Theorem funforn 6746

Description: A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)

Assertion

Ref	Expression
funforn	⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)

Proof of Theorem funforn

Step	Hyp	Ref	Expression
1		funfn 6515	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6745	. 2 ⊢ (𝐴 Fn dom 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)
3	1, 2	bitri 276	1 ⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)

Syntax hints:   ↔ wb 207  dom cdm 5618  ran crn 5619  Fun wfun 6479   Fn wfn 6480  –onto→wfo 6483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-cleq 2731  df-fn 6488  df-fo 6491

This theorem is referenced by:  fimacnvinrn  7012  imacosupp  8149  ordtypelem8  9430  wdomima2g  9491  imadomg  10447  gruima  10716  oppglsm  19608  1stcrestlem  23435  dfac14  23601  qtoptop2  23682  fsupprnfi  32784  imadomfi  42487  rn1st  45717