Theorem funresd 6519

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 6518	. 2 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
3	1, 2	syl 17	1 ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴))

Syntax hints:   → wi 4   ↾ cres 5613  Fun wfun 6470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-in 3904  df-ss 3914  df-br 5087  df-opab 5149  df-rel 5618  df-cnv 5619  df-co 5620  df-res 5623  df-fun 6478

This theorem is referenced by:  fnssresb  6598  respreima  6994  fssrescdmd  7054  frrlem11  8221  frrlem12  8222  frrlem15  9645  gsumzadd  19829  gsum2dlem2  19878  nogesgn1ores  27608  noinfres  27656  noinfbnd2lem1  27664  cyclnumvtx  29773  trlsegvdeglem2  30193  sspg  30700  ssps  30702  sspn  30708  fresf1o  32605  fsupprnfi  32665  gsumhashmul  33033  limsupresxr  45804  liminfresxr  45805  afvco2  47207