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Theorem fnresin2 6695
Description: Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
Assertion
Ref Expression
fnresin2 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐵𝐴)) Fn (𝐵𝐴))

Proof of Theorem fnresin2
StepHypRef Expression
1 inss2 4246 . 2 (𝐵𝐴) ⊆ 𝐴
2 fnssres 6692 . 2 ((𝐹 Fn 𝐴 ∧ (𝐵𝐴) ⊆ 𝐴) → (𝐹 ↾ (𝐵𝐴)) Fn (𝐵𝐴))
31, 2mpan2 691 1 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐵𝐴)) Fn (𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cin 3962  wss 3963  cres 5691   Fn wfn 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-fun 6565  df-fn 6566
This theorem is referenced by:  resfnfinfin  9375  resfifsupp  9435  hashresfn  14376
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