Theorem fssresd 5548

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 5547	. 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 3214   |` cres 4758   -->wf 5355

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 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-br 4116  df-opab 4178  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-res 4768  df-fun 5361  df-fn 5362  df-f 5363

This theorem is referenced by:  gsumsplit1r  13667  gfsump1  14114  gfsumcl  14116  znf1o  14931  cnrest  15232  cnptopresti  15235  cnptoprest  15236  psmetres2  15330  xmetres2  15376  metres2  15378  xmetresbl  15437  rescncf  15578  wlkres  16506  trilpolemlt1  16967