Theorem funresd 6609

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 6608	. 2 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
3	1, 2	syl 17	1 ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴))

Syntax hints:   → wi 4   ↾ cres 5687  Fun wfun 6555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-in 3958  df-ss 3968  df-br 5144  df-opab 5206  df-rel 5692  df-cnv 5693  df-co 5694  df-res 5697  df-fun 6563

This theorem is referenced by:  fnssresb  6690  respreima  7086  fssrescdmd  7146  frrlem11  8321  frrlem12  8322  frrlem15  9797  gsumzadd  19940  gsum2dlem2  19989  nogesgn1ores  27719  noinfres  27767  noinfbnd2lem1  27775  cyclnumvtx  29820  trlsegvdeglem2  30240  sspg  30747  ssps  30749  sspn  30755  fresf1o  32641  fsupprnfi  32701  gsumhashmul  33064  limsupresxr  45781  liminfresxr  45782  afvco2  47188