Theorem fssresd 5561

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 5560	. 2 |- ( ( F : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B )
4	1, 2, 3	syl2anc 415	1 |- ( ph -> ( F |` C ) : C --> B )

Syntax hints:   -> wi 4   C_ wss 3220   |` cres 4771   -->wf 5368

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-fun 5374  df-fn 5375  df-f 5376

This theorem is referenced by:  gzsumsplit1r  13692  gsump1  14134  gsumclfi  14136  gsummptfidmadd  14138  gsumsubmclfi  14140  znf1o  14958  cnrest  15259  cnptopresti  15262  cnptoprest  15263  psmetres2  15357  xmetres2  15403  metres2  15405  xmetresbl  15464  rescncf  15605  wlkres  16534  trilpolemlt1  16995