Theorem ctiunctlemuom 13271

Description: Lemma for ctiunct 13275. (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 ) }

Assertion

Ref	Expression
ctiunctlemuom	|- ( ph -> U C_ om )

Proof of Theorem ctiunctlemuom

Step	Hyp	Ref	Expression
1		ctiunct.u	. . 3 |- U = { z e. om | ( ( 1st ` ( J ` z ) ) e. S /\ ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T ) }
2		ssrab2 3327	. . 3 |- { z e. om | ( ( 1st ` ( J ` z ) ) e. S /\ ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T ) } C_ om
3	1, 2	eqsstri 3274	. 2 |- U C_ om
4	3	a1i 9	1 |- ( ph -> U C_ om )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 842   = wceq 1398   e. wcel 2205   A.wral 2522   {crab 2526   [_csb 3141   C_ wss 3214   omcom 4717   X. cxp 4752   -onto->wfo 5355   -1-1-onto->wf1o 5356   ` cfv 5357   1stc1st 6345   2ndc2nd 6346

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 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rab 2531  df-in 3220  df-ss 3227

This theorem is referenced by:  ctiunct  13275