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Theorem cdleme31fv2 33876
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 4423 . . . . . . . . 9  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
43notbid 295 . . . . . . . 8  |-  ( x  =  X  ->  ( -.  x  .<_  W  <->  -.  X  .<_  W ) )
54anbi2d 708 . . . . . . 7  |-  ( x  =  X  ->  (
( P  =/=  Q  /\  -.  x  .<_  W )  <-> 
( P  =/=  Q  /\  -.  X  .<_  W ) ) )
65notbid 295 . . . . . 6  |-  ( x  =  X  ->  ( -.  ( P  =/=  Q  /\  -.  x  .<_  W )  <->  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) ) )
76biimparc 489 . . . . 5  |-  ( ( -.  ( P  =/= 
Q  /\  -.  X  .<_  W )  /\  x  =  X )  ->  -.  ( P  =/=  Q  /\  -.  x  .<_  W ) )
87adantll 718 . . . 4  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  -.  ( P  =/= 
Q  /\  -.  x  .<_  W ) )
98iffalsed 3920 . . 3  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x )  =  x )
10 simpr 462 . . 3  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  x  =  X )
119, 10eqtrd 2463 . 2  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x )  =  X )
12 simpl 458 . 2  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  X  e.  B )
132, 11, 12, 12fvmptd 5966 1  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  ( F `  X )  =  X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   ifcif 3909   class class class wbr 4420    |-> cmpt 4479   ` cfv 5597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-iota 5561  df-fun 5599  df-fv 5605
This theorem is referenced by:  cdleme31id  33877  cdleme32fvcl  33923  cdleme32e  33928  cdleme32le  33930  cdleme48gfv  34020  cdleme50ldil  34031
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