Theorem xpimasn

Description: Direct image of a singleton by a Cartesian product. (Contributed by Thierry Arnoux, 14-Jan-2018) (Proof shortened by BJ, 6-Apr-2019)

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

Proof

Step	Hyp	Ref	Expression
1		disjsn	$$⊢A\cap \left\{X\right\}=\varnothing \leftrightarrow \neg X\in A$$
2	1	necon3abii	$$⊢A\cap \left\{X\right\}\ne \varnothing \leftrightarrow \neg \neg X\in A$$
3		notnotb	$$⊢X\in A\leftrightarrow \neg \neg X\in A$$
4	2 3	bitr4i	$$⊢A\cap \left\{X\right\}\ne \varnothing \leftrightarrow X\in A$$
5		xpima2	$$⊢A\cap \left\{X\right\}\ne \varnothing \to \left(A\times B\right)\left[\left\{X\right\}\right]=B$$
6	4 5	sylbir	$$⊢X\in A\to \left(A\times B\right)\left[\left\{X\right\}\right]=B$$