Theorem fnssresd 6692

Description: Restriction of a function to a subclass of its domain. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

Hypotheses

Ref	Expression
fnssresd.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
fnssresd.2	⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Assertion

Ref	Expression
fnssresd	⊢ (𝜑 → (𝐹 ↾ 𝐵) Fn 𝐵)

Proof of Theorem fnssresd

Step	Hyp	Ref	Expression
1		fnssresd.1	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		fnssresd.2	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
3		fnssres 6691	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵)
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → (𝐹 ↾ 𝐵) Fn 𝐵)

Syntax hints:   → wi 4   ⊆ wss 3951   ↾ cres 5687   Fn wfn 6556

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  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-fun 6563  df-fn 6564

This theorem is referenced by:  rescnvimafod  7093  fssrescdmd  7146  fpwwe2lem7  10677  pfxccat1  14740  mdetrsca  22609  2ndresdju  32659  fdifsupp  32694  fdifsuppconst  32698  ply1gsumz  33619  dimkerim  33678  rmulccn  33927  subfacp1lem3  35187  satfn  35360  eqresfnbd  42273  tfsconcatrev  43361  ofoafg  43367  xlimconst2  45850  dvnprodlem1  45961  fcoreslem4  47078  isubgredg  47852