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Theorem fnresin1 6685
Description: Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
Assertion
Ref Expression
fnresin1 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))

Proof of Theorem fnresin1
StepHypRef Expression
1 inss1 4231 . 2 (𝐴𝐵) ⊆ 𝐴
2 fnssres 6683 . 2 ((𝐹 Fn 𝐴 ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))
31, 2mpan2 689 1 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cin 3948  wss 3949  cres 5684   Fn wfn 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-res 5694  df-fun 6555  df-fn 6556
This theorem is referenced by:  frrlem4  8301  wfrlem4OLD  8339  fnresin  32432
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