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

Theorem fofun 6029
 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 6028 . 2 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
2 ffun 5961 . 2 (𝐹:𝐴𝐵 → Fun 𝐹)
31, 2syl 17 1 (𝐹:𝐴onto𝐵 → Fun 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  Fun wfun 5798  ⟶wf 5800  –onto→wfo 5802 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554  df-fn 5807  df-f 5808  df-fo 5810 This theorem is referenced by:  foimacnv  6067  resdif  6070  fococnv2  6075  fornex  7028  fodomfi2  8766  fin1a2lem7  9111  brdom3  9231  1stf1  16655  1stf2  16656  2ndf1  16658  2ndf2  16659  1stfcl  16660  2ndfcl  16661  qtopcld  21326  qtopcmap  21332  elfm3  21564  bcthlem4  22932  uniiccdif  23152  grporn  26759  xppreima  28829  qtophaus  29231  bdayfun  31075  poimirlem26  32605  poimirlem27  32606  ovoliunnfl  32621  voliunnfl  32623
 Copyright terms: Public domain W3C validator