Theorem funforn 5622

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 5407	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 5621	. 2 ⊢ (𝐴 Fn dom 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)
3	1, 2	bitri 184	1 ⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)

This proof depends on syntax axioms:   ↔ wb 105  dom cdm 4774  ran crn 4775  Fun wfun 5371   Fn wfn 5372  –onto→wfo 5375

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1502  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-fn 5380  df-fo 5383

This theorem is used by:  imacosuppfn  6508  dvrecap  15814