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Theorem funin 6406
 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 4183 . 2 (𝐹𝐺) ⊆ 𝐹
2 funss 6350 . 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 3912   ⊆ wss 3913  Fun wfun 6325 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-v 3475  df-in 3920  df-ss 3930  df-br 5043  df-opab 5105  df-rel 5538  df-cnv 5539  df-co 5540  df-fun 6333 This theorem is referenced by: (None)
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