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Theorem fnresin1 6166
 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 3976 . 2 (𝐴𝐵) ⊆ 𝐴
2 fnssres 6165 . 2 ((𝐹 Fn 𝐴 ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))
31, 2mpan2 709 1 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∩ cin 3714   ⊆ wss 3715   ↾ cres 5268   Fn wfn 6044 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-res 5278  df-fun 6051  df-fn 6052 This theorem is referenced by:  wfrlem4  7587  fnresin  29739  frrlem4  32089
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