Theorem funforn 6679

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 6448	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6678	. 2 ⊢ (𝐴 Fn dom 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)
3	1, 2	bitri 274	1 ⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)

Syntax hints:   ↔ wb 205  dom cdm 5580  ran crn 5581  Fun wfun 6412   Fn wfn 6413  –onto→wfo 6416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-cleq 2730  df-fn 6421  df-fo 6424

This theorem is referenced by:  fimacnvinrn  6931  imacosupp  7996  ordtypelem8  9214  wdomima2g  9275  imadomg  10221  gruima  10489  oppglsm  19162  1stcrestlem  22511  dfac14  22677  qtoptop2  22758  fsupprnfi  30928