Theorem xpima2

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

	Ref	Expression
Assertion	xpima2	$$⊢A\cap C\ne \varnothing \to \left(A\times B\right)\left[C\right]=B$$

Proof

Step	Hyp	Ref	Expression
1		xpima	$$⊢\left(A\times B\right)\left[C\right]=if\left(A\cap C=\varnothing ,\varnothing ,B\right)$$
2		ifnefalse	$$⊢A\cap C\ne \varnothing \to if\left(A\cap C=\varnothing ,\varnothing ,B\right)=B$$
3	1 2	eqtrid	$$⊢A\cap C\ne \varnothing \to \left(A\times B\right)\left[C\right]=B$$