Theorem fofun 6566

Description: An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)

Assertion

Ref	Expression
fofun	⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹)

Proof of Theorem fofun

Step	Hyp	Ref	Expression
1		fof 6565	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2	1	ffund 6491	1 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹)

Syntax hints:   → wi 4  Fun wfun 6318  –onto→wfo 6322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-fn 6327  df-f 6328  df-fo 6330

This theorem is referenced by:  foimacnv  6607  resdif  6610  fococnv2  6615  fornex  7639  fodomfi2  9471  fin1a2lem7  9817  brdom3  9939  1stf1  17434  1stf2  17435  2ndf1  17437  2ndf2  17438  1stfcl  17439  2ndfcl  17440  qtopcld  22318  qtopcmap  22324  elfm3  22555  bcthlem4  23931  uniiccdif  24182  grporn  28304  xppreima  30408  fsuppcurry1  30487  fsuppcurry2  30488  qtophaus  31189  bdayimaon  33310  nosupno  33316  bdayfun  33355  poimirlem26  35083  poimirlem27  35084  ovoliunnfl  35099  voliunnfl  35101