Theorem funforn 6799

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 6798	. 2 ⊢ (𝐴 Fn dom 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)
3	1, 2	bitri 278	1 ⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)

Syntax hints:   ↔ wb 209  dom cdm 5661  ran crn 5662  Fun wfun 6530   Fn wfn 6531  –onto→wfo 6534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-cleq 2755  df-fn 6539  df-fo 6542

This theorem is referenced by:  fimacnvinrn  7066  imacosupp  8201  ordtypelem8  9483  wdomima2g  9544  imadomg  10513  gruima  10782  oppglsm  19707  1stcrestlem  23609  dfac14  23775  qtoptop2  23856  fsupprnfi  33037  imadomfi  42769  rn1st  45988