Theorem fnssresd 6657

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 6656	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵)
4	1, 2, 3	syl2anc 595	1 ⊢ (𝜑 → (𝐹 ↾ 𝐵) Fn 𝐵)

Syntax hints:   → wi 4   ⊆ wss 3913   ↾ cres 5661   Fn wfn 6528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-res 5671  df-fun 6535  df-fn 6536

This theorem is referenced by:  rescnvimafod  7066  fssrescdmd  7120  fpwwe2lem7  10618  pfxccat1  14735  mdetrsca  22725  2ndresdju  32931  fdifsupp  32967  fdifsuppconst  32971  ply1gsumz  33830  esplyind  33906  dimkerim  33958  rmulccn  34259  subfacp1lem3  35569  satfn  35742  eqresfnbd  42886  tfsconcatrev  43960  ofoafg  43966  xlimconst2  46434  dvnprodlem1  46545  fcoreslem4  47685  isubgredg  48513