Theorem funforn 6751

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 6520	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6750	. 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 5622  ran crn 5623  Fun wfun 6484   Fn wfn 6485  –onto→wfo 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2726  df-fn 6493  df-fo 6496

This theorem is referenced by:  fimacnvinrn  7014  imacosupp  8149  ordtypelem8  9428  wdomima2g  9489  imadomg  10442  gruima  10711  oppglsm  19569  1stcrestlem  23394  dfac14  23560  qtoptop2  23641  fsupprnfi  32720  imadomfi  42195  rn1st  45459