Metamath Proof Explorer

Theorem f1oen2g

Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015)

		Ref	Expression
	Assertion	f1oen2g	$⊢ (A \in V \land B \in W \land F : A \underset{1-1 onto}{⟶} B) \to A \approx B$

Proof

Step	Hyp	Ref	Expression
1		f1of	$⊢ F : A \underset{1-1 onto}{⟶} B \to F : A ⟶ B$
2		fex2	$⊢ (F : A ⟶ B \land A \in V \land B \in W) \to F \in V$
3	1 2	syl3an1	$⊢ (F : A \underset{1-1 onto}{⟶} B \land A \in V \land B \in W) \to F \in V$
4	3	3coml	$⊢ (A \in V \land B \in W \land F : A \underset{1-1 onto}{⟶} B) \to F \in V$
5		simp3	$⊢ (A \in V \land B \in W \land F : A \underset{1-1 onto}{⟶} B) \to F : A \underset{1-1 onto}{⟶} B$
6		f1oen3g	$⊢ (F \in V \land F : A \underset{1-1 onto}{⟶} B) \to A \approx B$
7	4 5 6	syl2anc	$⊢ (A \in V \land B \in W \land F : A \underset{1-1 onto}{⟶} B) \to A \approx B$