f1oen4g

Description: The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024)

		Ref	Expression
	Assertion	f1oen4g	$⊢ ((F \in V \land A \in W \land B \in X) \land F : A \underset{1-1 onto}{⟶} B) \to A \approx B$

Proof

Step	Hyp	Ref	Expression
1		f1oeq1	$⊢ f = F \to (f : A \underset{1-1 onto}{⟶} B \leftrightarrow F : A \underset{1-1 onto}{⟶} B)$
2	1	spcegv	$⊢ F \in V \to (F : A \underset{1-1 onto}{⟶} B \to \exists f f : A \underset{1-1 onto}{⟶} B)$
3	2	imp	$⊢ (F \in V \land F : A \underset{1-1 onto}{⟶} B) \to \exists f f : A \underset{1-1 onto}{⟶} B$
4	3	3ad2antl1	$⊢ ((F \in V \land A \in W \land B \in X) \land F : A \underset{1-1 onto}{⟶} B) \to \exists f f : A \underset{1-1 onto}{⟶} B$
5		breng	$⊢ (A \in W \land B \in X) \to (A \approx B \leftrightarrow \exists f f : A \underset{1-1 onto}{⟶} B)$
6	5	3adant1	$⊢ (F \in V \land A \in W \land B \in X) \to (A \approx B \leftrightarrow \exists f f : A \underset{1-1 onto}{⟶} B)$
7	6	adantr	$⊢ ((F \in V \land A \in W \land B \in X) \land F : A \underset{1-1 onto}{⟶} B) \to (A \approx B \leftrightarrow \exists f f : A \underset{1-1 onto}{⟶} B)$
8	4 7	mpbird	$⊢ ((F \in V \land A \in W \land B \in X) \land F : A \underset{1-1 onto}{⟶} B) \to A \approx B$

Metamath Proof Explorer

Theorem f1oen4g

Proof