Metamath Proof Explorer

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 syl5eq 
 |-  ( ( A i^i C ) = (/) -> ( ( A X. B ) " C ) = (/) )