Theorem ctiunctlemu1st 13118

Description: Lemma for ctiunct 13124. (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 5653	. . . . . . 7 |- ( z = N -> ( 1st ` ( J ` z ) ) = ( 1st ` ( J ` N ) ) )
3	2	eleq1d 2300	. . . . . 6 |- ( z = N -> ( ( 1st ` ( J ` z ) ) e. S <-> ( 1st ` ( J ` N ) ) e. S ) )
4		2fveq3 5653	. . . . . . 7 |- ( z = N -> ( 2nd ` ( J ` z ) ) = ( 2nd ` ( J ` N ) ) )
5	2	fveq2d 5652	. . . . . . . 8 |- ( z = N -> ( F ` ( 1st ` ( J ` z ) ) ) = ( F ` ( 1st ` ( J ` N ) ) ) )
6	5	csbeq1d 3135	. . . . . . 7 |- ( z = N -> [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T = [_ ( F ` ( 1st ` ( J ` N ) ) ) / x ]_ T )
7	4, 6	eleq12d 2302	. . . . . 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 2966	. . . 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 842   = wceq 1398   e. wcel 2202   A.wral 2511   {crab 2515   [_csb 3128   C_ wss 3201   omcom 4694   X. cxp 4729   -onto->wfo 5331   -1-1-onto->wf1o 5332   ` cfv 5333   1stc1st 6310   2ndc2nd 6311

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341

This theorem is referenced by:  ctiunctlemf  13122