Theorem funforn 6761

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 6530	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6760	. 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 5632  ran crn 5633  Fun wfun 6494   Fn wfn 6495  –onto→wfo 6498

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 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-fn 6503  df-fo 6506

This theorem is referenced by:  fimacnvinrn  7025  imacosupp  8161  ordtypelem8  9442  wdomima2g  9503  imadomg  10456  gruima  10725  oppglsm  19583  1stcrestlem  23408  dfac14  23574  qtoptop2  23655  fsupprnfi  32782  imadomfi  42372  rn1st  45631