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Metamath Proof Explorer

Theorem f1dmex

Description: If the codomain of a one-to-one function exists, so does its domain. This theorem is equivalent to the Axiom of Replacement ax-rep . (Contributed by NM, 4-Sep-2004)

		Ref	Expression
	Assertion	f1dmex	$⊢ (F : A \underset{1-1}{⟶} B \land B \in C) \to A \in V$

Proof

Step	Hyp	Ref	Expression
1		f1f	$⊢ F : A \underset{1-1}{⟶} B \to F : A ⟶ B$
2	1	frnd	$⊢ F : A \underset{1-1}{⟶} B \to ran F \subseteq B$
3		ssexg	$⊢ (ran F \subseteq B \land B \in C) \to ran F \in V$
4	2 3	sylan	$⊢ (F : A \underset{1-1}{⟶} B \land B \in C) \to ran F \in V$
5	4	ex	$⊢ F : A \underset{1-1}{⟶} B \to (B \in C \to ran F \in V)$
6		f1cnv	$⊢ F : A \underset{1-1}{⟶} B \to F^{-1} : ran F \underset{1-1 onto}{⟶} A$
7		f1ofo	$⊢ F^{-1} : ran F \underset{1-1 onto}{⟶} A \to F^{-1} : ran F \underset{onto}{⟶} A$
8	6 7	syl	$⊢ F : A \underset{1-1}{⟶} B \to F^{-1} : ran F \underset{onto}{⟶} A$
9		focdmex	$⊢ ran F \in V \to (F^{-1} : ran F \underset{onto}{⟶} A \to A \in V)$
10	5 8 9	syl6ci	$⊢ F : A \underset{1-1}{⟶} B \to (B \in C \to A \in V)$
11	10	imp	$⊢ (F : A \underset{1-1}{⟶} B \land B \in C) \to A \in V$