Theorem funresd 39476

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 5929	. 2 |- ( Fun F -> Fun ( F |` A ) )
3	1, 2	syl 17	1 |- ( ph -> Fun ( F |` A ) )

Syntax hints:   -> wi 4   |` cres 5116   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-res 5126  df-fun 5890

This theorem is referenced by:  limsupresxr  39998  liminfresxr  39999