Theorem funforn 6813

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 6579	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6812	. 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 5677  ran crn 5678  Fun wfun 6538   Fn wfn 6539  –onto→wfo 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-cleq 2725  df-fn 6547  df-fo 6550

This theorem is referenced by:  fimacnvinrn  7074  imacosupp  8194  ordtypelem8  9520  wdomima2g  9581  imadomg  10529  gruima  10797  oppglsm  19510  1stcrestlem  22956  dfac14  23122  qtoptop2  23203  fsupprnfi  31914  rn1st  43978