Theorem feqresmptf 39433

Description: Express a restricted function as a mapping. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
feqresmptf.1	|- F/_ x F
feqresmptf.2	|- ( ph -> F : A --> B )
feqresmptf.3	|- ( ph -> C C_ A )

Assertion

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

Distinct variable group:   x, C

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

Proof of Theorem feqresmptf

Step	Hyp	Ref	Expression
1		nfcv 2764	. . 3 |- F/_ x C
2		feqresmptf.1	. . . 4 |- F/_ x F
3	2, 1	nfres 5398	. . 3 |- F/_ x ( F |` C )
4		feqresmptf.2	. . . 4 |- ( ph -> F : A --> B )
5		feqresmptf.3	. . . 4 |- ( ph -> C C_ A )
6	4, 5	fssresd 6071	. . 3 |- ( ph -> ( F |` C ) : C --> B )
7	1, 3, 6	feqmptdf 6251	. 2 |- ( ph -> ( F |` C ) = ( x e. C |-> ( ( F |` C ) ` x ) ) )
8		fvres 6207	. . 3 |- ( x e. C -> ( ( F |` C ) ` x ) = ( F ` x ) )
9	8	mpteq2ia 4740	. 2 |- ( x e. C |-> ( ( F |` C ) ` x ) ) = ( x e. C |-> ( F ` x ) )
10	7, 9	syl6eq 2672	1 |- ( ph -> ( F |` C ) = ( x e. C |-> ( F ` x ) ) )

Syntax hints:   -> wi 4   = wceq 1483   F/_wnfc 2751   C_ wss 3574   |-> cmpt 4729   |` cres 5116   -->wf 5884   ` cfv 5888

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-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-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-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by: (None)