Theorem ctiunctlemu1st 12591

Description: Lemma for ctiunct 12597. (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 ) }
ctiunctlem.n	|- ( ph -> N e. U )

Assertion

Ref	Expression
ctiunctlemu1st	|- ( ph -> ( 1st ` ( J ` N ) ) e. S )

Distinct variable groups:   z, F   z, J   z, N   z, S   z, T   x, z

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

Proof of Theorem ctiunctlemu1st

Step	Hyp	Ref	Expression
1		ctiunctlem.n	. . . 4 |- ( ph -> N e. U )
2		2fveq3 5559	. . . . . . 7 |- ( z = N -> ( 1st ` ( J ` z ) ) = ( 1st ` ( J ` N ) ) )
3	2	eleq1d 2262	. . . . . 6 |- ( z = N -> ( ( 1st ` ( J ` z ) ) e. S <-> ( 1st ` ( J ` N ) ) e. S ) )
4		2fveq3 5559	. . . . . . 7 |- ( z = N -> ( 2nd ` ( J ` z ) ) = ( 2nd ` ( J ` N ) ) )
5	2	fveq2d 5558	. . . . . . . 8 |- ( z = N -> ( F ` ( 1st ` ( J ` z ) ) ) = ( F ` ( 1st ` ( J ` N ) ) ) )
6	5	csbeq1d 3087	. . . . . . 7 |- ( z = N -> [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T = [_ ( F ` ( 1st ` ( J ` N ) ) ) / x ]_ T )
7	4, 6	eleq12d 2264	. . . . . 6 |- ( z = N -> ( ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T <-> ( 2nd ` ( J ` N ) ) e. [_ ( F ` ( 1st ` ( J ` N ) ) ) / x ]_ T ) )
8	3, 7	anbi12d 473	. . . . 5 |- ( z = N -> ( ( ( 1st ` ( J ` z ) ) e. S /\ ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T ) <-> ( ( 1st ` ( J ` N ) ) e. S /\ ( 2nd ` ( J ` N ) ) e. [_ ( F ` ( 1st ` ( J ` N ) ) ) / x ]_ T ) ) )
9		ctiunct.u	. . . . 5 |- U = { z e. om | ( ( 1st ` ( J ` z ) ) e. S /\ ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T ) }
10	8, 9	elrab2 2919	. . . 4 |- ( N e. U <-> ( N e. om /\ ( ( 1st ` ( J ` N ) ) e. S /\ ( 2nd ` ( J ` N ) ) e. [_ ( F ` ( 1st ` ( J ` N ) ) ) / x ]_ T ) ) )
11	1, 10	sylib 122	. . 3 |- ( ph -> ( N e. om /\ ( ( 1st ` ( J ` N ) ) e. S /\ ( 2nd ` ( J ` N ) ) e. [_ ( F ` ( 1st ` ( J ` N ) ) ) / x ]_ T ) ) )
12	11	simprd 114	. 2 |- ( ph -> ( ( 1st ` ( J ` N ) ) e. S /\ ( 2nd ` ( J ` N ) ) e. [_ ( F ` ( 1st ` ( J ` N ) ) ) / x ]_ T ) )
13	12	simpld 112	1 |- ( ph -> ( 1st ` ( J ` N ) ) e. S )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 835   = wceq 1364   e. wcel 2164   A.wral 2472   {crab 2476   [_csb 3080   C_ wss 3153   omcom 4622   X. cxp 4657   -onto->wfo 5252   -1-1-onto->wf1o 5253   ` cfv 5254   1stc1st 6191   2ndc2nd 6192

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262

This theorem is referenced by:  ctiunctlemf  12595