Theorem funresd 5394

Description: A restriction of a function is a function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypothesis

Ref	Expression
funresd.1	|- ( ph -> Fun F )

Assertion

Ref	Expression
funresd	|- ( ph -> Fun ( F |` A ) )

Proof of Theorem funresd

Step	Hyp	Ref	Expression
1		funresd.1	. 2 |- ( ph -> Fun F )
2		funres 5393	. 2 |- ( Fun F -> Fun ( F |` A ) )
3	1, 2	syl 14	1 |- ( ph -> Fun ( F |` A ) )

Syntax hints:   -> wi 4   |` cres 4751   Fun wfun 5346

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-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-ss 3224  df-br 4110  df-opab 4172  df-rel 4756  df-cnv 4757  df-co 4758  df-res 4761  df-fun 5354

This theorem is referenced by:  trlsegvdeglem2  16456