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Theorem funin 5879
 Description: The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53. (Contributed by NM, 19-Mar-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
funin (Fun 𝐹 → Fun (𝐹𝐺))

Proof of Theorem funin
StepHypRef Expression
1 inss1 3795 . 2 (𝐹𝐺) ⊆ 𝐹
2 funss 5822 . 2 ((𝐹𝐺) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹𝐺)))
31, 2ax-mp 5 1 (Fun 𝐹 → Fun (𝐹𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∩ cin 3539   ⊆ wss 3540  Fun wfun 5798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-fun 5806 This theorem is referenced by: (None)
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