Theorem funresd 6543

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 6542	. 2 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
3	1, 2	syl 17	1 ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴))

Syntax hints:   → wi 4   ↾ cres 5634  Fun wfun 6494

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 3444  df-in 3910  df-ss 3920  df-br 5101  df-opab 5163  df-rel 5639  df-cnv 5640  df-co 5641  df-res 5644  df-fun 6502

This theorem is referenced by:  fnssresb  6622  respreima  7020  fssrescdmd  7081  frrlem11  8248  frrlem12  8249  frrlem15  9681  gsumzadd  19863  gsum2dlem2  19912  nogesgn1ores  27654  noinfres  27702  noinfbnd2lem1  27710  cyclnumvtx  29885  trlsegvdeglem2  30308  sspg  30815  ssps  30817  sspn  30823  fresf1o  32720  fsupprnfi  32781  gsumhashmul  33160  limsupresxr  46121  liminfresxr  46122  afvco2  47533