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

Theorem cdlemk40 35590
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 3111 . . . . 5  |-  g  e. 
_V
2 cdlemk40.x . . . . . 6  |-  X  =  ( iota_ z  e.  T  ph )
3 riotaex 6242 . . . . . 6  |-  ( iota_ z  e.  T  ph )  e.  _V
42, 3eqeltri 2546 . . . . 5  |-  X  e. 
_V
51, 4ifex 4003 . . . 4  |-  if ( F  =  N , 
g ,  X )  e.  _V
65csbex 4575 . . 3  |-  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
)  e.  _V
7 cdlemk40.u . . . 4  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
87fvmpts 5945 . . 3  |-  ( ( G  e.  T  /\  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
)  e.  _V )  ->  ( U `  G
)  =  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
) )
96, 8mpan2 671 . 2  |-  ( G  e.  T  ->  ( U `  G )  =  [_ G  /  g ]_ if ( F  =  N ,  g ,  X ) )
10 csbif 3984 . . 3  |-  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
)  =  if (
[. G  /  g ]. F  =  N ,  [_ G  /  g ]_ g ,  [_ G  /  g ]_ X
)
11 sbcg 3400 . . . 4  |-  ( G  e.  T  ->  ( [. G  /  g ]. F  =  N  <->  F  =  N ) )
12 csbvarg 3843 . . . 4  |-  ( G  e.  T  ->  [_ G  /  g ]_ g  =  G )
13 eqidd 2463 . . . 4  |-  ( G  e.  T  ->  [_ G  /  g ]_ X  =  [_ G  /  g ]_ X )
1411, 12, 13ifbieq12d 3961 . . 3  |-  ( G  e.  T  ->  if ( [. G  /  g ]. F  =  N ,  [_ G  /  g ]_ g ,  [_ G  /  g ]_ X
)  =  if ( F  =  N ,  G ,  [_ G  / 
g ]_ X ) )
1510, 14syl5eq 2515 . 2  |-  ( G  e.  T  ->  [_ G  /  g ]_ if ( F  =  N ,  g ,  X
)  =  if ( F  =  N ,  G ,  [_ G  / 
g ]_ X ) )
169, 15eqtrd 2503 1  |-  ( G  e.  T  ->  ( U `  G )  =  if ( F  =  N ,  G ,  [_ G  /  g ]_ X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   _Vcvv 3108   [.wsbc 3326   [_csb 3430   ifcif 3934    |-> cmpt 4500   ` cfv 5581   iota_crio 6237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-riota 6238
This theorem is referenced by:  cdlemk40t  35591  cdlemk40f  35592
  Copyright terms: Public domain W3C validator