dff1o4

Description: Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998) (Proof shortened by Andrew Salmon, 22-Oct-2011)

		Ref	Expression
	Assertion	dff1o4	$⊢ F : A \underset{1-1 onto}{⟶} B \leftrightarrow (F Fn A \land F^{-1} Fn B)$

Step	Hyp	Ref	Expression
1		dff1o2	$⊢ F : A \underset{1-1 onto}{⟶} B \leftrightarrow (F Fn A \land Fun F^{-1} \land ran F = B)$
2		3anass	$⊢ (F Fn A \land Fun F^{-1} \land ran F = B) \leftrightarrow (F Fn A \land (Fun F^{-1} \land ran F = B))$
3		df-rn	$⊢ ran F = dom F^{-1}$
4	3	eqeq1i	$⊢ ran F = B \leftrightarrow dom F^{-1} = B$
5	4	anbi2i	$⊢ (Fun F^{-1} \land ran F = B) \leftrightarrow (Fun F^{-1} \land dom F^{-1} = B)$
6		df-fn	$⊢ F^{-1} Fn B \leftrightarrow (Fun F^{-1} \land dom F^{-1} = B)$
7	5 6	bitr4i	$⊢ (Fun F^{-1} \land ran F = B) \leftrightarrow F^{-1} Fn B$
8	7	anbi2i	$⊢ (F Fn A \land (Fun F^{-1} \land ran F = B)) \leftrightarrow (F Fn A \land F^{-1} Fn B)$
9	1 2 8	3bitri	$⊢ F : A \underset{1-1 onto}{⟶} B \leftrightarrow (F Fn A \land F^{-1} Fn B)$

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