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Theorem funin 5238
 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 3327 . 2 (𝐹𝐺) ⊆ 𝐹
2 funss 5186 . 2 ((𝐹𝐺) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹𝐺)))
31, 2ax-mp 5 1 (Fun 𝐹 → Fun (𝐹𝐺))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∩ cin 3101   ⊆ wss 3102  Fun wfun 5161 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-br 3966  df-opab 4026  df-rel 4590  df-cnv 4591  df-co 4592  df-fun 5169 This theorem is referenced by: (None)
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