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

Theorem funin 5401
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 3465 . 2  |-  ( F  i^i  G )  C_  F
2 funss 5355 . 2  |-  ( ( F  i^i  G ) 
C_  F  ->  ( Fun  F  ->  Fun  ( F  i^i  G ) ) )
31, 2ax-mp 8 1  |-  ( Fun 
F  ->  Fun  ( F  i^i  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    i^i cin 3227    C_ wss 3228   Fun wfun 5331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-v 2866  df-in 3235  df-ss 3242  df-br 4105  df-opab 4159  df-rel 4778  df-cnv 4779  df-co 4780  df-fun 5339
  Copyright terms: Public domain W3C validator