Theorem fnssresd 6624

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 6623	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵)
4	1, 2, 3	syl2anc 585	1 ⊢ (𝜑 → (𝐹 ↾ 𝐵) Fn 𝐵)

Syntax hints:   → wi 4   ⊆ wss 3903   ↾ cres 5634   Fn wfn 6495

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  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-res 5644  df-fun 6502  df-fn 6503

This theorem is referenced by:  rescnvimafod  7027  fssrescdmd  7081  fpwwe2lem7  10560  pfxccat1  14637  mdetrsca  22559  2ndresdju  32738  fdifsupp  32774  fdifsuppconst  32778  ply1gsumz  33691  esplyind  33751  dimkerim  33804  rmulccn  34105  subfacp1lem3  35395  satfn  35568  eqresfnbd  42601  tfsconcatrev  43702  ofoafg  43708  xlimconst2  46190  dvnprodlem1  46301  fcoreslem4  47423  isubgredg  48223