Theorem feqresmpt 5227

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 5066	. . . 4 |- ( ( F : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B )
4	1, 2, 3	syl2anc 391	. . 3 |- ( ph -> ( F |` C ) : C --> B )
5	4	feqmptd 5226	. 2 |- ( ph -> ( F |` C ) = ( x e. C |-> ( ( F |` C ) ` x ) ) )
6		fvres 5198	. . 3 |- ( x e. C -> ( ( F |` C ) ` x ) = ( F ` x ) )
7	6	mpteq2ia 3843	. 2 |- ( x e. C |-> ( ( F |` C ) ` x ) ) = ( x e. C |-> ( F ` x ) )
8	5, 7	syl6eq 2088	1 |- ( ph -> ( F |` C ) = ( x e. C |-> ( F ` x ) ) )

Syntax hints:   -> wi 4   = wceq 1243   C_ wss 2917   |-> cmpt 3818   |` cres 4347   -->wf 4898   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910

This theorem is referenced by: (None)