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Theorem cdleme31id 30508
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 18-Apr-2013.)
Hypothesis
Ref Expression
cdleme31fv2.f  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
Assertion
Ref Expression
cdleme31id  |-  ( ( X  e.  B  /\  P  =  Q )  ->  ( F `  X
)  =  X )
Distinct variable groups:    x, B    x, 
.<_    x, P    x, Q    x, W    x, X
Allowed substitution hints:    F( x)    O( x)

Proof of Theorem cdleme31id
StepHypRef Expression
1 simpl 444 . . 3  |-  ( ( P  =/=  Q  /\  -.  X  .<_  W )  ->  P  =/=  Q
)
21necon2bi 2596 . 2  |-  ( P  =  Q  ->  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )
3 cdleme31fv2.f . . 3  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
43cdleme31fv2 30507 . 2  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  ( F `  X )  =  X )
52, 4sylan2 461 1  |-  ( ( X  e.  B  /\  P  =  Q )  ->  ( F `  X
)  =  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   ifcif 3682   class class class wbr 4153    e. cmpt 4207   ` cfv 5394
This theorem is referenced by:  cdleme32fvaw  30553  cdleme42keg  30600  cdleme42mgN  30602  cdleme17d4  30611  cdleme48fvg  30614  cdleme50trn3  30667  cdlemg1idlemN  30686  cdlemg2idN  30710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402
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