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Theorem cdleme31id 34032
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 464 . . 3  |-  ( ( P  =/=  Q  /\  -.  X  .<_  W )  ->  P  =/=  Q
)
21necon2bi 2673 . 2  |-  ( P  =  Q  ->  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )
3 cdleme31fv2.f . . 3  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
43cdleme31fv2 34031 . 2  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  ( F `  X )  =  X )
52, 4sylan2 482 1  |-  ( ( X  e.  B  /\  P  =  Q )  ->  ( F `  X
)  =  X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   ifcif 3872   class class class wbr 4395    |-> cmpt 4454   ` cfv 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597
This theorem is referenced by:  cdleme32fvaw  34077  cdleme42keg  34124  cdleme42mgN  34126  cdleme17d4  34135  cdleme48fvg  34138  cdleme50trn3  34191  cdlemg1idlemN  34210  cdlemg2idN  34234
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