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Theorem cdleme31fv2 29861
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 23-Feb-2013.)
Hypothesis
Ref Expression
cdleme31fv2.f  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
Assertion
Ref Expression
cdleme31fv2  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  ( 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 cdleme31fv2
StepHypRef Expression
1 cdleme31fv2.f . . 3  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
21a1i 10 . 2  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) ) )
3 breq1 4027 . . . . . . . . 9  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
43notbid 285 . . . . . . . 8  |-  ( x  =  X  ->  ( -.  x  .<_  W  <->  -.  X  .<_  W ) )
54anbi2d 684 . . . . . . 7  |-  ( x  =  X  ->  (
( P  =/=  Q  /\  -.  x  .<_  W )  <-> 
( P  =/=  Q  /\  -.  X  .<_  W ) ) )
65notbid 285 . . . . . 6  |-  ( x  =  X  ->  ( -.  ( P  =/=  Q  /\  -.  x  .<_  W )  <->  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) ) )
76biimparc 473 . . . . 5  |-  ( ( -.  ( P  =/= 
Q  /\  -.  X  .<_  W )  /\  x  =  X )  ->  -.  ( P  =/=  Q  /\  -.  x  .<_  W ) )
87adantll 694 . . . 4  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  -.  ( P  =/= 
Q  /\  -.  x  .<_  W ) )
9 iffalse 3573 . . . 4  |-  ( -.  ( P  =/=  Q  /\  -.  x  .<_  W )  ->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x
)  =  x )
108, 9syl 15 . . 3  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x )  =  x )
11 simpr 447 . . 3  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  x  =  X )
1210, 11eqtrd 2316 . 2  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x )  =  X )
13 simpl 443 . 2  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  X  e.  B )
142, 12, 13, 13fvmptd 5568 1  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  ( F `  X )  =  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1685    =/= wne 2447   ifcif 3566   class class class wbr 4024    e. cmpt 4078   ` cfv 5221
This theorem is referenced by:  cdleme31id  29862  cdleme32fvcl  29908  cdleme32e  29913  cdleme32le  29915  cdleme48gfv  30005  cdleme50ldil  30016
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229
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