Theorem ctiunctlemu1st 12651

Description: Lemma for ctiunct 12657. (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 5563	. . . . . . 7 |- ( z = N -> ( 1st ` ( J ` z ) ) = ( 1st ` ( J ` N ) ) )
3	2	eleq1d 2265	. . . . . 6 |- ( z = N -> ( ( 1st ` ( J ` z ) ) e. S <-> ( 1st ` ( J ` N ) ) e. S ) )
4		2fveq3 5563	. . . . . . 7 |- ( z = N -> ( 2nd ` ( J ` z ) ) = ( 2nd ` ( J ` N ) ) )
5	2	fveq2d 5562	. . . . . . . 8 |- ( z = N -> ( F ` ( 1st ` ( J ` z ) ) ) = ( F ` ( 1st ` ( J ` N ) ) ) )
6	5	csbeq1d 3091	. . . . . . 7 |- ( z = N -> [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T = [_ ( F ` ( 1st ` ( J ` N ) ) ) / x ]_ T )
7	4, 6	eleq12d 2267	. . . . . 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 2923	. . . 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 2167   A.wral 2475   {crab 2479   [_csb 3084   C_ wss 3157   omcom 4626   X. cxp 4661   -onto->wfo 5256   -1-1-onto->wf1o 5257   ` cfv 5258   1stc1st 6196   2ndc2nd 6197

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-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266

This theorem is referenced by:  ctiunctlemf  12655