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Metamath Proof Explorer

Theorem f1dom3g

Description: The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg does not require the Axiom of Replacement nor the Axiom of Power Sets. (Contributed by BTernaryTau, 9-Sep-2024)

		Ref	Expression
	Assertion	f1dom3g	$⊢ (F \in V \land B \in W \land F : A \underset{1-1}{⟶} B) \to A ≼ B$

Proof

Step	Hyp	Ref	Expression
1		f1eq1	$⊢ f = F \to (f : A \underset{1-1}{⟶} B \leftrightarrow F : A \underset{1-1}{⟶} B)$
2	1	spcegv	$⊢ F \in V \to (F : A \underset{1-1}{⟶} B \to \exists f f : A \underset{1-1}{⟶} B)$
3	2	imp	$⊢ (F \in V \land F : A \underset{1-1}{⟶} B) \to \exists f f : A \underset{1-1}{⟶} B$
4	3	3adant2	$⊢ (F \in V \land B \in W \land F : A \underset{1-1}{⟶} B) \to \exists f f : A \underset{1-1}{⟶} B$
5		brdomg	$⊢ B \in W \to (A ≼ B \leftrightarrow \exists f f : A \underset{1-1}{⟶} B)$
6	5	3ad2ant2	$⊢ (F \in V \land B \in W \land F : A \underset{1-1}{⟶} B) \to (A ≼ B \leftrightarrow \exists f f : A \underset{1-1}{⟶} B)$
7	4 6	mpbird	$⊢ (F \in V \land B \in W \land F : A \underset{1-1}{⟶} B) \to A ≼ B$