Theorem funforn 6337

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 6130	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6336	. 2 ⊢ (𝐴 Fn dom 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)
3	1, 2	bitri 267	1 ⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)

Syntax hints:   ↔ wb 198  dom cdm 5311  ran crn 5312  Fun wfun 6094   Fn wfn 6095  –onto→wfo 6098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-ext 2776

This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-cleq 2791  df-fn 6103  df-fo 6106

This theorem is referenced by:  fimacnvinrn  6573  imacosupp  7572  ordtypelem8  8671  wdomima2g  8732  imadomg  9643  gruima  9911  oppglsm  18367  1stcrestlem  21581  dfac14  21747  qtoptop2  21828