Theorem funin 5163

Description: The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 19-Mar-2004.) (Revised by set.mm contributors, 18-Sep-2011.)

Assertion

Ref	Expression
funin	|- (Fun F -> Fun (F i^i G))

Proof of Theorem funin

Step	Hyp	Ref	Expression
1		inss1 3475	. 2 |- (F i^i G) C_ F
2		funss 5126	. 2 |- ((F i^i G) C_ F -> (Fun F -> Fun (F i^i G)))
3	1, 2	ax-mp 8	1 |- (Fun F -> Fun (F i^i G))

Syntax hints:   -> wi 4   i^i cin 3208   C_ wss 3257  Fun wfun 4775

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 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-opab 4623  df-br 4640  df-co 4726  df-cnv 4785  df-fun 4789

This theorem is referenced by: (None)