Theorem fssresd 5521

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 5520	. 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 3201   |` cres 4733   -->wf 5329

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-fun 5335  df-fn 5336  df-f 5337

This theorem is referenced by:  gsumsplit1r  13561  znf1o  14747  cnrest  15046  cnptopresti  15049  cnptoprest  15050  psmetres2  15144  xmetres2  15190  metres2  15192  xmetresbl  15251  rescncf  15392  wlkres  16320  trilpolemlt1  16773  gfsump1  16815  gfsumcl  16816