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Theorem cdlemk40 31106
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 2791 . . . . . 6  |-  g  e. 
_V
2 cdlemk40.x . . . . . . 7  |-  X  =  ( iota_ z  e.  T ph )
3 riotaex 6308 . . . . . . 7  |-  ( iota_ z  e.  T ph )  e.  _V
42, 3eqeltri 2353 . . . . . 6  |-  X  e. 
_V
51, 4ifex 3623 . . . . 5  |-  if ( F  =  N , 
g ,  X )  e.  _V
65ax-gen 1533 . . . 4  |-  A. g if ( F  =  N ,  g ,  X
)  e.  _V
7 csbexg 3091 . . . 4  |-  ( ( G  e.  T  /\  A. g if ( F  =  N ,  g ,  X )  e. 
_V )  ->  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
)  e.  _V )
86, 7mpan2 652 . . 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 5603 . . 3  |-  ( ( G  e.  T  /\  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
)  e.  _V )  ->  ( U `  G
)  =  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
) )
118, 10mpdan 649 . 2  |-  ( G  e.  T  ->  ( U `  G )  =  [_ G  /  g ]_ if ( F  =  N ,  g ,  X ) )
12 csbifg 3593 . 2  |-  ( G  e.  T  ->  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
)  =  if (
[. G  /  g ]. F  =  N ,  [_ G  /  g ]_ g ,  [_ G  /  g ]_ X
) )
13 sbcg 3056 . . 3  |-  ( G  e.  T  ->  ( [. G  /  g ]. F  =  N  <->  F  =  N ) )
14 csbvarg 3108 . . 3  |-  ( G  e.  T  ->  [_ G  /  g ]_ g  =  G )
15 eqidd 2284 . . 3  |-  ( G  e.  T  ->  [_ G  /  g ]_ X  =  [_ G  /  g ]_ X )
1613, 14, 15ifbieq12d 3587 . 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 2319 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 1527    = wceq 1623    e. wcel 1684   _Vcvv 2788   [.wsbc 2991   [_csb 3081   ifcif 3565    e. cmpt 4077   ` cfv 5255   iota_crio 6297
This theorem is referenced by:  cdlemk40t  31107  cdlemk40f  31108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-riota 6304
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