Theorem funforn 6596

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 6384	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6595	. 2 ⊢ (𝐴 Fn dom 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)
3	1, 2	bitri 277	1 ⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)

Syntax hints:   ↔ wb 208  dom cdm 5554  ran crn 5555  Fun wfun 6348   Fn wfn 6349  –onto→wfo 6352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-9 2120  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-cleq 2814  df-fn 6357  df-fo 6360

This theorem is referenced by:  fimacnvinrn  6839  imacosupp  7873  imacosuppOLD  7874  ordtypelem8  8988  wdomima2g  9049  imadomg  9955  gruima  10223  oppglsm  18766  1stcrestlem  22059  dfac14  22225  qtoptop2  22306