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Theorem cdlemk40 37056
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 3037 . . . . 5  |-  g  e. 
_V
2 cdlemk40.x . . . . . 6  |-  X  =  ( iota_ z  e.  T  ph )
3 riotaex 6162 . . . . . 6  |-  ( iota_ z  e.  T  ph )  e.  _V
42, 3eqeltri 2466 . . . . 5  |-  X  e. 
_V
51, 4ifex 3925 . . . 4  |-  if ( F  =  N , 
g ,  X )  e.  _V
65csbex 4500 . . 3  |-  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
)  e.  _V
7 cdlemk40.u . . . 4  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
87fvmpts 5859 . . 3  |-  ( ( G  e.  T  /\  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
)  e.  _V )  ->  ( U `  G
)  =  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
) )
96, 8mpan2 669 . 2  |-  ( G  e.  T  ->  ( U `  G )  =  [_ G  /  g ]_ if ( F  =  N ,  g ,  X ) )
10 csbif 3907 . . 3  |-  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
)  =  if (
[. G  /  g ]. F  =  N ,  [_ G  /  g ]_ g ,  [_ G  /  g ]_ X
)
11 sbcg 3318 . . . 4  |-  ( G  e.  T  ->  ( [. G  /  g ]. F  =  N  <->  F  =  N ) )
12 csbvarg 3770 . . . 4  |-  ( G  e.  T  ->  [_ G  /  g ]_ g  =  G )
1311, 12ifbieq1d 3880 . . 3  |-  ( G  e.  T  ->  if ( [. G  /  g ]. F  =  N ,  [_ G  /  g ]_ g ,  [_ G  /  g ]_ X
)  =  if ( F  =  N ,  G ,  [_ G  / 
g ]_ X ) )
1410, 13syl5eq 2435 . 2  |-  ( G  e.  T  ->  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
)  =  if ( F  =  N ,  G ,  [_ G  / 
g ]_ X ) )
159, 14eqtrd 2423 1  |-  ( G  e.  T  ->  ( U `  G )  =  if ( F  =  N ,  G ,  [_ G  /  g ]_ X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1826   _Vcvv 3034   [.wsbc 3252   [_csb 3348   ifcif 3857    |-> cmpt 4425   ` cfv 5496   iota_crio 6157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-riota 6158
This theorem is referenced by:  cdlemk40t  37057  cdlemk40f  37058
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