Theorem funforn 6797

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 6564	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
2		dffn4 6796	. 2 ⊢ (𝐴 Fn dom 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)
3	1, 2	bitri 278	1 ⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)

This proof depends on syntax axioms:   ↔ wb 209  dom cdm 5655  ran crn 5656  Fun wfun 6527   Fn wfn 6528  –onto→wfo 6531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2752  df-fn 6536  df-fo 6539

This theorem is used by:  fimacnvinrn  7065  imacosupp  8208  ordtypelem8  9498  wdomima2g  9559  imadomg  10538  gruima  10812  oppglsm  19770  1stcrestlem  23678  dfac14  23845  qtoptop2  23926  fsupprnfi  33165  imadomfi  42869  rn1st  46103