f1ocnv2d

Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015)

	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	f1ocnv2d	$⊢ φ \to (F : A \underset{1-1 onto}{⟶} B \land F^{-1} = (y \in B ⟼ D))$

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		eleq1a	$⊢ C \in B \to (y = C \to y \in B)$
6	2 5	syl	$⊢ (φ \land x \in A) \to (y = C \to y \in B)$
7	6	impr	$⊢ (φ \land (x \in A \land y = C)) \to y \in B$
8	4	biimpar	$⊢ ((φ \land (x \in A \land y \in B)) \land y = C) \to x = D$
9	8	exp42	$⊢ φ \to (x \in A \to (y \in B \to (y = C \to x = D)))$
10	9	com34	$⊢ φ \to (x \in A \to (y = C \to (y \in B \to x = D)))$
11	10	imp32	$⊢ (φ \land (x \in A \land y = C)) \to (y \in B \to x = D)$
12	7 11	jcai	$⊢ (φ \land (x \in A \land y = C)) \to (y \in B \land x = D)$
13		eleq1a	$⊢ D \in A \to (x = D \to x \in A)$
14	3 13	syl	$⊢ (φ \land y \in B) \to (x = D \to x \in A)$
15	14	impr	$⊢ (φ \land (y \in B \land x = D)) \to x \in A$
16	4	biimpa	$⊢ ((φ \land (x \in A \land y \in B)) \land x = D) \to y = C$
17	16	exp42	$⊢ φ \to (x \in A \to (y \in B \to (x = D \to y = C)))$
18	17	com23	$⊢ φ \to (y \in B \to (x \in A \to (x = D \to y = C)))$
19	18	com34	$⊢ φ \to (y \in B \to (x = D \to (x \in A \to y = C)))$
20	19	imp32	$⊢ (φ \land (y \in B \land x = D)) \to (x \in A \to y = C)$
21	15 20	jcai	$⊢ (φ \land (y \in B \land x = D)) \to (x \in A \land y = C)$
22	12 21	impbida	$⊢ φ \to ((x \in A \land y = C) \leftrightarrow (y \in B \land x = D))$
23	1 2 3 22	f1ocnvd	$⊢ φ \to (F : A \underset{1-1 onto}{⟶} B \land F^{-1} = (y \in B ⟼ D))$

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