Theorem funforn 6035

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 5833	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6034	. 2 ⊢ (𝐴 Fn dom 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)
3	1, 2	bitri 263	1 ⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)

Syntax hints:   ↔ wb 195  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  –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-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-cleq 2603  df-fn 5807  df-fo 5810

This theorem is referenced by:  fimacnvinrn  6256  imacosupp  7222  ordtypelem8  8313  wdomima2g  8374  imadomg  9237  gruima  9503  oppglsm  17880  1stcrestlem  21065  dfac14  21231  qtoptop2  21312