Theorem iunsn

Description: Indexed union of a singleton. Compare dfiun2 and rnmpt . (Contributed by Steven Nguyen, 7-Jun-2023)

	Ref	Expression
Assertion	iunsn	$$⊢\bigcup _{x\in A}\left\{B\right\}=\left\{y|\exists x\in Ay=B\right\}$$

Proof

Step	Hyp	Ref	Expression
1		df-iun	$$⊢\bigcup _{x\in A}\left\{B\right\}=\left\{y|\exists x\in Ay\in \left\{B\right\}\right\}$$
2		velsn	$$⊢y\in \left\{B\right\}\leftrightarrow y=B$$
3	2	rexbii	$$⊢\exists x\in Ay\in \left\{B\right\}\leftrightarrow \exists x\in Ay=B$$
4	3	abbii	$$⊢\left\{y|\exists x\in Ay\in \left\{B\right\}\right\}=\left\{y|\exists x\in Ay=B\right\}$$
5	1 4	eqtri	$$⊢\bigcup _{x\in A}\left\{B\right\}=\left\{y|\exists x\in Ay=B\right\}$$