Theorem fnssresd 6662

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 6661	. 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 3926   ↾ cres 5656   Fn wfn 6526

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  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-res 5666  df-fun 6533  df-fn 6534

This theorem is referenced by:  rescnvimafod  7063  fssrescdmd  7116  fpwwe2lem7  10651  pfxccat1  14720  mdetrsca  22541  2ndresdju  32627  fdifsupp  32662  fdifsuppconst  32666  ply1gsumz  33608  dimkerim  33667  rmulccn  33959  subfacp1lem3  35204  satfn  35377  eqresfnbd  42283  tfsconcatrev  43372  ofoafg  43378  xlimconst2  45864  dvnprodlem1  45975  fcoreslem4  47095  isubgredg  47879