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Theorem cdlemk40f 31654
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
cdlemk40f  |-  ( ( F  =/=  N  /\  G  e.  T )  ->  ( U `  G
)  =  [_ G  /  g ]_ X
)
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 cdlemk40f
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 31652 . 2  |-  ( G  e.  T  ->  ( U `  G )  =  if ( F  =  N ,  G ,  [_ G  /  g ]_ X ) )
4 ifnefalse 3740 . 2  |-  ( F  =/=  N  ->  if ( F  =  N ,  G ,  [_ G  /  g ]_ X
)  =  [_ G  /  g ]_ X
)
53, 4sylan9eqr 2490 1  |-  ( ( F  =/=  N  /\  G  e.  T )  ->  ( U `  G
)  =  [_ G  /  g ]_ X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   [_csb 3244   ifcif 3732    e. cmpt 4259   ` cfv 5447   iota_crio 6535
This theorem is referenced by:  cdlemk43N  31698  cdlemk35u  31699  cdlemk55u1  31700  cdlemk39u1  31702  cdlemk19u1  31704
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-iota 5411  df-fun 5449  df-fv 5455  df-riota 6542
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