Theorem funresd 6533

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 6532	. 2 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
3	1, 2	syl 17	1 ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴))

Syntax hints:   → wi 4   ↾ cres 5624  Fun wfun 6484

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 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-in 3897  df-ss 3907  df-br 5087  df-opab 5149  df-rel 5629  df-cnv 5630  df-co 5631  df-res 5634  df-fun 6492

This theorem is referenced by:  fnssresb  6612  respreima  7010  fssrescdmd  7071  frrlem11  8237  frrlem12  8238  frrlem15  9670  gsumzadd  19886  gsum2dlem2  19935  nogesgn1ores  27657  noinfres  27705  noinfbnd2lem1  27713  cyclnumvtx  29888  trlsegvdeglem2  30311  sspg  30819  ssps  30821  sspn  30827  fresf1o  32724  fsupprnfi  32785  gsumhashmul  33148  limsupresxr  46209  liminfresxr  46210  afvco2  47621