Theorem ctiunctlemf 11987

Description: Lemma for ctiunct 11989. (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 274	. . . . . . 7 |- ( ( ph /\ n e. U ) -> F : S -onto-> A )
3		fof 5353	. . . . . . 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 274	. . . . . . 7 |- ( ( ph /\ n e. U ) -> S C_ om )
7		ctiunct.sdc	. . . . . . . 8 |- ( ph -> A. n e. om DECID n e. S )
8	7	adantr 274	. . . . . . 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 469	. . . . . . 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 469	. . . . . . 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 469	. . . . . . 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 274	. . . . . . 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 109	. . . . . . 7 |- ( ( ph /\ n e. U ) -> n e. U )
19	6, 8, 2, 10, 12, 14, 16, 17, 18	ctiunctlemu1st 11983	. . . . . 6 |- ( ( ph /\ n e. U ) -> ( 1st ` ( J ` n ) ) e. S )
20	4, 19	ffvelrnd 5564	. . . . 5 |- ( ( ph /\ n e. U ) -> ( F ` ( 1st ` ( J ` n ) ) ) e. A )
21		fof 5353	. . . . . . . . . . 11 |- ( G : T -onto-> B -> G : T --> B )
22	13, 21	syl 14	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> G : T --> B )
23	22	ralrimiva 2508	. . . . . . . . 9 |- ( ph -> A. x e. A G : T --> B )
24	23	adantr 274	. . . . . . . 8 |- ( ( ph /\ n e. U ) -> A. x e. A G : T --> B )
25		rspsbc 2995	. . . . . . . 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 5279	. . . . . . . 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 146	. . . . . 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 11984	. . . . . 6 |- ( ( ph /\ n e. U ) -> ( 2nd ` ( J ` n ) ) e. [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ T )
31	29, 30	ffvelrnd 5564	. . . . 5 |- ( ( ph /\ n e. U ) -> ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) e. [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ B )
32		csbeq1 3010	. . . . . . 7 |- ( y = ( F ` ( 1st ` ( J ` n ) ) ) -> [_ y / x ]_ B = [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ B )
33	32	eleq2d 2210	. . . . . 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 2793	. . . . 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 409	. . . 4 |- ( ( ph /\ n e. U ) -> E. y e. A ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) e. [_ y / x ]_ B )
36		eliun 3825	. . . 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 133	. . 3 |- ( ( ph /\ n e. U ) -> ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) e. U_ y e. A [_ y / x ]_ B )
38		nfcv 2282	. . . 4 |- F/_ y B
39		nfcsb1v 3040	. . . 4 |- F/_ x [_ y / x ]_ B
40		csbeq1a 3016	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
41	38, 39, 40	cbviun 3858	. . 3 |- U_ x e. A B = U_ y e. A [_ y / x ]_ B
42	37, 41	eleqtrrdi 2234	. 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 5582	1 |- ( ph -> H : U --> U_ x e. A B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104  DECID wdc 820   = wceq 1332   e. wcel 1481   A.wral 2417   E.wrex 2418   {crab 2421   [.wsbc 2913   [_csb 3007   C_ wss 3076   U_ciun 3821   |-> cmpt 3997   omcom 4512   X. cxp 4545   -->wf 5127   -onto->wfo 5129   -1-1-onto->wf1o 5130   ` cfv 5131   1stc1st 6044   2ndc2nd 6045

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fo 5137  df-fv 5139

This theorem is referenced by:  ctiunctlemfo  11988