Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme31fv2 Structured version   Unicode version

Theorem cdleme31fv2 36535
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 4442 . . . . . . . . 9  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
43notbid 292 . . . . . . . 8  |-  ( x  =  X  ->  ( -.  x  .<_  W  <->  -.  X  .<_  W ) )
54anbi2d 701 . . . . . . 7  |-  ( x  =  X  ->  (
( P  =/=  Q  /\  -.  x  .<_  W )  <-> 
( P  =/=  Q  /\  -.  X  .<_  W ) ) )
65notbid 292 . . . . . 6  |-  ( x  =  X  ->  ( -.  ( P  =/=  Q  /\  -.  x  .<_  W )  <->  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) ) )
76biimparc 485 . . . . 5  |-  ( ( -.  ( P  =/= 
Q  /\  -.  X  .<_  W )  /\  x  =  X )  ->  -.  ( P  =/=  Q  /\  -.  x  .<_  W ) )
87adantll 711 . . . 4  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  -.  ( P  =/= 
Q  /\  -.  x  .<_  W ) )
98iffalsed 3940 . . 3  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x )  =  x )
10 simpr 459 . . 3  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  x  =  X )
119, 10eqtrd 2495 . 2  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x )  =  X )
12 simpl 455 . 2  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  X  e.  B )
132, 11, 12, 12fvmptd 5936 1  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  ( F `  X )  =  X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   ifcif 3929   class class class wbr 4439    |-> cmpt 4497   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578
This theorem is referenced by:  cdleme31id  36536  cdleme32fvcl  36582  cdleme32e  36587  cdleme32le  36589  cdleme48gfv  36679  cdleme50ldil  36690
  Copyright terms: Public domain W3C validator