Theorem funresd 5414

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 5413	. 2 |- ( Fun F -> Fun ( F |` A ) )
3	1, 2	syl 14	1 |- ( ph -> Fun ( F |` A ) )

Syntax hints:   -> wi 4   |` cres 4771   Fun wfun 5366

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-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-br 4126  df-opab 4188  df-rel 4776  df-cnv 4777  df-co 4778  df-res 4781  df-fun 5374

This theorem is referenced by:  trlsegvdeglem2  16616