Theorem funresd 6535

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 6534	. 2 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
3	1, 2	syl 17	1 ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴))

Syntax hints:   → wi 4   ↾ cres 5626  Fun wfun 6486

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 3432  df-in 3897  df-ss 3907  df-br 5087  df-opab 5149  df-rel 5631  df-cnv 5632  df-co 5633  df-res 5636  df-fun 6494

This theorem is referenced by:  fnssresb  6614  respreima  7012  fssrescdmd  7073  frrlem11  8239  frrlem12  8240  frrlem15  9672  gsumzadd  19888  gsum2dlem2  19937  nogesgn1ores  27652  noinfres  27700  noinfbnd2lem1  27708  cyclnumvtx  29883  trlsegvdeglem2  30306  sspg  30814  ssps  30816  sspn  30822  fresf1o  32719  fsupprnfi  32780  gsumhashmul  33143  limsupresxr  46212  liminfresxr  46213  afvco2  47636