Theorem funforn 6803

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 6570	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6802	. 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)

This proof depends on syntax axioms:   ↔ wb 209  dom cdm 5663  ran crn 5664  Fun wfun 6534   Fn wfn 6535  –onto→wfo 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2757  df-fn 6543  df-fo 6546

This theorem is used by:  fimacnvinrn  7070  imacosupp  8211  ordtypelem8  9494  wdomima2g  9555  imadomg  10533  gruima  10804  oppglsm  19758  1stcrestlem  23661  dfac14  23828  qtoptop2  23909  fsupprnfi  33110  imadomfi  42829  rn1st  46048