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Theorem funin 5267
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 3347 . 2  |-  ( F  i^i  G )  C_  F
2 funss 5215 . 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 set class
Syntax hints:    -> wi 4    i^i cin 3120    C_ wss 3121   Fun wfun 5190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-br 3988  df-opab 4049  df-rel 4616  df-cnv 4617  df-co 4618  df-fun 5198
This theorem is referenced by: (None)
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