |-  ( t = x -> ( ( 2nd ` t ) = N <-> ( 2nd ` x ) = N ) )
413 fveq2 
 |-  ( t = x -> ( 1st ` t ) = ( 1st ` x ) )
414 413 csbeq1d 
 |-  ( t = x -> [_ ( 1st ` t ) / s ]_ C = [_ ( 1st ` x ) / s ]_ C )
415 414 eqeq2d 
 |-  ( t = x -> ( i = [_ ( 1st ` t ) / s ]_ C <-> i = [_ ( 1st ` x ) / s ]_ C ) )
416 415 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 ) )
417 416 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 ) )
418 2fveq3 
 |-  ( t = x -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` x ) ) )
419 418 fveq1d 
 |-  ( t = x -> ( ( 1st ` ( 1st ` t ) ) ` N ) = ( ( 1st ` ( 1st ` x ) ) ` N ) )
420 419 eqeq1d 
 |-  ( t = x -> ( ( ( 1st ` ( 1st ` t ) ) ` N ) = 0 <-> ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 ) )
421 2fveq3 
 |-  ( t = x -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` x ) ) )
422 421 fveq1d 
 |-  ( t = x -> ( ( 2nd ` ( 1st ` t ) ) ` N ) = ( ( 2nd ` ( 1st ` x ) ) ` N ) )
423 422 eqeq1d 
 |-  ( t = x -> ( ( ( 2nd ` ( 1st ` t ) ) ` N ) = N <-> ( ( 2nd ` ( 1st ` x ) ) ` N ) = N ) )
424 417 420 423 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 ) ) )
425 412 424 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 ) ) ) )
426 425 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 ) )
427 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 ) } ) )
428 427 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 ) ) )
429 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 ) } ) ) )
430 csbeq1a 
 |-  ( s = ( 1st ` x ) -> C = [_ ( 1st ` x ) / s ]_ C )
431 430 eqcoms 
 |-  ( ( 1st ` x ) = s -> C = [_ ( 1st ` x ) / s ]_ C )
432 431 eqcomd 
 |-  ( ( 1st ` x ) = s -> [_ ( 1st ` x ) / s ]_ C = C )
433 432 eqeq2d 
 |-  ( ( 1st ` x ) = s -> ( i = [_ ( 1st ` x ) / s ]_ C <-> i = C ) )
434 433 rexbidv 
 |-  ( ( 1st ` x ) = s -> ( E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` x ) / s ]_ C <-> E. j e. ( 0 ... ( N - 1 ) ) i = C ) )
435 434 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 ) )
436 fveq2 
 |-  ( ( 1st ` x ) = s -> ( 1st ` ( 1st ` x ) ) = ( 1st ` s ) )
437 436 fveq1d 
 |-  ( ( 1st ` x ) = s -> ( ( 1st ` ( 1st ` x ) ) ` N ) = ( ( 1st ` s ) ` N ) )
438 437 eqeq1d 
 |-  ( ( 1st ` x ) = s -> ( ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 <-> ( ( 1st ` s ) ` N ) = 0 ) )
439 fveq2 
 |-  ( ( 1st ` x ) = s -> ( 2nd ` ( 1st ` x ) ) = ( 2nd ` s ) )
440 439 fveq1d 
 |-  ( ( 1st ` x ) = s -> ( ( 2nd ` ( 1st ` x ) ) ` N ) = ( ( 2nd ` s ) ` N ) )
441 440 eqeq1d 
 |-  ( ( 1st ` x ) = s -> ( ( ( 2nd ` ( 1st ` x ) ) ` N ) = N <-> ( ( 2nd ` s ) ` N ) = N ) )
442 435 438 441 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 ) ) )
443 429 442 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 ) ) ) )
444 428 443 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 ) ) ) )
445 444 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 ) ) ) )
446 445 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 ) ) ) )
447 446 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 ) ) )
448 nn0fz0 
 |-  ( N e. NN0 <-> N e. ( 0 ... N ) )
449 175 448 sylib 
 |-  ( ph -> N e. ( 0 ... N ) )
450 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 ) ) )
451 449 450 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 ) ) )
452 451 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 ) ) )
453 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 ) )
454 op2ndg 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( 2nd ` <. s , N >. ) = N )
455 454 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 ) ) ) )
456 op1stg 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( 1st ` <. s , N >. ) = s )
457 csbeq1a 
 |-  ( s = ( 1st ` <. s , N >. ) -> C = [_ ( 1st ` <. s , N >. ) / s ]_ C )
458 457 eqcoms 
 |-  ( ( 1st ` <. s , N >. ) = s -> C = [_ ( 1st ` <. s , N >. ) / s ]_ C )
459 458 eqcomd 
 |-  ( ( 1st ` <. s , N >. ) = s -> [_ ( 1st ` <. s , N >. ) / s ]_ C = C )
460 456 459 syl 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> [_ ( 1st ` <. s , N >. ) / s ]_ C = C )
461 460 eqeq2d 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( i = [_ ( 1st ` <. s , N >. ) / s ]_ C <-> i = C ) )
462 461 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 ) )
463 462 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 ) )
464 456 fveq2d 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( 1st ` ( 1st ` <. s , N >. ) ) = ( 1st ` s ) )
465 464 fveq1d 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) = ( ( 1st ` s ) ` N ) )
466 465 eqeq1d 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) = 0 <-> ( ( 1st ` s ) ` N ) = 0 ) )
467 456 fveq2d 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( 2nd ` ( 1st ` <. s , N >. ) ) = ( 2nd ` s ) )
468 467 fveq1d 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) = ( ( 2nd ` s ) ` N ) )
469 468 eqeq1d 
 |-  ( ( s e. _V /\ N e. ( 0 ... N ) ) -> ( ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) = N <-> ( ( 2nd ` s ) ` N ) = N ) )
470 463 466 469 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 ) ) )
471 456 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 ) ) )
472 455 470 471 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 ) ) )
473 49 453 472 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 ) ) )
474 473 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 ) )
475 fveqeq2 
 |-  ( x = <. s , N >. -> ( ( 2nd ` x ) = N <-> ( 2nd ` <. s , N >. ) = N ) )
476 fveq2 
 |-  ( x = <. s , N >. -> ( 1st ` x ) = ( 1st ` <. s , N >. ) )
477 476 csbeq1d 
 |-  ( x = <. s , N >. -> [_ ( 1st ` x ) / s ]_ C = [_ ( 1st ` <. s , N >. ) / s ]_ C )
478 477 eqeq2d 
 |-  ( x = <. s , N >. -> ( i = [_ ( 1st ` x ) / s ]_ C <-> i = [_ ( 1st ` <. s , N >. ) / s ]_ C ) )
479 478 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 ) )
480 479 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 ) )
481 2fveq3 
 |-  ( x = <. s , N >. -> ( 1st ` ( 1st ` x ) ) = ( 1st ` ( 1st ` <. s , N >. ) ) )
482 481 fveq1d 
 |-  ( x = <. s , N >. -> ( ( 1st ` ( 1st ` x ) ) ` N ) = ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) )
483 482 eqeq1d 
 |-  ( x = <. s , N >. -> ( ( ( 1st ` ( 1st ` x ) ) ` N ) = 0 <-> ( ( 1st ` ( 1st ` <. s , N >. ) ) ` N ) = 0 ) )
484 2fveq3 
 |-  ( x = <. s , N >. -> ( 2nd ` ( 1st ` x ) ) = ( 2nd ` ( 1st ` <. s , N >. ) ) )
485 484 fveq1d 
 |-  ( x = <. s , N >. -> ( ( 2nd ` ( 1st ` x ) ) ` N ) = ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) )
486 485 eqeq1d 
 |-  ( x = <. s , N >. -> ( ( ( 2nd ` ( 1st ` x ) ) ` N ) = N <-> ( ( 2nd ` ( 1st ` <. s , N >. ) ) ` N ) = N ) )
487 480 483 486 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 ) ) )
488 475 487 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 ) ) ) )
489 fveqeq2 
 |-  ( x = <. s , N >. -> ( ( 1st ` x ) = s <-> ( 1st ` <. s , N >. ) = s ) )
490 488 489 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 ) ) )
491 490 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 ) )
492 474 491 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 ) )
493 452 492 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 ) )
494 493 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 ) ) )
495 447 494 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 ) ) ) )
496 426 495 bitrid 
 |-  ( 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 ) ) ) )
497 496 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 ) ) } )
498 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 } )
499 404 409 498 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 }
500 nfv 
 |-  F/ s ( 2nd ` t ) = N
501 nfcv 
 |-  F/_ s ( 0 ... ( N - 1 ) )
502 nfcsb1v 
 |-  F/_ s [_ ( 1st ` t ) / s ]_ C
503 502 nfeq2 
 |-  F/ s i = [_ ( 1st ` t ) / s ]_ C
504 501 503 nfrexw 
 |-  F/ s E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C
505 501 504 nfralw 
 |-  F/ s A. i e. ( 0 ... ( N - 1 ) ) E. j e. ( 0 ... ( N - 1 ) ) i = [_ ( 1st ` t ) / s ]_ C
506 nfv 
 |-  F/ s ( ( 1st ` ( 1st ` t ) ) ` N ) = 0
507 nfv 
 |-  F/ s ( ( 2nd ` ( 1st ` t ) ) ` N ) = N
508 505 506 507 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 )
509 500 508 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 ) )
510 nfcv 
 |-  F/_ s ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) )
511 509 510 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 ) ) }
512 nfv 
 |-  F/ s ( 1st ` x ) = y
513 511 512 nfrexw 
 |-  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
514 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
515 eqeq2 
 |-  ( y = s -> ( ( 1st ` x ) = y <-> ( 1st ` x ) = s ) )
516 515 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 ) )
517 513 514 516 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 }
518 499 517 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 }
519 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 ) ) }
520 497 518 519 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 ) } )
521 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 ) } ) )
522 520 521 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 ) } ) )
523 411 522 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 ) } )
524 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 ) } )
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 ) ) } --> { 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 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 ) )
527 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 ) )
528 526 527 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 ) ) )
529 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 )
530 529 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 ) )
531 530 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 }
532 531 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 } )
533 412 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 ) )
534 532 533 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 ) )
535 531 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 } )
536 fveqeq2 
 |-  ( t = y -> ( ( 2nd ` t ) = N <-> ( 2nd ` y ) = N ) )
537 536 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 ) )
538 535 537 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 ) )
539 eqtr3 
 |-  ( ( ( 2nd ` x ) = N /\ ( 2nd ` y ) = N ) -> ( 2nd ` x ) = ( 2nd ` y ) )
540 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 ) )
541 540 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 ) )
542 541 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 ) )
543 542 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 ) )
544 539 543 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 ) )
545 544 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 ) )
546 534 538 545 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 ) )
547 528 546 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 ) )
548 547 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 )
549 525 548 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 ) ) )
550 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 ) ) )
551 549 550 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 ) } )
552 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 ) } ) )
553 551 523 552 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 ) } )
554 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 )
555 32 554 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
556 555 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
557 556 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 ) } )
558 553 557 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 ) } )
559 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 )
560 29 559 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
561 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 ) } ) )
562 555 560 561 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 ) } )
563 558 562 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 ) } ) )
564 563 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 ) } ) ) )
565 205 401 564 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 ) } ) ) )
566 168 565 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 ) } ) ) )