MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funin Structured version   Visualization version   GIF version

Theorem funin 6629
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 4229 . 2 (𝐹𝐺) ⊆ 𝐹
2 funss 6572 . 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 3946  wss 3947  Fun wfun 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-in 3954  df-ss 3964  df-br 5149  df-opab 5211  df-rel 5685  df-cnv 5686  df-co 5687  df-fun 6550
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator