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Theorem funin 5661
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 3714 . 2  |-  ( F  i^i  G )  C_  F
2 funss 5612 . 2  |-  ( ( F  i^i  G ) 
C_  F  ->  ( Fun  F  ->  Fun  ( F  i^i  G ) ) )
31, 2ax-mp 5 1  |-  ( Fun 
F  ->  Fun  ( F  i^i  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    i^i cin 3470    C_ wss 3471   Fun wfun 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-in 3478  df-ss 3485  df-br 4457  df-opab 4516  df-rel 5015  df-cnv 5016  df-co 5017  df-fun 5596
This theorem is referenced by: (None)
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