Theorem funforn 2792

Description: A function maps its domain onto its range.

Assertion

Ref	Expression
funforn	⊢ (Fun A ↔ A:dom A–onto→ran A)

Proof of Theorem funforn

Step	Hyp	Ref	Expression
1		cleqid 1102	. . . 4 ⊢ dom A = dom A
2	1	biantru 543	. . 3 ⊢ (Fun A ↔ (Fun A ∧ dom A = dom A))
3		df-fn 2433	. . 3 ⊢ (A Fn dom A ↔ (Fun A ∧ dom A = dom A))
4	2, 3	bitr4 154	. 2 ⊢ (Fun A ↔ A Fn dom A)
5		fnforn 2791	. 2 ⊢ (A Fn dom A ↔ A:dom A–onto→ran A)
6	4, 5	bitr 151	1 ⊢ (Fun A ↔ A:dom A–onto→ran A)

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091  dom cdm 2410  ran crn 2411  Fun wfun 2416   Fn wfn 2417  –onto→wfo 2420

This theorem is referenced by:  imadomg 3616

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-gen 677  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-cleq 1097  df-fn 2433  df-fo 2436

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