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Theorem ctiunctlemfo 12372
Description: Lemma for ctiunct 12373. (Contributed by Jim Kingdon, 28-Oct-2023.)
Hypotheses
Ref Expression
ctiunct.som |- ( ph -> S C_ om )
ctiunct.sdc |- ( ph -> A. n e. om DECID n e. S )
ctiunct.f |- ( ph -> F : S -onto-> A )
ctiunct.tom |- ( ( ph /\ x e. A ) -> T C_ om )
ctiunct.tdc |- ( ( ph /\ x e. A ) -> A. n e. om DECID n e. T )
ctiunct.g |- ( ( ph /\ x e. A ) -> G : T -onto-> B )
ctiunct.j |- ( ph -> J : om -1-1-onto-> ( om X. om ) )
ctiunct.u |- U = { z e. om | ( ( 1st ` ( J ` z ) ) e. S /\ ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T ) }
ctiunct.h |- H = ( n e. U |-> ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) )
ctiunct.nfh |- F/_ x H
ctiunct.nfu |- F/_ x U
Assertion
Ref Expression
ctiunctlemfo |- ( ph -> H : U -onto-> U_ x e. A B )
Distinct variable groups:   A, n, x   B, n   n, F, x, z   n, G   n, J, x, z   z, S   z, T   U, n   ph, n, x   z, n
Allowed substitution hints:   ph( z)   A( z)   B( x, z)   S( x, n)   T( x, n)   U( x, z)   G( x, z)   H( x, z, n)
Proof of Theorem ctiunctlemfo
Dummy variables r w u v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ctiunct.som . . 3 |- ( ph -> S C_ om )
2 ctiunct.sdc . . 3 |- ( ph -> A. n e. om DECID n e. S )
3 ctiunct.f . . 3 |- ( ph -> F : S -onto-> A )
4 ctiunct.tom . . 3 |- ( ( ph /\ x e. A ) -> T C_ om )
5 ctiunct.tdc . . 3 |- ( ( ph /\ x e. A ) -> A. n e. om DECID n e. T )
6 ctiunct.g . . 3 |- ( ( ph /\ x e. A ) -> G : T -onto-> B )
7 ctiunct.j . . 3 |- ( ph -> J : om -1-1-onto-> ( om X. om ) )
8 ctiunct.u . . 3 |- U = { z e. om | ( ( 1st ` ( J ` z ) ) e. S /\ ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T ) }
9 ctiunct.h . . 3 |- H = ( n e. U |-> ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) )
101, 2, 3, 4, 5, 6, 7, 8, 9ctiunctlemf 12371 . 2 |- ( ph -> H : U --> U_ x e. A B )
11 nfv 1516 . . . . 5 |- F/ x ph
12 nfiu1 3896 . . . . . 6 |- F/_ x U_ x e. A B
1312nfcri 2302 . . . . 5 |- F/ x u e. U_ x e. A B
1411, 13nfan 1553 . . . 4 |- F/ x ( ph /\ u e. U_ x e. A B )
15 ctiunct.nfu . . . . 5 |- F/_ x U
16 ctiunct.nfh . . . . . . 7 |- F/_ x H
17 nfcv 2308 . . . . . . 7 |- F/_ x v
1816, 17nffv 5496 . . . . . 6 |- F/_ x ( H ` v )
1918nfeq2 2320 . . . . 5 |- F/ x u = ( H ` v )
2015, 19nfrexxy 2505 . . . 4 |- F/ x E. v e. U u = ( H ` v )
21 simplll 523 . . . . . . 7 |- ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) -> ph )
22 simplr 520 . . . . . . 7 |- ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) -> x e. A )
2321, 22, 6syl2anc 409 . . . . . 6 |- ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) -> G : T -onto-> B )
24 foelrn 5721 . . . . . 6 |- ( ( G : T -onto-> B /\ u e. B ) -> E. w e. T u = ( G ` w ) )
2523, 24sylancom 417 . . . . 5 |- ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) -> E. w e. T u = ( G ` w ) )
263ad4antr 486 . . . . . . 7 |- ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) -> F : S -onto-> A )
27 simpllr 524 . . . . . . 7 |- ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) -> x e. A )
28 foelrn 5721 . . . . . . 7 |- ( ( F : S -onto-> A /\ x e. A ) -> E. r e. S x = ( F ` r ) )
2926, 27, 28syl2anc 409 . . . . . 6 |- ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) -> E. r e. S x = ( F ` r ) )
30 f1ocnv 5445 . . . . . . . . . . . 12 |- ( J : om -1-1-onto-> ( om X. om ) -> `' J : ( om X. om ) -1-1-onto-> om )
317, 30syl 14 . . . . . . . . . . 11 |- ( ph -> `' J : ( om X. om ) -1-1-onto-> om )
32 f1of 5432 . . . . . . . . . . 11 |- ( `' J : ( om X. om ) -1-1-onto-> om -> `' J : ( om X. om ) --> om )
3331, 32syl 14 . . . . . . . . . 10 |- ( ph -> `' J : ( om X. om ) --> om )
3433ad5antr 488 . . . . . . . . 9 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> `' J : ( om X. om ) --> om )
351ad5antr 488 . . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> S C_ om )
36 simprl 521 . . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> r e. S )
3735, 36sseldd 3143 . . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> r e. om )
38 simp-5l 533 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> ph )
3927adantr 274 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> x e. A )
4038, 39, 4syl2anc 409 . . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> T C_ om )
41 simplrl 525 . . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> w e. T )
4240, 41sseldd 3143 . . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> w e. om )
4337, 42opelxpd 4637 . . . . . . . . 9 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> <. r , w >. e. ( om X. om ) )
4434, 43ffvelrnd 5621 . . . . . . . 8 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> ( `' J ` <. r , w >. ) e. om )
4538, 7syl 14 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> J : om -1-1-onto-> ( om X. om ) )
46 f1ocnvfv2 5746 . . . . . . . . . . . . 13 |- ( ( J : om -1-1-onto-> ( om X. om ) /\ <. r , w >. e. ( om X. om ) ) -> ( J ` ( `' J ` <. r , w >. ) ) = <. r , w >. )
4745, 43, 46syl2anc 409 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> ( J ` ( `' J ` <. r , w >. ) ) = <. r , w >. )
4847fveq2d 5490 . . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) = ( 1st ` <. r , w >. ) )
49 vex 2729 . . . . . . . . . . . 12 |- r e. _V
50 vex 2729 . . . . . . . . . . . 12 |- w e. _V
5149, 50op1st 6114 . . . . . . . . . . 11 |- ( 1st ` <. r , w >. ) = r
5248, 51eqtrdi 2215 . . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) = r )
5352, 36eqeltrd 2243 . . . . . . . . 9 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) e. S )
5447fveq2d 5490 . . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> ( 2nd ` ( J ` ( `' J ` <. r , w >. ) ) ) = ( 2nd ` <. r , w >. ) )
5549, 50op2nd 6115 . . . . . . . . . . 11 |- ( 2nd ` <. r , w >. ) = w
5654, 55eqtrdi 2215 . . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> ( 2nd ` ( J ` ( `' J ` <. r , w >. ) ) ) = w )
5752fveq2d 5490 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) = ( F ` r ) )
58 simprr 522 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> x = ( F ` r ) )
5957, 58eqtr4d 2201 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) = x )
6059csbeq1d 3052 . . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> [_ ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) / x ]_ T = [_ x / x ]_ T )
61 csbid 3053 . . . . . . . . . . 11 |- [_ x / x ]_ T = T
6260, 61eqtrdi 2215 . . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> [_ ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) / x ]_ T = T )
6341, 56, 623eltr4d 2250 . . . . . . . . 9 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> ( 2nd ` ( J ` ( `' J ` <. r , w >. ) ) ) e. [_ ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) / x ]_ T )
6453, 63jca 304 . . . . . . . 8 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> ( ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) e. S /\ ( 2nd ` ( J ` ( `' J ` <. r , w >. ) ) ) e. [_ ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) / x ]_ T ) )
65 2fveq3 5491 . . . . . . . . . . 11 |- ( z = ( `' J ` <. r , w >. ) -> ( 1st ` ( J ` z ) ) = ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) )
6665eleq1d 2235 . . . . . . . . . 10 |- ( z = ( `' J ` <. r , w >. ) -> ( ( 1st ` ( J ` z ) ) e. S <-> ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) e. S ) )
67 2fveq3 5491 . . . . . . . . . . 11 |- ( z = ( `' J ` <. r , w >. ) -> ( 2nd ` ( J ` z ) ) = ( 2nd ` ( J ` ( `' J ` <. r , w >. ) ) ) )
6865fveq2d 5490 . . . . . . . . . . . 12 |- ( z = ( `' J ` <. r , w >. ) -> ( F ` ( 1st ` ( J ` z ) ) ) = ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) )
6968csbeq1d 3052 . . . . . . . . . . 11 |- ( z = ( `' J ` <. r , w >. ) -> [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T = [_ ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) / x ]_ T )
7067, 69eleq12d 2237 . . . . . . . . . 10 |- ( z = ( `' J ` <. r , w >. ) -> ( ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T <-> ( 2nd ` ( J ` ( `' J ` <. r , w >. ) ) ) e. [_ ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) / x ]_ T ) )
7166, 70anbi12d 465 . . . . . . . . 9 |- ( z = ( `' J ` <. r , w >. ) -> ( ( ( 1st ` ( J ` z ) ) e. S /\ ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T ) <-> ( ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) e. S /\ ( 2nd ` ( J ` ( `' J ` <. r , w >. ) ) ) e. [_ ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) / x ]_ T ) ) )
7271, 8elrab2 2885 . . . . . . . 8 |- ( ( `' J ` <. r , w >. ) e. U <-> ( ( `' J ` <. r , w >. ) e. om /\ ( ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) e. S /\ ( 2nd ` ( J ` ( `' J ` <. r , w >. ) ) ) e. [_ ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) / x ]_ T ) ) )
7344, 64, 72sylanbrc 414 . . . . . . 7 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> ( `' J ` <. r , w >. ) e. U )
7459csbeq1d 3052 . . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> [_ ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) / x ]_ G = [_ x / x ]_ G )
75 csbid 3053 . . . . . . . . . 10 |- [_ x / x ]_ G = G
7674, 75eqtrdi 2215 . . . . . . . . 9 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> [_ ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) / x ]_ G = G )
7776, 56fveq12d 5493 . . . . . . . 8 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> ( [_ ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) / x ]_ G ` ( 2nd ` ( J ` ( `' J ` <. r , w >. ) ) ) ) = ( G ` w ) )
78 2fveq3 5491 . . . . . . . . . . . 12 |- ( n = ( `' J ` <. r , w >. ) -> ( 1st ` ( J ` n ) ) = ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) )
7978fveq2d 5490 . . . . . . . . . . 11 |- ( n = ( `' J ` <. r , w >. ) -> ( F ` ( 1st ` ( J ` n ) ) ) = ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) )
8079csbeq1d 3052 . . . . . . . . . 10 |- ( n = ( `' J ` <. r , w >. ) -> [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G = [_ ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) / x ]_ G )
81 2fveq3 5491 . . . . . . . . . 10 |- ( n = ( `' J ` <. r , w >. ) -> ( 2nd ` ( J ` n ) ) = ( 2nd ` ( J ` ( `' J ` <. r , w >. ) ) ) )
8280, 81fveq12d 5493 . . . . . . . . 9 |- ( n = ( `' J ` <. r , w >. ) -> ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) = ( [_ ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) / x ]_ G ` ( 2nd ` ( J ` ( `' J ` <. r , w >. ) ) ) ) )
83 simplrr 526 . . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> u = ( G ` w ) )
8483, 77eqtr4d 2201 . . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> u = ( [_ ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) / x ]_ G ` ( 2nd ` ( J ` ( `' J ` <. r , w >. ) ) ) ) )
85 simpllr 524 . . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> u e. B )
8684, 85eqeltrrd 2244 . . . . . . . . 9 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> ( [_ ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) / x ]_ G ` ( 2nd ` ( J ` ( `' J ` <. r , w >. ) ) ) ) e. B )
879, 82, 73, 86fvmptd3 5579 . . . . . . . 8 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> ( H ` ( `' J ` <. r , w >. ) ) = ( [_ ( F ` ( 1st ` ( J ` ( `' J ` <. r , w >. ) ) ) ) / x ]_ G ` ( 2nd ` ( J ` ( `' J ` <. r , w >. ) ) ) ) )
8877, 87, 833eqtr4rd 2209 . . . . . . 7 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> u = ( H ` ( `' J ` <. r , w >. ) ) )
89 fveq2 5486 . . . . . . . 8 |- ( v = ( `' J ` <. r , w >. ) -> ( H ` v ) = ( H ` ( `' J ` <. r , w >. ) ) )
9089rspceeqv 2848 . . . . . . 7 |- ( ( ( `' J ` <. r , w >. ) e. U /\ u = ( H ` ( `' J ` <. r , w >. ) ) ) -> E. v e. U u = ( H ` v ) )
9173, 88, 90syl2anc 409 . . . . . 6 |- ( ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) /\ ( r e. S /\ x = ( F ` r ) ) ) -> E. v e. U u = ( H ` v ) )
9229, 91rexlimddv 2588 . . . . 5 |- ( ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) /\ ( w e. T /\ u = ( G ` w ) ) ) -> E. v e. U u = ( H ` v ) )
9325, 92rexlimddv 2588 . . . 4 |- ( ( ( ( ph /\ u e. U_ x e. A B ) /\ x e. A ) /\ u e. B ) -> E. v e. U u = ( H ` v ) )
94 eliun 3870 . . . . . 6 |- ( u e. U_ x e. A B <-> E. x e. A u e. B )
9594biimpi 119 . . . . 5 |- ( u e. U_ x e. A B -> E. x e. A u e. B )
9695adantl 275 . . . 4 |- ( ( ph /\ u e. U_ x e. A B ) -> E. x e. A u e. B )
9714, 20, 93, 96r19.29af2 2606 . . 3 |- ( ( ph /\ u e. U_ x e. A B ) -> E. v e. U u = ( H ` v ) )
9897ralrimiva 2539 . 2 |- ( ph -> A. u e. U_ x e. A B E. v e. U u = ( H ` v ) )
99 dffo3 5632 . 2 |- ( H : U -onto-> U_ x e. A B <-> ( H : U --> U_ x e. A B /\ A. u e. U_ x e. A B E. v e. U u = ( H ` v ) ) )
10010, 98, 99sylanbrc 414 1 |- ( ph -> H : U -onto-> U_ x e. A B )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103  DECID wdc 824   = wceq 1343   e. wcel 2136   F/_wnfc 2295   A.wral 2444   E.wrex 2445   {crab 2448   [_csb 3045   C_ wss 3116   <.cop 3579   U_ciun 3866   |-> cmpt 4043   omcom 4567   X. cxp 4602   `'ccnv 4603   -->wf 5184   -onto->wfo 5186   -1-1-onto->wf1o 5187   ` cfv 5188   1stc1st 6106   2ndc2nd 6107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109
This theorem is referenced by:  ctiunct  12373
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