Theorem iotajust

Description: Soundness justification theorem for df-iota . (Contributed by Andrew Salmon, 29-Jun-2011)

	Ref	Expression
Assertion	iotajust	$$⊢\bigcup \left\{y|\left\{x|\phi \right\}=\left\{y\right\}\right\}=\bigcup \left\{z|\left\{x|\phi \right\}=\left\{z\right\}\right\}$$

Proof

Step	Hyp	Ref	Expression
1		sneq	$$⊢y=w\to \left\{y\right\}=\left\{w\right\}$$
2	1	eqeq2d	$$⊢y=w\to \left(\left\{x|\phi \right\}=\left\{y\right\}\leftrightarrow \left\{x|\phi \right\}=\left\{w\right\}\right)$$
3	2	cbvabv	$$⊢\left\{y|\left\{x|\phi \right\}=\left\{y\right\}\right\}=\left\{w|\left\{x|\phi \right\}=\left\{w\right\}\right\}$$
4		sneq	$$⊢w=z\to \left\{w\right\}=\left\{z\right\}$$
5	4	eqeq2d	$$⊢w=z\to \left(\left\{x|\phi \right\}=\left\{w\right\}\leftrightarrow \left\{x|\phi \right\}=\left\{z\right\}\right)$$
6	5	cbvabv	$$⊢\left\{w|\left\{x|\phi \right\}=\left\{w\right\}\right\}=\left\{z|\left\{x|\phi \right\}=\left\{z\right\}\right\}$$
7	3 6	eqtri	$$⊢\left\{y|\left\{x|\phi \right\}=\left\{y\right\}\right\}=\left\{z|\left\{x|\phi \right\}=\left\{z\right\}\right\}$$
8	7	unieqi	$$⊢\bigcup \left\{y|\left\{x|\phi \right\}=\left\{y\right\}\right\}=\bigcup \left\{z|\left\{x|\phi \right\}=\left\{z\right\}\right\}$$