Theorem bj-xpima1sn

Description: The image of a singleton by a direct product, empty case. [Change and relabel xpimasn accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019)

	Ref	Expression
Assertion	bj-xpima1sn	|- ( -. X e. A -> ( ( A X. B ) " { X } ) = (/) )

Proof

Step	Hyp	Ref	Expression
1		bj-xpimasn	|- ( ( A X. B ) " { X } ) = if ( X e. A , B , (/) )
2		iffalse	|- ( -. X e. A -> if ( X e. A , B , (/) ) = (/) )
3	1 2	eqtrid	|- ( -. X e. A -> ( ( A X. B ) " { X } ) = (/) )