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Theorem cdleme31id 29833
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 445 . . 3  |-  ( ( P  =/=  Q  /\  -.  X  .<_  W )  ->  P  =/=  Q
)
21necon2bi 2467 . 2  |-  ( P  =  Q  ->  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )
3 cdleme31fv2.f . . 3  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
43cdleme31fv2 29832 . 2  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  ( F `  X )  =  X )
52, 4sylan2 462 1  |-  ( ( X  e.  B  /\  P  =  Q )  ->  ( F `  X
)  =  X )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2421   ifcif 3539   class class class wbr 3997    e. cmpt 4051   ` cfv 4673
This theorem is referenced by:  cdleme32fvaw  29878  cdleme42keg  29925  cdleme42mgN  29927  cdleme17d4  29936  cdleme48fvg  29939  cdleme50trn3  29992  cdlemg1idlemN  30011  cdlemg2idN  30035
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fv 4689
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