Metamath Proof Explorer

Theorem ineccnvmo2

Description: Equivalence of a double universal quantification restricted to the range and an "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 4-Sep-2021)

		Ref	Expression
	Assertion	ineccnvmo2	\|- ( A. x e. ran F A. y e. ran F ( x = y \/ ( [ x ] `' F i^i [ y ] `' F ) = (/) ) <-> A. u E* x u F x )

Proof

Step	Hyp	Ref	Expression
1		ineccnvmo	\|- ( A. x e. ran F A. y e. ran F ( x = y \/ ( [ x ] `' F i^i [ y ] `' F ) = (/) ) <-> A. u E* x e. ran F u F x )
2		alrmomorn	\|- ( A. u E* x e. ran F u F x <-> A. u E* x u F x )
3	1 2	bitri	\|- ( A. x e. ran F A. y e. ran F ( x = y \/ ( [ x ] `' F i^i [ y ] `' F ) = (/) ) <-> A. u E* x u F x )

Step

Hyp

Ref

Expression

ineccnvmo

 |-  ( A. x e. ran F A. y e. ran F ( x = y \/ ( [ x ] `' F i^i [ y ] `' F ) = (/) ) <-> A. u E* x e. ran F u F x )

alrmomorn

 |-  ( A. u E* x e. ran F u F x <-> A. u E* x u F x )

1 2

bitri

 |-  ( A. x e. ran F A. y e. ran F ( x = y \/ ( [ x ] `' F i^i [ y ] `' F ) = (/) ) <-> A. u E* x u F x )