Theorem ctiunctlemf 12809

Description: Lemma for ctiunct 12811. (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 5498	. . . . . . 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 12805	. . . . . 6 |- ( ( ph /\ n e. U ) -> ( 1st ` ( J ` n ) ) e. S )
20	4, 19	ffvelcdmd 5716	. . . . 5 |- ( ( ph /\ n e. U ) -> ( F ` ( 1st ` ( J ` n ) ) ) e. A )
21		fof 5498	. . . . . . . . . . 11 |- ( G : T -onto-> B -> G : T --> B )
22	13, 21	syl 14	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> G : T --> B )
23	22	ralrimiva 2579	. . . . . . . . 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 3081	. . . . . . . 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 5424	. . . . . . . 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 12806	. . . . . 6 |- ( ( ph /\ n e. U ) -> ( 2nd ` ( J ` n ) ) e. [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ T )
31	29, 30	ffvelcdmd 5716	. . . . 5 |- ( ( ph /\ n e. U ) -> ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) e. [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ B )
32		csbeq1 3096	. . . . . . 7 |- ( y = ( F ` ( 1st ` ( J ` n ) ) ) -> [_ y / x ]_ B = [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ B )
33	32	eleq2d 2275	. . . . . 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 2877	. . . . 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 3931	. . . 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 2348	. . . 4 |- F/_ y B
39		nfcsb1v 3126	. . . 4 |- F/_ x [_ y / x ]_ B
40		csbeq1a 3102	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
41	38, 39, 40	cbviun 3964	. . 3 |- U_ x e. A B = U_ y e. A [_ y / x ]_ B
42	37, 41	eleqtrrdi 2299	. 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 5734	1 |- ( ph -> H : U --> U_ x e. A B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 836   = wceq 1373   e. wcel 2176   A.wral 2484   E.wrex 2485   {crab 2488   [.wsbc 2998   [_csb 3093   C_ wss 3166   U_ciun 3927   |-> cmpt 4105   omcom 4638   X. cxp 4673   -->wf 5267   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271   1stc1st 6224   2ndc2nd 6225

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279

This theorem is referenced by:  ctiunctlemfo  12810