Theorem funforn 6827

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 6596	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6826	. 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 5685  ran crn 5686  Fun wfun 6555   Fn wfn 6556  –onto→wfo 6559

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 2007  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-fn 6564  df-fo 6567

This theorem is referenced by:  fimacnvinrn  7091  imacosupp  8234  ordtypelem8  9565  wdomima2g  9626  imadomg  10574  gruima  10842  oppglsm  19660  1stcrestlem  23460  dfac14  23626  qtoptop2  23707  fsupprnfi  32701  imadomfi  42003  rn1st  45280