Theorem funforn 6743

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 6512	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6742	. 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 5619  ran crn 5620  Fun wfun 6476   Fn wfn 6477  –onto→wfo 6480

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 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2721  df-fn 6485  df-fo 6488

This theorem is referenced by:  fimacnvinrn  7005  imacosupp  8142  ordtypelem8  9417  wdomima2g  9478  imadomg  10428  gruima  10696  oppglsm  19521  1stcrestlem  23337  dfac14  23503  qtoptop2  23584  fsupprnfi  32634  imadomfi  41975  rn1st  45251