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Theorem cdleme31fv2 34376
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 11 . 2  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) ) )
3 breq1 4404 . . . . . . . . 9  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
43notbid 294 . . . . . . . 8  |-  ( x  =  X  ->  ( -.  x  .<_  W  <->  -.  X  .<_  W ) )
54anbi2d 703 . . . . . . 7  |-  ( x  =  X  ->  (
( P  =/=  Q  /\  -.  x  .<_  W )  <-> 
( P  =/=  Q  /\  -.  X  .<_  W ) ) )
65notbid 294 . . . . . 6  |-  ( x  =  X  ->  ( -.  ( P  =/=  Q  /\  -.  x  .<_  W )  <->  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) ) )
76biimparc 487 . . . . 5  |-  ( ( -.  ( P  =/= 
Q  /\  -.  X  .<_  W )  /\  x  =  X )  ->  -.  ( P  =/=  Q  /\  -.  x  .<_  W ) )
87adantll 713 . . . 4  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  -.  ( P  =/= 
Q  /\  -.  x  .<_  W ) )
9 iffalse 3908 . . . 4  |-  ( -.  ( P  =/=  Q  /\  -.  x  .<_  W )  ->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x
)  =  x )
108, 9syl 16 . . 3  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x )  =  x )
11 simpr 461 . . 3  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  x  =  X )
1210, 11eqtrd 2495 . 2  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x )  =  X )
13 simpl 457 . 2  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  X  e.  B )
142, 12, 13, 13fvmptd 5889 1  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  ( F `  X )  =  X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   ifcif 3900   class class class wbr 4401    |-> cmpt 4459   ` cfv 5527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535
This theorem is referenced by:  cdleme31id  34377  cdleme32fvcl  34423  cdleme32e  34428  cdleme32le  34430  cdleme48gfv  34520  cdleme50ldil  34531
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