Theorem feqresmpt 5632

Description: Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)

Hypotheses

Ref	Expression
feqmptd.1	|- ( ph -> F : A --> B )
feqresmpt.2	|- ( ph -> C C_ A )

Assertion

Ref	Expression
feqresmpt	|- ( ph -> ( F |` C ) = ( x e. C |-> ( F ` x ) ) )

Distinct variable groups:   x, A   x, C   x, F

Allowed substitution hints:   ph( x)   B( x)

Proof of Theorem feqresmpt

Step	Hyp	Ref	Expression
1		feqmptd.1	. . . 4 |- ( ph -> F : A --> B )
2		feqresmpt.2	. . . 4 |- ( ph -> C C_ A )
3		fssres 5450	. . . 4 |- ( ( F : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B )
4	1, 2, 3	syl2anc 411	. . 3 |- ( ph -> ( F |` C ) : C --> B )
5	4	feqmptd 5631	. 2 |- ( ph -> ( F |` C ) = ( x e. C |-> ( ( F |` C ) ` x ) ) )
6		fvres 5599	. . 3 |- ( x e. C -> ( ( F |` C ) ` x ) = ( F ` x ) )
7	6	mpteq2ia 4129	. 2 |- ( x e. C |-> ( ( F |` C ) ` x ) ) = ( x e. C |-> ( F ` x ) )
8	5, 7	eqtrdi 2253	1 |- ( ph -> ( F |` C ) = ( x e. C |-> ( F ` x ) ) )

Syntax hints:   -> wi 4   = wceq 1372   C_ wss 3165   |-> cmpt 4104   |` cres 4676   -->wf 5266   ` cfv 5270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fv 5278

This theorem is referenced by:  dvmulxxbr  15145