Theorem funresd 6541

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 6540	. 2 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
3	1, 2	syl 17	1 ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴))

Syntax hints:   → wi 4   ↾ cres 5633  Fun wfun 6492

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-in 3896  df-ss 3906  df-br 5086  df-opab 5148  df-rel 5638  df-cnv 5639  df-co 5640  df-res 5643  df-fun 6500

This theorem is referenced by:  fnssresb  6620  respreima  7018  fssrescdmd  7079  frrlem11  8246  frrlem12  8247  frrlem15  9681  gsumzadd  19897  gsum2dlem2  19946  nogesgn1ores  27638  noinfres  27686  noinfbnd2lem1  27694  cyclnumvtx  29868  trlsegvdeglem2  30291  sspg  30799  ssps  30801  sspn  30807  fresf1o  32704  fsupprnfi  32765  gsumhashmul  33128  limsupresxr  46194  liminfresxr  46195  afvco2  47624