Metamath Proof Explorer

Theorem dfdisjALTV5

Description: Alternate definition of the disjoint relation predicate, cf. dffunALTV5 . (Contributed by Peter Mazsa, 5-Sep-2021)

		Ref	Expression
	Assertion	dfdisjALTV5	\|- ( Disj R <-> ( A. u e. dom R A. v e. dom R ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) )

Step	Hyp	Ref	Expression
1		dfdisjALTV2	\|- ( Disj R <-> ( ,~ `' R C_ _I /\ Rel R ) )
2		cosscnvssid5	\|- ( ( ,~ `' R C_ _I /\ Rel R ) <-> ( A. u e. dom R A. v e. dom R ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) )
3	1 2	bitri	\|- ( Disj R <-> ( A. u e. dom R A. v e. dom R ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) )

Step

Hyp

Ref

Expression

 |-  ( Disj R <-> ( ,~ `' R C_ _I /\ Rel R ) )

 |-  ( ( ,~ `' R C_ _I /\ Rel R ) <-> ( A. u e. dom R A. v e. dom R ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) )

1 2

 |-  ( Disj R <-> ( A. u e. dom R A. v e. dom R ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) )