Theorem funforn 5347

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 5148	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 5346	. 2 ⊢ (𝐴 Fn dom 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)
3	1, 2	bitri 183	1 ⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)

Syntax hints:   ↔ wb 104  dom cdm 4534  ran crn 4535  Fun wfun 5112   Fn wfn 5113  –onto→wfo 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1425  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-cleq 2130  df-fn 5121  df-fo 5124

This theorem is referenced by:  dvrecap  12835