Theorem feqresmpt 5616

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 5434	. . . 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 5615	. 2 |- ( ph -> ( F |` C ) = ( x e. C |-> ( ( F |` C ) ` x ) ) )
6		fvres 5583	. . 3 |- ( x e. C -> ( ( F |` C ) ` x ) = ( F ` x ) )
7	6	mpteq2ia 4120	. 2 |- ( x e. C |-> ( ( F |` C ) ` x ) ) = ( x e. C |-> ( F ` x ) )
8	5, 7	eqtrdi 2245	1 |- ( ph -> ( F |` C ) = ( x e. C |-> ( F ` x ) ) )

Syntax hints:   -> wi 4   = wceq 1364   C_ wss 3157   |-> cmpt 4095   |` cres 4666   -->wf 5255   ` cfv 5259

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267

This theorem is referenced by:  dvmulxxbr  14948