Theorem funresd 6621

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 6620	. 2 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
3	1, 2	syl 17	1 ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴))

Syntax hints:   → wi 4   ↾ cres 5702  Fun wfun 6567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-in 3983  df-ss 3993  df-br 5167  df-opab 5229  df-rel 5707  df-cnv 5708  df-co 5709  df-res 5712  df-fun 6575

This theorem is referenced by:  fnssresb  6702  respreima  7099  fssrescdmd  7160  frrlem11  8337  frrlem12  8338  frrlem15  9826  gsumzadd  19964  gsum2dlem2  20013  nogesgn1ores  27737  noinfres  27785  noinfbnd2lem1  27793  trlsegvdeglem2  30253  sspg  30760  ssps  30762  sspn  30768  fresf1o  32650  fsupprnfi  32704  gsumhashmul  33040  limsupresxr  45687  liminfresxr  45688  afvco2  47091