f1oeng

Description: The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998)

		Ref	Expression
	Assertion	f1oeng	$⊢ (A \in C \land F : A \underset{1-1 onto}{⟶} B) \to A \approx B$

Step	Hyp	Ref	Expression
1		fornex	$⊢ A \in C \to (F : A \underset{onto}{⟶} B \to B \in V)$
2		f1ofo	$⊢ F : A \underset{1-1 onto}{⟶} B \to F : A \underset{onto}{⟶} B$
3	1 2	impel	$⊢ (A \in C \land F : A \underset{1-1 onto}{⟶} B) \to B \in V$
4		f1oen2g	$⊢ (A \in C \land B \in V \land F : A \underset{1-1 onto}{⟶} B) \to A \approx B$
5	4	3com23	$⊢ (A \in C \land F : A \underset{1-1 onto}{⟶} B \land B \in V) \to A \approx B$
6	3 5	mpd3an3	$⊢ (A \in C \land F : A \underset{1-1 onto}{⟶} B) \to A \approx B$

Metamath Proof Explorer