Theorem funresd 6576

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 6575	. 2 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
3	1, 2	syl 18	1 ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↾ cres 5657  Fun wfun 6527

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 3906  df-ss 3916  df-br 5104  df-opab 5168  df-rel 5662  df-cnv 5663  df-co 5664  df-res 5667  df-fun 6535

This theorem is used by:  fnssresb  6654  respreima  7058  fssrescdmd  7120  frrlem11  8295  frrlem12  8296  frrlem15  9739  gsumzadd  20049  gsum2dlem2  20098  nogesgn1ores  27910  noinfres  27958  noinfbnd2lem1  27966  cyclnumvtx  30267  trlsegvdeglem2  30701  sspg  31209  ssps  31211  sspn  31217  fresf1o  33104  fsupprnfi  33164  gsumhashmul  33507  limsupresxr  46594  liminfresxr  46595  afvco2  48064