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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 (FG))

Proof of Theorem funin
StepHypRef Expression
1 inss1 3475 . 2 (FG) F
2 funss 5126 . 2 ((FG) F → (Fun F → Fun (FG)))
31, 2ax-mp 8 1 (Fun F → Fun (FG))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∩ cin 3208   ⊆ 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)
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