Theorem inecmo3

Description: Equivalence of a double universal quantification restricted to the domain and an "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 5-Sep-2021)

	Ref	Expression
Assertion	inecmo3	|- ( ( A. u e. dom R A. v e. dom R ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) <-> ( A. x E* u u R x /\ Rel R ) )

Proof

Step	Hyp	Ref	Expression
1		inecmo2	|- ( ( A. u e. dom R A. v e. dom R ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) <-> ( A. x E* u e. dom R u R x /\ Rel R ) )
2		alrmomodm	|- ( Rel R -> ( A. x E* u e. dom R u R x <-> A. x E* u u R x ) )
3	2	pm5.32ri	|- ( ( A. x E* u e. dom R u R x /\ Rel R ) <-> ( A. x E* u u R x /\ Rel R ) )
4	1 3	bitri	|- ( ( A. u e. dom R A. v e. dom R ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) <-> ( A. x E* u u R x /\ Rel R ) )