Theorem xpima1

Description: Direct image by a Cartesian product (case of empty intersection with the domain). (Contributed by Thierry Arnoux, 16-Dec-2017)

	Ref	Expression
Assertion	xpima1	|- ( ( A i^i C ) = (/) -> ( ( A X. B ) " C ) = (/) )

Proof

Step	Hyp	Ref	Expression
1		xpima	|- ( ( A X. B ) " C ) = if ( ( A i^i C ) = (/) , (/) , B )
2		iftrue	|- ( ( A i^i C ) = (/) -> if ( ( A i^i C ) = (/) , (/) , B ) = (/) )
3	1 2	eqtrid	|- ( ( A i^i C ) = (/) -> ( ( A X. B ) " C ) = (/) )