Theorem funforn 6811

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 6577	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6810	. 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 5675  ran crn 5676  Fun wfun 6536   Fn wfn 6537  –onto→wfo 6540

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 1911  ax-6 1969  ax-7 2009  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-cleq 2722  df-fn 6545  df-fo 6548

This theorem is referenced by:  fimacnvinrn  7072  imacosupp  8196  ordtypelem8  9522  wdomima2g  9583  imadomg  10531  gruima  10799  oppglsm  19551  1stcrestlem  23176  dfac14  23342  qtoptop2  23423  fsupprnfi  32181  rn1st  44276