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Theorem cdlemk40t 31777
Description: TODO: fix comment. (Contributed by NM, 31-Jul-2013.)
Hypotheses
Ref Expression
cdlemk40.x  |-  X  =  ( iota_ z  e.  T ph )
cdlemk40.u  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
Assertion
Ref Expression
cdlemk40t  |-  ( ( F  =  N  /\  G  e.  T )  ->  ( U `  G
)  =  G )
Distinct variable groups:    g, F    g, N    T, g
Allowed substitution hints:    ph( z, g)    T( z)    U( z, g)    F( z)    G( z, g)    N( z)    X( z, g)

Proof of Theorem cdlemk40t
StepHypRef Expression
1 cdlemk40.x . . 3  |-  X  =  ( iota_ z  e.  T ph )
2 cdlemk40.u . . 3  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
31, 2cdlemk40 31776 . 2  |-  ( G  e.  T  ->  ( U `  G )  =  if ( F  =  N ,  G ,  [_ G  /  g ]_ X ) )
4 iftrue 3747 . 2  |-  ( F  =  N  ->  if ( F  =  N ,  G ,  [_ G  /  g ]_ X
)  =  G )
53, 4sylan9eqr 2492 1  |-  ( ( F  =  N  /\  G  e.  T )  ->  ( U `  G
)  =  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   [_csb 3253   ifcif 3741    e. cmpt 4268   ` cfv 5456   iota_crio 6544
This theorem is referenced by:  cdlemk35u  31823  cdlemk55u  31825  cdlemk39u  31827  cdlemk19u  31829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-riota 6551
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