Theorem ctiunctlemf 12680

Description: Lemma for ctiunct 12682. (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 ) ) ) )

Assertion

Ref	Expression
ctiunctlemf	|- ( ph -> H : U --> U_ x e. A B )

Distinct variable groups:   A, n, x   B, n   x, F, z   x, J, z   z, S   z, T   U, n   ph, n, x   x, z, n

Allowed substitution hints:   ph( z)   A( z)   B( x, z)   S( x, n)   T( x, n)   U( x, z)   F( n)   G( x, z, n)   H( x, z, n)   J( n)

Proof of Theorem ctiunctlemf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ctiunct.f	. . . . . . . 8 |- ( ph -> F : S -onto-> A )
2	1	adantr 276	. . . . . . 7 |- ( ( ph /\ n e. U ) -> F : S -onto-> A )
3		fof 5483	. . . . . . 7 |- ( F : S -onto-> A -> F : S --> A )
4	2, 3	syl 14	. . . . . 6 |- ( ( ph /\ n e. U ) -> F : S --> A )
5		ctiunct.som	. . . . . . . 8 |- ( ph -> S C_ om )
6	5	adantr 276	. . . . . . 7 |- ( ( ph /\ n e. U ) -> S C_ om )
7		ctiunct.sdc	. . . . . . . 8 |- ( ph -> A. n e. om DECID n e. S )
8	7	adantr 276	. . . . . . 7 |- ( ( ph /\ n e. U ) -> A. n e. om DECID n e. S )
9		ctiunct.tom	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> T C_ om )
10	9	adantlr 477	. . . . . . 7 |- ( ( ( ph /\ n e. U ) /\ x e. A ) -> T C_ om )
11		ctiunct.tdc	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> A. n e. om DECID n e. T )
12	11	adantlr 477	. . . . . . 7 |- ( ( ( ph /\ n e. U ) /\ x e. A ) -> A. n e. om DECID n e. T )
13		ctiunct.g	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> G : T -onto-> B )
14	13	adantlr 477	. . . . . . 7 |- ( ( ( ph /\ n e. U ) /\ x e. A ) -> G : T -onto-> B )
15		ctiunct.j	. . . . . . . 8 |- ( ph -> J : om -1-1-onto-> ( om X. om ) )
16	15	adantr 276	. . . . . . 7 |- ( ( ph /\ n e. U ) -> J : om -1-1-onto-> ( om X. om ) )
17		ctiunct.u	. . . . . . 7 |- U = { z e. om | ( ( 1st ` ( J ` z ) ) e. S /\ ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T ) }
18		simpr 110	. . . . . . 7 |- ( ( ph /\ n e. U ) -> n e. U )
19	6, 8, 2, 10, 12, 14, 16, 17, 18	ctiunctlemu1st 12676	. . . . . 6 |- ( ( ph /\ n e. U ) -> ( 1st ` ( J ` n ) ) e. S )
20	4, 19	ffvelcdmd 5701	. . . . 5 |- ( ( ph /\ n e. U ) -> ( F ` ( 1st ` ( J ` n ) ) ) e. A )
21		fof 5483	. . . . . . . . . . 11 |- ( G : T -onto-> B -> G : T --> B )
22	13, 21	syl 14	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> G : T --> B )
23	22	ralrimiva 2570	. . . . . . . . 9 |- ( ph -> A. x e. A G : T --> B )
24	23	adantr 276	. . . . . . . 8 |- ( ( ph /\ n e. U ) -> A. x e. A G : T --> B )
25		rspsbc 3072	. . . . . . . 8 |- ( ( F ` ( 1st ` ( J ` n ) ) ) e. A -> ( A. x e. A G : T --> B -> [. ( F ` ( 1st ` ( J ` n ) ) ) / x ]. G : T --> B ) )
26	20, 24, 25	sylc 62	. . . . . . 7 |- ( ( ph /\ n e. U ) -> [. ( F ` ( 1st ` ( J ` n ) ) ) / x ]. G : T --> B )
27		sbcfg 5409	. . . . . . . 8 |- ( ( F ` ( 1st ` ( J ` n ) ) ) e. A -> ( [. ( F ` ( 1st ` ( J ` n ) ) ) / x ]. G : T --> B <-> [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G : [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ T --> [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ B ) )
28	20, 27	syl 14	. . . . . . 7 |- ( ( ph /\ n e. U ) -> ( [. ( F ` ( 1st ` ( J ` n ) ) ) / x ]. G : T --> B <-> [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G : [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ T --> [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ B ) )
29	26, 28	mpbid 147	. . . . . 6 |- ( ( ph /\ n e. U ) -> [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G : [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ T --> [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ B )
30	6, 8, 2, 10, 12, 14, 16, 17, 18	ctiunctlemu2nd 12677	. . . . . 6 |- ( ( ph /\ n e. U ) -> ( 2nd ` ( J ` n ) ) e. [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ T )
31	29, 30	ffvelcdmd 5701	. . . . 5 |- ( ( ph /\ n e. U ) -> ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) e. [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ B )
32		csbeq1 3087	. . . . . . 7 |- ( y = ( F ` ( 1st ` ( J ` n ) ) ) -> [_ y / x ]_ B = [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ B )
33	32	eleq2d 2266	. . . . . 6 |- ( y = ( F ` ( 1st ` ( J ` n ) ) ) -> ( ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) e. [_ y / x ]_ B <-> ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) e. [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ B ) )
34	33	rspcev 2868	. . . . 5 |- ( ( ( F ` ( 1st ` ( J ` n ) ) ) e. A /\ ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) e. [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ B ) -> E. y e. A ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) e. [_ y / x ]_ B )
35	20, 31, 34	syl2anc 411	. . . 4 |- ( ( ph /\ n e. U ) -> E. y e. A ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) e. [_ y / x ]_ B )
36		eliun 3921	. . . 4 |- ( ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) e. U_ y e. A [_ y / x ]_ B <-> E. y e. A ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) e. [_ y / x ]_ B )
37	35, 36	sylibr 134	. . 3 |- ( ( ph /\ n e. U ) -> ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) e. U_ y e. A [_ y / x ]_ B )
38		nfcv 2339	. . . 4 |- F/_ y B
39		nfcsb1v 3117	. . . 4 |- F/_ x [_ y / x ]_ B
40		csbeq1a 3093	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
41	38, 39, 40	cbviun 3954	. . 3 |- U_ x e. A B = U_ y e. A [_ y / x ]_ B
42	37, 41	eleqtrrdi 2290	. 2 |- ( ( ph /\ n e. U ) -> ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) e. U_ x e. A B )
43		ctiunct.h	. 2 |- H = ( n e. U |-> ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) )
44	42, 43	fmptd 5719	1 |- ( ph -> H : U --> U_ x e. A B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 835   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   {crab 2479   [.wsbc 2989   [_csb 3084   C_ wss 3157   U_ciun 3917   |-> cmpt 4095   omcom 4627   X. cxp 4662   -->wf 5255   -onto->wfo 5257   -1-1-onto->wf1o 5258   ` cfv 5259   1stc1st 6205   2ndc2nd 6206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fo 5265  df-fv 5267

This theorem is referenced by:  ctiunctlemfo  12681