Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemk40 Unicode version

Theorem cdlemk40 31728
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 2804 . . . . . 6  |-  g  e. 
_V
2 cdlemk40.x . . . . . . 7  |-  X  =  ( iota_ z  e.  T ph )
3 riotaex 6324 . . . . . . 7  |-  ( iota_ z  e.  T ph )  e.  _V
42, 3eqeltri 2366 . . . . . 6  |-  X  e. 
_V
51, 4ifex 3636 . . . . 5  |-  if ( F  =  N , 
g ,  X )  e.  _V
65ax-gen 1536 . . . 4  |-  A. g if ( F  =  N ,  g ,  X
)  e.  _V
7 csbexg 3104 . . . 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 5619 . . 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 3606 . 2  |-  ( G  e.  T  ->  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
)  =  if (
[. G  /  g ]. F  =  N ,  [_ G  /  g ]_ g ,  [_ G  /  g ]_ X
) )
13 sbcg 3069 . . 3  |-  ( G  e.  T  ->  ( [. G  /  g ]. F  =  N  <->  F  =  N ) )
14 csbvarg 3121 . . 3  |-  ( G  e.  T  ->  [_ G  /  g ]_ g  =  G )
15 eqidd 2297 . . 3  |-  ( G  e.  T  ->  [_ G  /  g ]_ X  =  [_ G  /  g ]_ X )
1613, 14, 15ifbieq12d 3600 . 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 2332 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 1530    = wceq 1632    e. wcel 1696   _Vcvv 2801   [.wsbc 3004   [_csb 3094   ifcif 3578    e. cmpt 4093   ` cfv 5271   iota_crio 6313
This theorem is referenced by:  cdlemk40t  31729  cdlemk40f  31730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-riota 6320
  Copyright terms: Public domain W3C validator