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

Theorem funin 5286
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  ( F  i^i  G ) )

Proof of Theorem funin
StepHypRef Expression
1 inss1 3392 . 2  |-  ( F  i^i  G )  C_  F
2 funss 5241 . 2  |-  ( ( F  i^i  G ) 
C_  F  ->  ( Fun  F  ->  Fun  ( F  i^i  G ) ) )
31, 2ax-mp 10 1  |-  ( Fun 
F  ->  Fun  ( F  i^i  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    i^i cin 3154    C_ wss 3155   Fun wfun 5217
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-v 2793  df-in 3162  df-ss 3169  df-br 4027  df-opab 4081  df-rel 4697  df-cnv 4698  df-co 4699  df-fun 5225
  Copyright terms: Public domain W3C validator