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Theorem fnresin2 6628
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 4190 . 2 (𝐵𝐴) ⊆ 𝐴
2 fnssres 6625 . 2 ((𝐹 Fn 𝐴 ∧ (𝐵𝐴) ⊆ 𝐴) → (𝐹 ↾ (𝐵𝐴)) Fn (𝐵𝐴))
31, 2mpan2 690 1 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐵𝐴)) Fn (𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cin 3910  wss 3911  cres 5636   Fn wfn 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-fun 6499  df-fn 6500
This theorem is referenced by:  resfnfinfin  9279  resfifsupp  9338  hashresfn  14246
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