Theorem cdleme31fv1 35679

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 10-Feb-2013.)

Hypotheses

Ref	Expression
cdleme31.o	|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
cdleme31.f	|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
cdleme31.c	|- C = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) -> z = ( N .\/ ( X ./\ W ) ) ) )

Assertion

Ref	Expression
cdleme31fv1	|- ( ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) -> ( F ` X ) = C )

Distinct variable groups:   x, B   x, C   x, .<_   x, P   x, Q   x, W   x, s, z, X

Allowed substitution hints:   A( x, z, s)   B( z, s)   C( z, s)   P( z, s)   Q( z, s)   F( x, z, s)   .\/ ( x, z, s)   .<_ ( z, s)   ./\ ( x, z, s)   N( x, z, s)   O( x, z, s)   W( z, s)

Proof of Theorem cdleme31fv1

Step	Hyp	Ref	Expression
1		cdleme31.o	. . 3 |- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
2		cdleme31.f	. . 3 |- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
3		cdleme31.c	. . 3 |- C = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) -> z = ( N .\/ ( X ./\ W ) ) ) )
4	1, 2, 3	cdleme31fv 35678	. 2 |- ( X e. B -> ( F ` X ) = if ( ( P =/= Q /\ -. X .<_ W ) , C , X ) )
5		iftrue 4092	. 2 |- ( ( P =/= Q /\ -. X .<_ W ) -> if ( ( P =/= Q /\ -. X .<_ W ) , C , X ) = C )
6	4, 5	sylan9eq 2676	1 |- ( ( X e. B /\ ( P =/= Q /\ -. X .<_ W ) ) -> ( F ` X ) = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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  df-riota 6611  df-ov 6653

This theorem is referenced by:  cdleme31fv1s  35680  cdleme32fvcl  35728  cdleme32a  35729  cdleme42b  35766