f1domfi2

Description: If the domain of a one-to-one function is finite, then the function's domain is dominated by its codomain when the latter is a set. This theorem is proved without using the Axiom of Power Sets (unlike f1dom2g ). (Contributed by BTernaryTau, 24-Nov-2024)

		Ref	Expression
	Assertion	f1domfi2	$⊢ (A \in Fin \land B \in V \land F : A \underset{1-1}{⟶} B) \to A ≼ B$

Proof

Step	Hyp	Ref	Expression
1		f1fn	$⊢ F : A \underset{1-1}{⟶} B \to F Fn A$
2		fnfi	$⊢ (F Fn A \land A \in Fin) \to F \in Fin$
3	1 2	sylan	$⊢ (F : A \underset{1-1}{⟶} B \land A \in Fin) \to F \in Fin$
4	3	ancoms	$⊢ (A \in Fin \land F : A \underset{1-1}{⟶} B) \to F \in Fin$
5	4	3adant2	$⊢ (A \in Fin \land B \in V \land F : A \underset{1-1}{⟶} B) \to F \in Fin$
6		f1dom3g	$⊢ (F \in Fin \land B \in V \land F : A \underset{1-1}{⟶} B) \to A ≼ B$
7	5 6	syld3an1	$⊢ (A \in Fin \land B \in V \land F : A \underset{1-1}{⟶} B) \to A ≼ B$

Metamath Proof Explorer

Theorem f1domfi2

Proof