Theorem bj-xpimasn

Description: The image of a singleton, general case. [Change and relabel xpimasn accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019)

	Ref	Expression
Assertion	bj-xpimasn	|- ( ( A X. B ) " { X } ) = if ( X e. A , B , (/) )

Proof

Step	Hyp	Ref	Expression
1		xpima	|- ( ( A X. B ) " { X } ) = if ( ( A i^i { X } ) = (/) , (/) , B )
2		disjsn	|- ( ( A i^i { X } ) = (/) <-> -. X e. A )
3		eqid	|- B = B
4	2 3	ifbieq2i	|- if ( ( A i^i { X } ) = (/) , (/) , B ) = if ( -. X e. A , (/) , B )
5		ifnot	|- if ( -. X e. A , (/) , B ) = if ( X e. A , B , (/) )
6	1 4 5	3eqtri	|- ( ( A X. B ) " { X } ) = if ( X e. A , B , (/) )