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Theorem cdlemk40f 31034
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 31032 . 2  |-  ( G  e.  T  ->  ( U `  G )  =  if ( F  =  N ,  G ,  [_ G  /  g ]_ X ) )
4 ifnefalse 3691 . 2  |-  ( F  =/=  N  ->  if ( F  =  N ,  G ,  [_ G  /  g ]_ X
)  =  [_ G  /  g ]_ X
)
53, 4sylan9eqr 2442 1  |-  ( ( F  =/=  N  /\  G  e.  T )  ->  ( U `  G
)  =  [_ G  /  g ]_ X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   [_csb 3195   ifcif 3683    e. cmpt 4208   ` cfv 5395   iota_crio 6479
This theorem is referenced by:  cdlemk43N  31078  cdlemk35u  31079  cdlemk55u1  31080  cdlemk39u1  31082  cdlemk19u1  31084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-iota 5359  df-fun 5397  df-fv 5403  df-riota 6486
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