Theorem bj-xpima2sn

Description: The image of a singleton by a direct product, nonempty case. [To replace xpimasn .] (Contributed by BJ, 6-Apr-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-xpima2sn	$$⊢X\in A\to \left(A\times B\right)\left[\left\{X\right\}\right]=B$$

Proof

Step	Hyp	Ref	Expression
1		bj-xpimasn	$$⊢\left(A\times B\right)\left[\left\{X\right\}\right]=if\left(X\in A,B,\varnothing \right)$$
2		iftrue	$$⊢X\in A\to if\left(X\in A,B,\varnothing \right)=B$$
3	1 2	syl5eq	$$⊢X\in A\to \left(A\times B\right)\left[\left\{X\right\}\right]=B$$