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Theorem cdlemk40 31776
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
cdlemk40  |-  ( G  e.  T  ->  ( U `  G )  =  if ( F  =  N ,  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 cdlemk40
StepHypRef Expression
1 vex 2961 . . . . . 6  |-  g  e. 
_V
2 cdlemk40.x . . . . . . 7  |-  X  =  ( iota_ z  e.  T ph )
3 riotaex 6555 . . . . . . 7  |-  ( iota_ z  e.  T ph )  e.  _V
42, 3eqeltri 2508 . . . . . 6  |-  X  e. 
_V
51, 4ifex 3799 . . . . 5  |-  if ( F  =  N , 
g ,  X )  e.  _V
65ax-gen 1556 . . . 4  |-  A. g if ( F  =  N ,  g ,  X
)  e.  _V
7 csbexg 3263 . . . 4  |-  ( ( G  e.  T  /\  A. g if ( F  =  N ,  g ,  X )  e. 
_V )  ->  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
)  e.  _V )
86, 7mpan2 654 . . 3  |-  ( G  e.  T  ->  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
)  e.  _V )
9 cdlemk40.u . . . 4  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
109fvmpts 5809 . . 3  |-  ( ( G  e.  T  /\  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
)  e.  _V )  ->  ( U `  G
)  =  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
) )
118, 10mpdan 651 . 2  |-  ( G  e.  T  ->  ( U `  G )  =  [_ G  /  g ]_ if ( F  =  N ,  g ,  X ) )
12 csbifg 3769 . 2  |-  ( G  e.  T  ->  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
)  =  if (
[. G  /  g ]. F  =  N ,  [_ G  /  g ]_ g ,  [_ G  /  g ]_ X
) )
13 sbcg 3228 . . 3  |-  ( G  e.  T  ->  ( [. G  /  g ]. F  =  N  <->  F  =  N ) )
14 csbvarg 3280 . . 3  |-  ( G  e.  T  ->  [_ G  /  g ]_ g  =  G )
15 eqidd 2439 . . 3  |-  ( G  e.  T  ->  [_ G  /  g ]_ X  =  [_ G  /  g ]_ X )
1613, 14, 15ifbieq12d 3763 . 2  |-  ( G  e.  T  ->  if ( [. G  /  g ]. F  =  N ,  [_ G  /  g ]_ g ,  [_ G  /  g ]_ X
)  =  if ( F  =  N ,  G ,  [_ G  / 
g ]_ X ) )
1711, 12, 163eqtrd 2474 1  |-  ( G  e.  T  ->  ( U `  G )  =  if ( F  =  N ,  G ,  [_ G  /  g ]_ X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550    = wceq 1653    e. wcel 1726   _Vcvv 2958   [.wsbc 3163   [_csb 3253   ifcif 3741    e. cmpt 4268   ` cfv 5456   iota_crio 6544
This theorem is referenced by:  cdlemk40t  31777  cdlemk40f  31778
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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