Theorem eccnvepres3

Description: Condition for a restricted converse epsilon coset of a set to be the set itself. (Contributed by Peter Mazsa, 11-May-2021)

	Ref	Expression
Assertion	eccnvepres3	|- ( B e. dom ( `' _E |` A ) -> [ B ] ( `' _E |` A ) = B )

Proof

Step	Hyp	Ref	Expression
1		resdmres	|- ( `' _E |` dom ( `' _E |` A ) ) = ( `' _E |` A )
2	1	eceq2i	|- [ B ] ( `' _E |` dom ( `' _E |` A ) ) = [ B ] ( `' _E |` A )
3		eccnvepres2	|- ( B e. dom ( `' _E |` A ) -> [ B ] ( `' _E |` dom ( `' _E |` A ) ) = B )
4	2 3	eqtr3id	|- ( B e. dom ( `' _E |` A ) -> [ B ] ( `' _E |` A ) = B )