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Theorem funin 6577
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 4188 . 2 (𝐹𝐺) ⊆ 𝐹
2 funss 6520 . 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 3909  wss 3910  Fun wfun 6490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3447  df-in 3917  df-ss 3927  df-br 5106  df-opab 5168  df-rel 5640  df-cnv 5641  df-co 5642  df-fun 6498
This theorem is referenced by: (None)
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