Theorem fssresd 5513

Description: Restriction of a function with a subclass of its domain, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

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

Assertion

Ref	Expression
fssresd	|- ( ph -> ( F |` C ) : C --> B )

Proof of Theorem fssresd

Step	Hyp	Ref	Expression
1		fssresd.1	. 2 |- ( ph -> F : A --> B )
2		fssresd.2	. 2 |- ( ph -> C C_ A )
3		fssres 5512	. 2 |- ( ( F : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> ( F |` C ) : C --> B )

Syntax hints:   -> wi 4   C_ wss 3200   |` cres 4727   -->wf 5322

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-fun 5328  df-fn 5329  df-f 5330

This theorem is referenced by:  gsumsplit1r  13480  znf1o  14664  cnrest  14958  cnptopresti  14961  cnptoprest  14962  psmetres2  15056  xmetres2  15102  metres2  15104  xmetresbl  15163  rescncf  15304  wlkres  16229  trilpolemlt1  16645