Theorem iunxsnf

Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	iunxsnf.1	|- F/_ x C
	iunxsnf.2	|- A e. _V
	iunxsnf.3	|- ( x = A -> B = C )
Assertion	iunxsnf	|- U_ x e. { A } B = C

Proof

Step	Hyp	Ref	Expression
1		iunxsnf.1	|- F/_ x C
2		iunxsnf.2	|- A e. _V
3		iunxsnf.3	|- ( x = A -> B = C )
4	1 3	iunxsngf	|- ( A e. _V -> U_ x e. { A } B = C )
5	2 4	ax-mp	|- U_ x e. { A } B = C