Theorem funforn 6768

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 6536	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6767	. 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 5638  ran crn 5639  Fun wfun 6495   Fn wfn 6496  –onto→wfo 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-cleq 2723  df-fn 6504  df-fo 6507

This theorem is referenced by:  fimacnvinrn  7027  imacosupp  8145  ordtypelem8  9470  wdomima2g  9531  imadomg  10479  gruima  10747  oppglsm  19438  1stcrestlem  22840  dfac14  23006  qtoptop2  23087  fsupprnfi  31674  rn1st  43623