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Metamath Proof Explorer

Theorem dfdisjALTV

Description: Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, see the comment of df-disjs why we need disjoint relations instead of converse functions anyway. (Contributed by Peter Mazsa, 27-Jul-2021)

		Ref	Expression
	Assertion	dfdisjALTV	$⊢ Disj R \leftrightarrow (FunALTV R^{-1} \land Rel R)$

Proof

Step	Hyp	Ref	Expression
1		df-disjALTV	$⊢ Disj R \leftrightarrow (CnvRefRel ≀ R^{-1} \land Rel R)$
2		relcnv	$⊢ Rel R^{-1}$
3		df-funALTV	$⊢ FunALTV R^{-1} \leftrightarrow (CnvRefRel ≀ R^{-1} \land Rel R^{-1})$
4	2 3	mpbiran2	$⊢ FunALTV R^{-1} \leftrightarrow CnvRefRel ≀ R^{-1}$
5	4	anbi1i	$⊢ (FunALTV R^{-1} \land Rel R) \leftrightarrow (CnvRefRel ≀ R^{-1} \land Rel R)$
6	1 5	bitr4i	$⊢ Disj R \leftrightarrow (FunALTV R^{-1} \land Rel R)$