Theorem funforn 6797

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 6566	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6796	. 2 ⊢ (𝐴 Fn dom 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)
3	1, 2	bitri 275	1 ⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)

Syntax hints:   ↔ wb 206  dom cdm 5654  ran crn 5655  Fun wfun 6525   Fn wfn 6526  –onto→wfo 6529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2727  df-fn 6534  df-fo 6537

This theorem is referenced by:  fimacnvinrn  7061  imacosupp  8208  ordtypelem8  9539  wdomima2g  9600  imadomg  10548  gruima  10816  oppglsm  19623  1stcrestlem  23390  dfac14  23556  qtoptop2  23637  fsupprnfi  32669  imadomfi  42015  rn1st  45297