Metamath Proof Explorer

Theorem f1o2d

Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014)

	Ref	Expression
Hypotheses	f1od.1	$⊢ F = (x \in A ⟼ C)$
	f1o2d.2	$⊢ (φ \land x \in A) \to C \in B$
	f1o2d.3	$⊢ (φ \land y \in B) \to D \in A$
	f1o2d.4	$⊢ (φ \land (x \in A \land y \in B)) \to (x = D \leftrightarrow y = C)$
Assertion	f1o2d	$⊢ φ \to F : A \underset{1-1 onto}{⟶} B$

Step	Hyp	Ref	Expression
1		f1od.1	$⊢ F = (x \in A ⟼ C)$
2		f1o2d.2	$⊢ (φ \land x \in A) \to C \in B$
3		f1o2d.3	$⊢ (φ \land y \in B) \to D \in A$
4		f1o2d.4	$⊢ (φ \land (x \in A \land y \in B)) \to (x = D \leftrightarrow y = C)$
5	1 2 3 4	f1ocnv2d	$⊢ φ \to (F : A \underset{1-1 onto}{⟶} B \land F^{-1} = (y \in B ⟼ D))$
6	5	simpld	$⊢ φ \to F : A \underset{1-1 onto}{⟶} B$