Theorem funforn 6572

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 6354	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6571	. 2 ⊢ (𝐴 Fn dom 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)
3	1, 2	bitri 278	1 ⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)

Syntax hints:   ↔ wb 209  dom cdm 5519  ran crn 5520  Fun wfun 6318   Fn wfn 6319  –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-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-fn 6327  df-fo 6330

This theorem is referenced by:  fimacnvinrn  6817  imacosupp  7857  imacosuppOLD  7858  ordtypelem8  8973  wdomima2g  9034  imadomg  9945  gruima  10213  oppglsm  18759  1stcrestlem  22057  dfac14  22223  qtoptop2  22304  fsupprnfi  30452