Step |
Hyp |
Ref |
Expression |
1 |
|
poimir.0 |
|- ( ph -> N e. NN ) |
2 |
|
poimirlem22.s |
|- S = { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } |
3 |
|
poimirlem22.1 |
|- ( ph -> F : ( 0 ... ( N - 1 ) ) --> ( ( 0 ... K ) ^m ( 1 ... N ) ) ) |
4 |
|
poimirlem12.2 |
|- ( ph -> T e. S ) |
5 |
|
poimirlem12.3 |
|- ( ph -> ( 2nd ` T ) = N ) |
6 |
|
poimirlem12.4 |
|- ( ph -> U e. S ) |
7 |
|
poimirlem12.5 |
|- ( ph -> ( 2nd ` U ) = N ) |
8 |
|
poimirlem12.6 |
|- ( ph -> M e. ( 0 ... ( N - 1 ) ) ) |
9 |
|
eldif |
|- ( y e. ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) \ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) <-> ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ -. y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) |
10 |
|
imassrn |
|- ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) C_ ran ( 2nd ` ( 1st ` T ) ) |
11 |
|
elrabi |
|- ( T e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , 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 ) ) ) |
12 |
11 2
|
eleq2s |
|- ( T e. S -> T e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) |
13 |
|
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 ) } ) ) |
14 |
4 12 13
|
3syl |
|- ( ph -> ( 1st ` T ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) |
15 |
|
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 ) } ) |
16 |
14 15
|
syl |
|- ( ph -> ( 2nd ` ( 1st ` T ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) |
17 |
|
fvex |
|- ( 2nd ` ( 1st ` T ) ) e. _V |
18 |
|
f1oeq1 |
|- ( f = ( 2nd ` ( 1st ` T ) ) -> ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) ) |
19 |
17 18
|
elab |
|- ( ( 2nd ` ( 1st ` T ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } <-> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
20 |
16 19
|
sylib |
|- ( ph -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
21 |
|
f1of |
|- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 1 ... N ) ) |
22 |
|
frn |
|- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 1 ... N ) -> ran ( 2nd ` ( 1st ` T ) ) C_ ( 1 ... N ) ) |
23 |
20 21 22
|
3syl |
|- ( ph -> ran ( 2nd ` ( 1st ` T ) ) C_ ( 1 ... N ) ) |
24 |
10 23
|
sstrid |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) C_ ( 1 ... N ) ) |
25 |
|
elrabi |
|- ( U e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , 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 ) ) ) |
26 |
25 2
|
eleq2s |
|- ( U e. S -> U e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) |
27 |
|
xp1st |
|- ( U e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( 1st ` U ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) |
28 |
6 26 27
|
3syl |
|- ( ph -> ( 1st ` U ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) |
29 |
|
xp2nd |
|- ( ( 1st ` U ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( 2nd ` ( 1st ` U ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) |
30 |
28 29
|
syl |
|- ( ph -> ( 2nd ` ( 1st ` U ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) |
31 |
|
fvex |
|- ( 2nd ` ( 1st ` U ) ) e. _V |
32 |
|
f1oeq1 |
|- ( f = ( 2nd ` ( 1st ` U ) ) -> ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) ) |
33 |
31 32
|
elab |
|- ( ( 2nd ` ( 1st ` U ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } <-> ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
34 |
30 33
|
sylib |
|- ( ph -> ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
35 |
|
f1ofo |
|- ( ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) ) |
36 |
|
foima |
|- ( ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) -> ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... N ) ) = ( 1 ... N ) ) |
37 |
34 35 36
|
3syl |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... N ) ) = ( 1 ... N ) ) |
38 |
24 37
|
sseqtrrd |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) C_ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... N ) ) ) |
39 |
38
|
ssdifd |
|- ( ph -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) \ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) C_ ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... N ) ) \ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) |
40 |
|
dff1o3 |
|- ( ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) /\ Fun `' ( 2nd ` ( 1st ` U ) ) ) ) |
41 |
40
|
simprbi |
|- ( ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> Fun `' ( 2nd ` ( 1st ` U ) ) ) |
42 |
|
imadif |
|- ( Fun `' ( 2nd ` ( 1st ` U ) ) -> ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... N ) \ ( 1 ... M ) ) ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... N ) ) \ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) |
43 |
34 41 42
|
3syl |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... N ) \ ( 1 ... M ) ) ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... N ) ) \ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) |
44 |
|
difun2 |
|- ( ( ( ( M + 1 ) ... N ) u. ( 1 ... M ) ) \ ( 1 ... M ) ) = ( ( ( M + 1 ) ... N ) \ ( 1 ... M ) ) |
45 |
|
elfznn0 |
|- ( M e. ( 0 ... ( N - 1 ) ) -> M e. NN0 ) |
46 |
|
nn0p1nn |
|- ( M e. NN0 -> ( M + 1 ) e. NN ) |
47 |
8 45 46
|
3syl |
|- ( ph -> ( M + 1 ) e. NN ) |
48 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
49 |
47 48
|
eleqtrdi |
|- ( ph -> ( M + 1 ) e. ( ZZ>= ` 1 ) ) |
50 |
1
|
nncnd |
|- ( ph -> N e. CC ) |
51 |
|
npcan1 |
|- ( N e. CC -> ( ( N - 1 ) + 1 ) = N ) |
52 |
50 51
|
syl |
|- ( ph -> ( ( N - 1 ) + 1 ) = N ) |
53 |
|
elfzuz3 |
|- ( M e. ( 0 ... ( N - 1 ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) ) |
54 |
|
peano2uz |
|- ( ( N - 1 ) e. ( ZZ>= ` M ) -> ( ( N - 1 ) + 1 ) e. ( ZZ>= ` M ) ) |
55 |
8 53 54
|
3syl |
|- ( ph -> ( ( N - 1 ) + 1 ) e. ( ZZ>= ` M ) ) |
56 |
52 55
|
eqeltrrd |
|- ( ph -> N e. ( ZZ>= ` M ) ) |
57 |
|
fzsplit2 |
|- ( ( ( M + 1 ) e. ( ZZ>= ` 1 ) /\ N e. ( ZZ>= ` M ) ) -> ( 1 ... N ) = ( ( 1 ... M ) u. ( ( M + 1 ) ... N ) ) ) |
58 |
49 56 57
|
syl2anc |
|- ( ph -> ( 1 ... N ) = ( ( 1 ... M ) u. ( ( M + 1 ) ... N ) ) ) |
59 |
|
uncom |
|- ( ( 1 ... M ) u. ( ( M + 1 ) ... N ) ) = ( ( ( M + 1 ) ... N ) u. ( 1 ... M ) ) |
60 |
58 59
|
eqtrdi |
|- ( ph -> ( 1 ... N ) = ( ( ( M + 1 ) ... N ) u. ( 1 ... M ) ) ) |
61 |
60
|
difeq1d |
|- ( ph -> ( ( 1 ... N ) \ ( 1 ... M ) ) = ( ( ( ( M + 1 ) ... N ) u. ( 1 ... M ) ) \ ( 1 ... M ) ) ) |
62 |
|
incom |
|- ( ( ( M + 1 ) ... N ) i^i ( 1 ... M ) ) = ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) |
63 |
8 45
|
syl |
|- ( ph -> M e. NN0 ) |
64 |
63
|
nn0red |
|- ( ph -> M e. RR ) |
65 |
64
|
ltp1d |
|- ( ph -> M < ( M + 1 ) ) |
66 |
|
fzdisj |
|- ( M < ( M + 1 ) -> ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) = (/) ) |
67 |
65 66
|
syl |
|- ( ph -> ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) = (/) ) |
68 |
62 67
|
syl5eq |
|- ( ph -> ( ( ( M + 1 ) ... N ) i^i ( 1 ... M ) ) = (/) ) |
69 |
|
disj3 |
|- ( ( ( ( M + 1 ) ... N ) i^i ( 1 ... M ) ) = (/) <-> ( ( M + 1 ) ... N ) = ( ( ( M + 1 ) ... N ) \ ( 1 ... M ) ) ) |
70 |
68 69
|
sylib |
|- ( ph -> ( ( M + 1 ) ... N ) = ( ( ( M + 1 ) ... N ) \ ( 1 ... M ) ) ) |
71 |
44 61 70
|
3eqtr4a |
|- ( ph -> ( ( 1 ... N ) \ ( 1 ... M ) ) = ( ( M + 1 ) ... N ) ) |
72 |
71
|
imaeq2d |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... N ) \ ( 1 ... M ) ) ) = ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) |
73 |
43 72
|
eqtr3d |
|- ( ph -> ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... N ) ) \ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) = ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) |
74 |
39 73
|
sseqtrd |
|- ( ph -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) \ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) C_ ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) |
75 |
74
|
sselda |
|- ( ( ph /\ y e. ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) \ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) -> y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) |
76 |
9 75
|
sylan2br |
|- ( ( ph /\ ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ -. y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) -> y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) |
77 |
|
fveq2 |
|- ( t = U -> ( 2nd ` t ) = ( 2nd ` U ) ) |
78 |
77
|
breq2d |
|- ( t = U -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` U ) ) ) |
79 |
78
|
ifbid |
|- ( t = U -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` U ) , y , ( y + 1 ) ) ) |
80 |
79
|
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 } ) ) ) ) |
81 |
|
2fveq3 |
|- ( t = U -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` U ) ) ) |
82 |
|
2fveq3 |
|- ( t = U -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` U ) ) ) |
83 |
82
|
imaeq1d |
|- ( t = U -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) ) |
84 |
83
|
xpeq1d |
|- ( t = U -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) ) |
85 |
82
|
imaeq1d |
|- ( t = U -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) ) |
86 |
85
|
xpeq1d |
|- ( t = U -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) |
87 |
84 86
|
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 } ) ) ) |
88 |
81 87
|
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 } ) ) ) ) |
89 |
88
|
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 } ) ) ) ) |
90 |
80 89
|
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 } ) ) ) ) |
91 |
90
|
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 } ) ) ) ) ) |
92 |
91
|
eqeq2d |
|- ( t = U -> ( F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) <-> F = ( 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 } ) ) ) ) ) ) |
93 |
92 2
|
elrab2 |
|- ( U e. S <-> ( U e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ F = ( 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 } ) ) ) ) ) ) |
94 |
93
|
simprbi |
|- ( U e. S -> F = ( 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 } ) ) ) ) ) |
95 |
6 94
|
syl |
|- ( ph -> F = ( 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 } ) ) ) ) ) |
96 |
|
breq1 |
|- ( y = M -> ( y < ( 2nd ` U ) <-> M < ( 2nd ` U ) ) ) |
97 |
|
id |
|- ( y = M -> y = M ) |
98 |
96 97
|
ifbieq1d |
|- ( y = M -> if ( y < ( 2nd ` U ) , y , ( y + 1 ) ) = if ( M < ( 2nd ` U ) , M , ( y + 1 ) ) ) |
99 |
1
|
nnred |
|- ( ph -> N e. RR ) |
100 |
|
peano2rem |
|- ( N e. RR -> ( N - 1 ) e. RR ) |
101 |
99 100
|
syl |
|- ( ph -> ( N - 1 ) e. RR ) |
102 |
|
elfzle2 |
|- ( M e. ( 0 ... ( N - 1 ) ) -> M <_ ( N - 1 ) ) |
103 |
8 102
|
syl |
|- ( ph -> M <_ ( N - 1 ) ) |
104 |
99
|
ltm1d |
|- ( ph -> ( N - 1 ) < N ) |
105 |
64 101 99 103 104
|
lelttrd |
|- ( ph -> M < N ) |
106 |
105 7
|
breqtrrd |
|- ( ph -> M < ( 2nd ` U ) ) |
107 |
106
|
iftrued |
|- ( ph -> if ( M < ( 2nd ` U ) , M , ( y + 1 ) ) = M ) |
108 |
98 107
|
sylan9eqr |
|- ( ( ph /\ y = M ) -> if ( y < ( 2nd ` U ) , y , ( y + 1 ) ) = M ) |
109 |
108
|
csbeq1d |
|- ( ( ph /\ y = M ) -> [_ 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 } ) ) ) = [_ M / j ]_ ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
110 |
|
oveq2 |
|- ( j = M -> ( 1 ... j ) = ( 1 ... M ) ) |
111 |
110
|
imaeq2d |
|- ( j = M -> ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) |
112 |
111
|
xpeq1d |
|- ( j = M -> ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) ) |
113 |
|
oveq1 |
|- ( j = M -> ( j + 1 ) = ( M + 1 ) ) |
114 |
113
|
oveq1d |
|- ( j = M -> ( ( j + 1 ) ... N ) = ( ( M + 1 ) ... N ) ) |
115 |
114
|
imaeq2d |
|- ( j = M -> ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) |
116 |
115
|
xpeq1d |
|- ( j = M -> ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) |
117 |
112 116
|
uneq12d |
|- ( j = M -> ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) = ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) |
118 |
117
|
oveq2d |
|- ( j = M -> ( ( 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 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
119 |
118
|
adantl |
|- ( ( ph /\ j = M ) -> ( ( 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 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
120 |
8 119
|
csbied |
|- ( ph -> [_ M / j ]_ ( ( 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 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
121 |
120
|
adantr |
|- ( ( ph /\ y = M ) -> [_ M / j ]_ ( ( 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 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
122 |
109 121
|
eqtrd |
|- ( ( ph /\ y = M ) -> [_ 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 } ) ) ) = ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
123 |
|
ovexd |
|- ( ph -> ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) e. _V ) |
124 |
95 122 8 123
|
fvmptd |
|- ( ph -> ( F ` M ) = ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
125 |
124
|
fveq1d |
|- ( ph -> ( ( F ` M ) ` y ) = ( ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ` y ) ) |
126 |
125
|
adantr |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( F ` M ) ` y ) = ( ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ` y ) ) |
127 |
|
imassrn |
|- ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) C_ ran ( 2nd ` ( 1st ` U ) ) |
128 |
|
f1of |
|- ( ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) --> ( 1 ... N ) ) |
129 |
|
frn |
|- ( ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) --> ( 1 ... N ) -> ran ( 2nd ` ( 1st ` U ) ) C_ ( 1 ... N ) ) |
130 |
34 128 129
|
3syl |
|- ( ph -> ran ( 2nd ` ( 1st ` U ) ) C_ ( 1 ... N ) ) |
131 |
127 130
|
sstrid |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) C_ ( 1 ... N ) ) |
132 |
131
|
sselda |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> y e. ( 1 ... N ) ) |
133 |
|
xp1st |
|- ( ( 1st ` U ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( 1st ` ( 1st ` U ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) ) |
134 |
|
elmapfn |
|- ( ( 1st ` ( 1st ` U ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` U ) ) Fn ( 1 ... N ) ) |
135 |
28 133 134
|
3syl |
|- ( ph -> ( 1st ` ( 1st ` U ) ) Fn ( 1 ... N ) ) |
136 |
135
|
adantr |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( 1st ` ( 1st ` U ) ) Fn ( 1 ... N ) ) |
137 |
|
1ex |
|- 1 e. _V |
138 |
|
fnconstg |
|- ( 1 e. _V -> ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) |
139 |
137 138
|
ax-mp |
|- ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) |
140 |
|
c0ex |
|- 0 e. _V |
141 |
|
fnconstg |
|- ( 0 e. _V -> ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) |
142 |
140 141
|
ax-mp |
|- ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) |
143 |
139 142
|
pm3.2i |
|- ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) /\ ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) |
144 |
|
imain |
|- ( Fun `' ( 2nd ` ( 1st ` U ) ) -> ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) ) |
145 |
34 41 144
|
3syl |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) ) |
146 |
67
|
imaeq2d |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) ) = ( ( 2nd ` ( 1st ` U ) ) " (/) ) ) |
147 |
|
ima0 |
|- ( ( 2nd ` ( 1st ` U ) ) " (/) ) = (/) |
148 |
146 147
|
eqtrdi |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) ) = (/) ) |
149 |
145 148
|
eqtr3d |
|- ( ph -> ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) = (/) ) |
150 |
|
fnun |
|- ( ( ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) /\ ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) /\ ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) = (/) ) -> ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) ) |
151 |
143 149 150
|
sylancr |
|- ( ph -> ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) ) |
152 |
|
imaundi |
|- ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... M ) u. ( ( M + 1 ) ... N ) ) ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) |
153 |
58
|
imaeq2d |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... N ) ) = ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... M ) u. ( ( M + 1 ) ... N ) ) ) ) |
154 |
153 37
|
eqtr3d |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... M ) u. ( ( M + 1 ) ... N ) ) ) = ( 1 ... N ) ) |
155 |
152 154
|
syl5eqr |
|- ( ph -> ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) = ( 1 ... N ) ) |
156 |
155
|
fneq2d |
|- ( ph -> ( ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) <-> ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( 1 ... N ) ) ) |
157 |
151 156
|
mpbid |
|- ( ph -> ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( 1 ... N ) ) |
158 |
157
|
adantr |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( 1 ... N ) ) |
159 |
|
ovexd |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( 1 ... N ) e. _V ) |
160 |
|
inidm |
|- ( ( 1 ... N ) i^i ( 1 ... N ) ) = ( 1 ... N ) |
161 |
|
eqidd |
|- ( ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) /\ y e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` U ) ) ` y ) = ( ( 1st ` ( 1st ` U ) ) ` y ) ) |
162 |
|
fvun2 |
|- ( ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) /\ ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) /\ ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) = (/) /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) ) -> ( ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = ( ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ` y ) ) |
163 |
139 142 162
|
mp3an12 |
|- ( ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) = (/) /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = ( ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ` y ) ) |
164 |
149 163
|
sylan |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = ( ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ` y ) ) |
165 |
140
|
fvconst2 |
|- ( y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) -> ( ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ` y ) = 0 ) |
166 |
165
|
adantl |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ` y ) = 0 ) |
167 |
164 166
|
eqtrd |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = 0 ) |
168 |
167
|
adantr |
|- ( ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) /\ y e. ( 1 ... N ) ) -> ( ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = 0 ) |
169 |
136 158 159 159 160 161 168
|
ofval |
|- ( ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) /\ y e. ( 1 ... N ) ) -> ( ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ` y ) = ( ( ( 1st ` ( 1st ` U ) ) ` y ) + 0 ) ) |
170 |
132 169
|
mpdan |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ` y ) = ( ( ( 1st ` ( 1st ` U ) ) ` y ) + 0 ) ) |
171 |
|
elmapi |
|- ( ( 1st ` ( 1st ` U ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` U ) ) : ( 1 ... N ) --> ( 0 ..^ K ) ) |
172 |
28 133 171
|
3syl |
|- ( ph -> ( 1st ` ( 1st ` U ) ) : ( 1 ... N ) --> ( 0 ..^ K ) ) |
173 |
172
|
ffvelrnda |
|- ( ( ph /\ y e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` U ) ) ` y ) e. ( 0 ..^ K ) ) |
174 |
|
elfzonn0 |
|- ( ( ( 1st ` ( 1st ` U ) ) ` y ) e. ( 0 ..^ K ) -> ( ( 1st ` ( 1st ` U ) ) ` y ) e. NN0 ) |
175 |
173 174
|
syl |
|- ( ( ph /\ y e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` U ) ) ` y ) e. NN0 ) |
176 |
175
|
nn0cnd |
|- ( ( ph /\ y e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` U ) ) ` y ) e. CC ) |
177 |
176
|
addid1d |
|- ( ( ph /\ y e. ( 1 ... N ) ) -> ( ( ( 1st ` ( 1st ` U ) ) ` y ) + 0 ) = ( ( 1st ` ( 1st ` U ) ) ` y ) ) |
178 |
132 177
|
syldan |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( ( 1st ` ( 1st ` U ) ) ` y ) + 0 ) = ( ( 1st ` ( 1st ` U ) ) ` y ) ) |
179 |
126 170 178
|
3eqtrd |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( F ` M ) ` y ) = ( ( 1st ` ( 1st ` U ) ) ` y ) ) |
180 |
76 179
|
syldan |
|- ( ( ph /\ ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ -. y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) -> ( ( F ` M ) ` y ) = ( ( 1st ` ( 1st ` U ) ) ` y ) ) |
181 |
|
fveq2 |
|- ( t = T -> ( 2nd ` t ) = ( 2nd ` T ) ) |
182 |
181
|
breq2d |
|- ( t = T -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` T ) ) ) |
183 |
182
|
ifbid |
|- ( t = T -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) ) |
184 |
183
|
csbeq1d |
|- ( t = T -> [_ if ( y < ( 2nd ` t ) , 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 ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
185 |
|
2fveq3 |
|- ( t = T -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` T ) ) ) |
186 |
|
2fveq3 |
|- ( t = T -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` T ) ) ) |
187 |
186
|
imaeq1d |
|- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) ) |
188 |
187
|
xpeq1d |
|- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) ) |
189 |
186
|
imaeq1d |
|- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) ) |
190 |
189
|
xpeq1d |
|- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) |
191 |
188 190
|
uneq12d |
|- ( t = T -> ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) = ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) |
192 |
185 191
|
oveq12d |
|- ( t = T -> ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
193 |
192
|
csbeq2dv |
|- ( t = T -> [_ if ( y < ( 2nd ` T ) , 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 ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
194 |
184 193
|
eqtrd |
|- ( t = T -> [_ if ( y < ( 2nd ` t ) , 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 ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
195 |
194
|
mpteq2dv |
|- ( t = T -> ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , 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 ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) |
196 |
195
|
eqeq2d |
|- ( t = T -> ( F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) <-> F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) ) |
197 |
196 2
|
elrab2 |
|- ( T e. S <-> ( T e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) ) |
198 |
197
|
simprbi |
|- ( T e. S -> F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) |
199 |
4 198
|
syl |
|- ( ph -> F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) |
200 |
|
breq1 |
|- ( y = M -> ( y < ( 2nd ` T ) <-> M < ( 2nd ` T ) ) ) |
201 |
200 97
|
ifbieq1d |
|- ( y = M -> if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) = if ( M < ( 2nd ` T ) , M , ( y + 1 ) ) ) |
202 |
105 5
|
breqtrrd |
|- ( ph -> M < ( 2nd ` T ) ) |
203 |
202
|
iftrued |
|- ( ph -> if ( M < ( 2nd ` T ) , M , ( y + 1 ) ) = M ) |
204 |
201 203
|
sylan9eqr |
|- ( ( ph /\ y = M ) -> if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) = M ) |
205 |
204
|
csbeq1d |
|- ( ( ph /\ y = M ) -> [_ if ( y < ( 2nd ` T ) , 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 / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
206 |
110
|
imaeq2d |
|- ( j = M -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) |
207 |
206
|
xpeq1d |
|- ( j = M -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) ) |
208 |
114
|
imaeq2d |
|- ( j = M -> ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) |
209 |
208
|
xpeq1d |
|- ( j = M -> ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) |
210 |
207 209
|
uneq12d |
|- ( j = M -> ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) = ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) |
211 |
210
|
oveq2d |
|- ( j = M -> ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
212 |
211
|
adantl |
|- ( ( ph /\ j = M ) -> ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
213 |
8 212
|
csbied |
|- ( ph -> [_ M / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
214 |
213
|
adantr |
|- ( ( ph /\ y = M ) -> [_ M / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
215 |
205 214
|
eqtrd |
|- ( ( ph /\ y = M ) -> [_ if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
216 |
|
ovexd |
|- ( ph -> ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) e. _V ) |
217 |
199 215 8 216
|
fvmptd |
|- ( ph -> ( F ` M ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
218 |
217
|
fveq1d |
|- ( ph -> ( ( F ` M ) ` y ) = ( ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ` y ) ) |
219 |
218
|
adantr |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( ( F ` M ) ` y ) = ( ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ` y ) ) |
220 |
24
|
sselda |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> y e. ( 1 ... N ) ) |
221 |
|
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 ) ) ) |
222 |
|
elmapfn |
|- ( ( 1st ` ( 1st ` T ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` T ) ) Fn ( 1 ... N ) ) |
223 |
14 221 222
|
3syl |
|- ( ph -> ( 1st ` ( 1st ` T ) ) Fn ( 1 ... N ) ) |
224 |
223
|
adantr |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( 1st ` ( 1st ` T ) ) Fn ( 1 ... N ) ) |
225 |
|
fnconstg |
|- ( 1 e. _V -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) |
226 |
137 225
|
ax-mp |
|- ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) |
227 |
|
fnconstg |
|- ( 0 e. _V -> ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) |
228 |
140 227
|
ax-mp |
|- ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) |
229 |
226 228
|
pm3.2i |
|- ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) |
230 |
|
dff1o3 |
|- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) /\ Fun `' ( 2nd ` ( 1st ` T ) ) ) ) |
231 |
230
|
simprbi |
|- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> Fun `' ( 2nd ` ( 1st ` T ) ) ) |
232 |
|
imain |
|- ( Fun `' ( 2nd ` ( 1st ` T ) ) -> ( ( 2nd ` ( 1st ` T ) ) " ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) ) |
233 |
20 231 232
|
3syl |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) ) |
234 |
67
|
imaeq2d |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) ) = ( ( 2nd ` ( 1st ` T ) ) " (/) ) ) |
235 |
|
ima0 |
|- ( ( 2nd ` ( 1st ` T ) ) " (/) ) = (/) |
236 |
234 235
|
eqtrdi |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) ) = (/) ) |
237 |
233 236
|
eqtr3d |
|- ( ph -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) = (/) ) |
238 |
|
fnun |
|- ( ( ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) /\ ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) = (/) ) -> ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) ) |
239 |
229 237 238
|
sylancr |
|- ( ph -> ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) ) |
240 |
|
imaundi |
|- ( ( 2nd ` ( 1st ` T ) ) " ( ( 1 ... M ) u. ( ( M + 1 ) ... N ) ) ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) |
241 |
58
|
imaeq2d |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( ( 1 ... M ) u. ( ( M + 1 ) ... N ) ) ) ) |
242 |
|
f1ofo |
|- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) ) |
243 |
|
foima |
|- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) = ( 1 ... N ) ) |
244 |
20 242 243
|
3syl |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) = ( 1 ... N ) ) |
245 |
241 244
|
eqtr3d |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( ( 1 ... M ) u. ( ( M + 1 ) ... N ) ) ) = ( 1 ... N ) ) |
246 |
240 245
|
syl5eqr |
|- ( ph -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) = ( 1 ... N ) ) |
247 |
246
|
fneq2d |
|- ( ph -> ( ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) <-> ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( 1 ... N ) ) ) |
248 |
239 247
|
mpbid |
|- ( ph -> ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( 1 ... N ) ) |
249 |
248
|
adantr |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( 1 ... N ) ) |
250 |
|
ovexd |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( 1 ... N ) e. _V ) |
251 |
|
eqidd |
|- ( ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) /\ y e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` y ) = ( ( 1st ` ( 1st ` T ) ) ` y ) ) |
252 |
|
fvun1 |
|- ( ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) /\ ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) = (/) /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) ) -> ( ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) ` y ) ) |
253 |
226 228 252
|
mp3an12 |
|- ( ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) = (/) /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) ` y ) ) |
254 |
237 253
|
sylan |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) ` y ) ) |
255 |
137
|
fvconst2 |
|- ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) -> ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) ` y ) = 1 ) |
256 |
255
|
adantl |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) ` y ) = 1 ) |
257 |
254 256
|
eqtrd |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = 1 ) |
258 |
257
|
adantr |
|- ( ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) /\ y e. ( 1 ... N ) ) -> ( ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = 1 ) |
259 |
224 249 250 250 160 251 258
|
ofval |
|- ( ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) /\ y e. ( 1 ... N ) ) -> ( ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ` y ) = ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) ) |
260 |
220 259
|
mpdan |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ` y ) = ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) ) |
261 |
219 260
|
eqtrd |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( ( F ` M ) ` y ) = ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) ) |
262 |
261
|
adantrr |
|- ( ( ph /\ ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ -. y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) -> ( ( F ` M ) ` y ) = ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) ) |
263 |
1
|
nngt0d |
|- ( ph -> 0 < N ) |
264 |
263 7
|
breqtrrd |
|- ( ph -> 0 < ( 2nd ` U ) ) |
265 |
1 2 6 264
|
poimirlem5 |
|- ( ph -> ( F ` 0 ) = ( 1st ` ( 1st ` U ) ) ) |
266 |
263 5
|
breqtrrd |
|- ( ph -> 0 < ( 2nd ` T ) ) |
267 |
1 2 4 266
|
poimirlem5 |
|- ( ph -> ( F ` 0 ) = ( 1st ` ( 1st ` T ) ) ) |
268 |
265 267
|
eqtr3d |
|- ( ph -> ( 1st ` ( 1st ` U ) ) = ( 1st ` ( 1st ` T ) ) ) |
269 |
268
|
fveq1d |
|- ( ph -> ( ( 1st ` ( 1st ` U ) ) ` y ) = ( ( 1st ` ( 1st ` T ) ) ` y ) ) |
270 |
269
|
adantr |
|- ( ( ph /\ ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ -. y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) -> ( ( 1st ` ( 1st ` U ) ) ` y ) = ( ( 1st ` ( 1st ` T ) ) ` y ) ) |
271 |
180 262 270
|
3eqtr3d |
|- ( ( ph /\ ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ -. y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) -> ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) = ( ( 1st ` ( 1st ` T ) ) ` y ) ) |
272 |
|
elmapi |
|- ( ( 1st ` ( 1st ` T ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0 ..^ K ) ) |
273 |
14 221 272
|
3syl |
|- ( ph -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0 ..^ K ) ) |
274 |
273
|
ffvelrnda |
|- ( ( ph /\ y e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` y ) e. ( 0 ..^ K ) ) |
275 |
|
elfzonn0 |
|- ( ( ( 1st ` ( 1st ` T ) ) ` y ) e. ( 0 ..^ K ) -> ( ( 1st ` ( 1st ` T ) ) ` y ) e. NN0 ) |
276 |
274 275
|
syl |
|- ( ( ph /\ y e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` y ) e. NN0 ) |
277 |
276
|
nn0red |
|- ( ( ph /\ y e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` y ) e. RR ) |
278 |
277
|
ltp1d |
|- ( ( ph /\ y e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` y ) < ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) ) |
279 |
277 278
|
gtned |
|- ( ( ph /\ y e. ( 1 ... N ) ) -> ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) =/= ( ( 1st ` ( 1st ` T ) ) ` y ) ) |
280 |
220 279
|
syldan |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) =/= ( ( 1st ` ( 1st ` T ) ) ` y ) ) |
281 |
280
|
neneqd |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> -. ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) = ( ( 1st ` ( 1st ` T ) ) ` y ) ) |
282 |
281
|
adantrr |
|- ( ( ph /\ ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ -. y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) -> -. ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) = ( ( 1st ` ( 1st ` T ) ) ` y ) ) |
283 |
271 282
|
pm2.65da |
|- ( ph -> -. ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ -. y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) |
284 |
|
iman |
|- ( ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) -> y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) <-> -. ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ -. y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) |
285 |
283 284
|
sylibr |
|- ( ph -> ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) -> y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) |
286 |
285
|
ssrdv |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) C_ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) |