Theorem cdleme31id 35682

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

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( P =/= Q /\ -. X .<_ W ) -> P =/= Q )
2	1	necon2bi 2824	. 2 |- ( P = Q -> -. ( P =/= Q /\ -. X .<_ W ) )
3		cdleme31fv2.f	. . 3 |- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
4	3	cdleme31fv2 35681	. 2 |- ( ( X e. B /\ -. ( P =/= Q /\ -. X .<_ W ) ) -> ( F ` X ) = X )
5	2, 4	sylan2 491	1 |- ( ( X e. B /\ P = Q ) -> ( F ` X ) = X )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  cdleme32fvaw  35727  cdleme42keg  35774  cdleme42mgN  35776  cdleme17d4  35785  cdleme48fvg  35788  cdleme50trn3  35841  cdlemg1idlemN  35860  cdlemg2idN  35884