Theorem funin 5152

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

Step	Hyp	Ref	Expression
1		inss1 3262	. 2 |- ( F i^i G ) C_ F
2		funss 5100	. 2 |- ( ( F i^i G ) C_ F -> ( Fun F -> Fun ( F i^i G ) ) )
3	1, 2	ax-mp 7	1 |- ( Fun F -> Fun ( F i^i G ) )

Syntax hints:   -> wi 4   i^i cin 3036   C_ wss 3037   Fun wfun 5075

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-in 3043  df-ss 3050  df-br 3896  df-opab 3950  df-rel 4506  df-cnv 4507  df-co 4508  df-fun 5083

This theorem is referenced by: (None)