Theorem cdlemk40 36205

Description: TODO: fix comment. (Contributed by NM, 31-Jul-2013.)

Hypotheses

Ref	Expression
cdlemk40.x	|- X = ( iota_ z e. T ph )
cdlemk40.u	|- U = ( g e. T |-> if ( F = N , g , X ) )

Assertion

Ref	Expression
cdlemk40	|- ( G e. T -> ( U ` G ) = if ( F = N , G , [_ G / g ]_ X ) )

Distinct variable groups:   g, F   g, N   T, g

Allowed substitution hints:   ph( z, g)   T( z)   U( z, g)   F( z)   G( z, g)   N( z)   X( z, g)

Proof of Theorem cdlemk40

Step	Hyp	Ref	Expression
1		vex 3203	. . . . 5 |- g e. _V
2		cdlemk40.x	. . . . . 6 |- X = ( iota_ z e. T ph )
3		riotaex 6615	. . . . . 6 |- ( iota_ z e. T ph ) e. _V
4	2, 3	eqeltri 2697	. . . . 5 |- X e. _V
5	1, 4	ifex 4156	. . . 4 |- if ( F = N , g , X ) e. _V
6	5	csbex 4793	. . 3 |- [_ G / g ]_ if ( F = N , g , X ) e. _V
7		cdlemk40.u	. . . 4 |- U = ( g e. T |-> if ( F = N , g , X ) )
8	7	fvmpts 6285	. . 3 |- ( ( G e. T /\ [_ G / g ]_ if ( F = N , g , X ) e. _V ) -> ( U ` G ) = [_ G / g ]_ if ( F = N , g , X ) )
9	6, 8	mpan2 707	. 2 |- ( G e. T -> ( U ` G ) = [_ G / g ]_ if ( F = N , g , X ) )
10		csbif 4138	. . 3 |- [_ G / g ]_ if ( F = N , g , X ) = if ( [. G / g ]. F = N , [_ G / g ]_ g , [_ G / g ]_ X )
11		sbcg 3503	. . . 4 |- ( G e. T -> ( [. G / g ]. F = N <-> F = N ) )
12		csbvarg 4003	. . . 4 |- ( G e. T -> [_ G / g ]_ g = G )
13	11, 12	ifbieq1d 4109	. . 3 |- ( G e. T -> if ( [. G / g ]. F = N , [_ G / g ]_ g , [_ G / g ]_ X ) = if ( F = N , G , [_ G / g ]_ X ) )
14	10, 13	syl5eq 2668	. 2 |- ( G e. T -> [_ G / g ]_ if ( F = N , g , X ) = if ( F = N , G , [_ G / g ]_ X ) )
15	9, 14	eqtrd 2656	1 |- ( G e. T -> ( U ` G ) = if ( F = N , G , [_ G / g ]_ X ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   [_csb 3533   ifcif 4086   |-> cmpt 4729   ` cfv 5888   iota_crio 6610

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-fal 1489  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-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  df-riota 6611

This theorem is referenced by:  cdlemk40t  36206  cdlemk40f  36207