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Theorem fnresin2 6694
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 4238 . 2 (𝐵𝐴) ⊆ 𝐴
2 fnssres 6691 . 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 3950  wss 3951  cres 5687   Fn wfn 6556
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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-fun 6563  df-fn 6564
This theorem is referenced by:  resfnfinfin  9377  resfifsupp  9437  hashresfn  14379
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