Theorem fnssresd 5446

Description: Restriction of a function to a subclass of its domain. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

Hypotheses

Ref	Expression
fnssresd.1	|- ( ph -> F Fn A )
fnssresd.2	|- ( ph -> B C_ A )

Assertion

Ref	Expression
fnssresd	|- ( ph -> ( F |` B ) Fn B )

Proof of Theorem fnssresd

Step	Hyp	Ref	Expression
1		fnssresd.1	. 2 |- ( ph -> F Fn A )
2		fnssresd.2	. 2 |- ( ph -> B C_ A )
3		fnssres 5445	. 2 |- ( ( F Fn A /\ B C_ A ) -> ( F |` B ) Fn B )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> ( F |` B ) Fn B )

Syntax hints:   -> wi 4   C_ wss 3200   |` cres 4727   Fn wfn 5321

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-res 4737  df-fun 5328  df-fn 5329

This theorem is referenced by:  pfxccat1  11282