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

Theorem funin 5479
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 3521 . 2  |-  ( F  i^i  G )  C_  F
2 funss 5431 . 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 3279    C_ wss 3280   Fun wfun 5407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-in 3287  df-ss 3294  df-br 4173  df-opab 4227  df-rel 4844  df-cnv 4845  df-co 4846  df-fun 5415
  Copyright terms: Public domain W3C validator