New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  funres11 GIF version

Theorem funres11 5164
 Description: The restriction of a one-to-one function is one-to-one. (Contributed by set.mm contributors, 25-Mar-1998.)
Assertion
Ref Expression
funres11 (Fun F → Fun (F A))

Proof of Theorem funres11
StepHypRef Expression
1 resss 4988 . 2 (F A) F
2 cnvss 4885 . 2 ((F A) F(F A) F)
3 funss 5126 . 2 ((F A) F → (Fun F → Fun (F A)))
41, 2, 3mp2b 9 1 (Fun F → Fun (F A))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊆ wss 3257  ◡ccnv 4771   ↾ cres 4774  Fun wfun 4775 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-res 4788  df-fun 4789 This theorem is referenced by:  f1ores  5300  resdif  5306
 Copyright terms: Public domain W3C validator