Theorem funresd 6579

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 6578	. 2 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
3	1, 2	syl 17	1 ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴))

Syntax hints:   → wi 4   ↾ cres 5656  Fun wfun 6525

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 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-in 3933  df-ss 3943  df-br 5120  df-opab 5182  df-rel 5661  df-cnv 5662  df-co 5663  df-res 5666  df-fun 6533

This theorem is referenced by:  fnssresb  6660  respreima  7056  fssrescdmd  7116  frrlem11  8295  frrlem12  8296  frrlem15  9771  gsumzadd  19903  gsum2dlem2  19952  nogesgn1ores  27638  noinfres  27686  noinfbnd2lem1  27694  cyclnumvtx  29782  trlsegvdeglem2  30202  sspg  30709  ssps  30711  sspn  30717  fresf1o  32609  fsupprnfi  32669  gsumhashmul  33055  limsupresxr  45795  liminfresxr  45796  afvco2  47205