Theorem funresd 6549

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

Hypothesis

Ref	Expression
funresd.1	⊢ (𝜑 → Fun 𝐹)

Assertion

Ref	Expression
funresd	⊢ (𝜑 → Fun (𝐹 ↾ 𝐴))

Proof of Theorem funresd

Step	Hyp	Ref	Expression
1		funresd.1	. 2 ⊢ (𝜑 → Fun 𝐹)
2		funres 6548	. 2 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
3	1, 2	syl 17	1 ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴))

Syntax hints:   → wi 4   ↾ cres 5640  Fun wfun 6495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3448  df-in 3920  df-ss 3930  df-br 5111  df-opab 5173  df-rel 5645  df-cnv 5646  df-co 5647  df-res 5650  df-fun 6503

This theorem is referenced by:  fnssresb  6628  respreima  7021  frrlem11  8232  frrlem12  8233  frrlem15  9702  gsumzadd  19713  gsum2dlem2  19762  nogesgn1ores  27059  noinfres  27107  noinfbnd2lem1  27115  trlsegvdeglem2  29228  sspg  29733  ssps  29735  sspn  29741  fresf1o  31612  fsupprnfi  31674  gsumhashmul  31968  limsupresxr  44127  liminfresxr  44128  afvco2  45528