MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fofun Structured version   Visualization version   GIF version

Theorem fofun 6806
Description: An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
Assertion
Ref Expression
fofun (𝐹:𝐴onto𝐵 → Fun 𝐹)

Proof of Theorem fofun
StepHypRef Expression
1 fof 6805 . 2 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
21ffund 6720 1 (𝐹:𝐴onto𝐵 → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  Fun wfun 6536  ontowfo 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3471  df-in 3951  df-ss 3961  df-fn 6545  df-f 6546  df-fo 6548
This theorem is referenced by:  foco  6819  foimacnv  6850  resdif  6854  fococnv2  6859  focdmex  7953  fodomfi2  10075  fin1a2lem7  10421  brdom3  10543  1stf1  18174  1stf2  18175  2ndf1  18177  2ndf2  18178  1stfcl  18179  2ndfcl  18180  qtopcld  23604  qtopcmap  23610  elfm3  23841  bcthlem4  25242  uniiccdif  25494  bdayimaon  27613  nosupno  27623  noinfno  27638  bdayfun  27692  noeta2  27704  precsexlem10  28101  precsexlem11  28102  grporn  30318  xppreima  32415  fsuppcurry1  32491  fsuppcurry2  32492  qtophaus  33373  poimirlem26  37054  poimirlem27  37055  ovoliunnfl  37070  voliunnfl  37072
  Copyright terms: Public domain W3C validator