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Theorem ctiunctlemuom 12437
Description: Lemma for ctiunct 12441. (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
StepHypRef 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 3241 . . 3  |-  { z  e.  om  |  ( ( 1st `  ( J `  z )
)  e.  S  /\  ( 2nd `  ( J `
 z ) )  e.  [_ ( F `
 ( 1st `  ( J `  z )
) )  /  x ]_ T ) }  C_  om
31, 2eqsstri 3188 . 2  |-  U  C_  om
43a1i 9 1  |-  ( ph  ->  U  C_  om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104  DECID wdc 834    = wceq 1353    e. wcel 2148   A.wral 2455   {crab 2459   [_csb 3058    C_ wss 3130   omcom 4590    X. cxp 4625   -onto->wfo 5215   -1-1-onto->wf1o 5216   ` cfv 5217   1stc1st 6139   2ndc2nd 6140
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-in 3136  df-ss 3143
This theorem is referenced by:  ctiunct  12441
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