Theorem feqresmpt 5475

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 5298	. . . 4 |- ( ( F : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B )
4	1, 2, 3	syl2anc 408	. . 3 |- ( ph -> ( F |` C ) : C --> B )
5	4	feqmptd 5474	. 2 |- ( ph -> ( F |` C ) = ( x e. C |-> ( ( F |` C ) ` x ) ) )
6		fvres 5445	. . 3 |- ( x e. C -> ( ( F |` C ) ` x ) = ( F ` x ) )
7	6	mpteq2ia 4014	. 2 |- ( x e. C |-> ( ( F |` C ) ` x ) ) = ( x e. C |-> ( F ` x ) )
8	5, 7	syl6eq 2188	1 |- ( ph -> ( F |` C ) = ( x e. C |-> ( F ` x ) ) )

Syntax hints:   -> wi 4   = wceq 1331   C_ wss 3071   |-> cmpt 3989   |` cres 4541   -->wf 5119   ` cfv 5123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131

This theorem is referenced by:  dvmulxxbr  12835