Theorem 1cossxrncnvepresex

Description: Sufficient condition for a restricted converse epsilon range Cartesian product to be a set. (Contributed by Peter Mazsa, 23-Sep-2021)

	Ref	Expression
Assertion	1cossxrncnvepresex	|- ( ( A e. V /\ R e. W ) -> ,~ ( R |X. ( `' _E |` A ) ) e. _V )

Proof

Step	Hyp	Ref	Expression
1		xrncnvepresex	|- ( ( A e. V /\ R e. W ) -> ( R |X. ( `' _E |` A ) ) e. _V )
2		cossex	|- ( ( R |X. ( `' _E |` A ) ) e. _V -> ,~ ( R |X. ( `' _E |` A ) ) e. _V )
3	1 2	syl	|- ( ( A e. V /\ R e. W ) -> ,~ ( R |X. ( `' _E |` A ) ) e. _V )