Theorem fnssresd 6613

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 6612	. 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 3898   ↾ cres 5623   Fn wfn 6484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-res 5633  df-fun 6491  df-fn 6492

This theorem is referenced by:  rescnvimafod  7015  fssrescdmd  7068  fpwwe2lem7  10539  pfxccat1  14616  mdetrsca  22538  2ndresdju  32653  fdifsupp  32690  fdifsuppconst  32694  ply1gsumz  33608  esplyind  33659  dimkerim  33712  rmulccn  34013  subfacp1lem3  35298  satfn  35471  eqresfnbd  42403  tfsconcatrev  43505  ofoafg  43511  xlimconst2  45995  dvnprodlem1  46106  fcoreslem4  47228  isubgredg  48028