Theorem ctiunctlemu1st 12920

Description: Lemma for ctiunct 12926. (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 5604	. . . . . . 7 |- ( z = N -> ( 1st ` ( J ` z ) ) = ( 1st ` ( J ` N ) ) )
3	2	eleq1d 2276	. . . . . 6 |- ( z = N -> ( ( 1st ` ( J ` z ) ) e. S <-> ( 1st ` ( J ` N ) ) e. S ) )
4		2fveq3 5604	. . . . . . 7 |- ( z = N -> ( 2nd ` ( J ` z ) ) = ( 2nd ` ( J ` N ) ) )
5	2	fveq2d 5603	. . . . . . . 8 |- ( z = N -> ( F ` ( 1st ` ( J ` z ) ) ) = ( F ` ( 1st ` ( J ` N ) ) ) )
6	5	csbeq1d 3108	. . . . . . 7 |- ( z = N -> [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T = [_ ( F ` ( 1st ` ( J ` N ) ) ) / x ]_ T )
7	4, 6	eleq12d 2278	. . . . . 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 2939	. . . 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 836   = wceq 1373   e. wcel 2178   A.wral 2486   {crab 2490   [_csb 3101   C_ wss 3174   omcom 4656   X. cxp 4691   -onto->wfo 5288   -1-1-onto->wf1o 5289   ` cfv 5290   1stc1st 6247   2ndc2nd 6248

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298

This theorem is referenced by:  ctiunctlemf  12924