Theorem funforn 6812

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 6811	. 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 205  dom cdm 5672  ran crn 5673  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 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-cleq 2719  df-fn 6545  df-fo 6548

This theorem is referenced by:  fimacnvinrn  7075  imacosupp  8208  ordtypelem8  9540  wdomima2g  9601  imadomg  10549  gruima  10817  oppglsm  19588  1stcrestlem  23343  dfac14  23509  qtoptop2  23590  fsupprnfi  32456  imadomfi  41410  rn1st  44573