Theorem sn-iotalemcor

Description: Corollary of sn-iotalem . Compare sb8iota . (Contributed by SN, 6-Nov-2024)

	Ref	Expression
Assertion	sn-iotalemcor	$$⊢\left(\iota x|\phi \right)=\left(\iota y|\left\{x|\phi \right\}=\left\{y\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		sn-iotalem	$$⊢\left\{y|\left\{x|\phi \right\}=\left\{y\right\}\right\}=\left\{z|\left\{y|\left\{x|\phi \right\}=\left\{y\right\}\right\}=\left\{z\right\}\right\}$$
2	1	unieqi	$$⊢\bigcup \left\{y|\left\{x|\phi \right\}=\left\{y\right\}\right\}=\bigcup \left\{z|\left\{y|\left\{x|\phi \right\}=\left\{y\right\}\right\}=\left\{z\right\}\right\}$$
3		df-iota	$$⊢\left(\iota x|\phi \right)=\bigcup \left\{y|\left\{x|\phi \right\}=\left\{y\right\}\right\}$$
4		df-iota	$$⊢\left(\iota y|\left\{x|\phi \right\}=\left\{y\right\}\right)=\bigcup \left\{z|\left\{y|\left\{x|\phi \right\}=\left\{y\right\}\right\}=\left\{z\right\}\right\}$$
5	2 3 4	3eqtr4i	$$⊢\left(\iota x|\phi \right)=\left(\iota y|\left\{x|\phi \right\}=\left\{y\right\}\right)$$