Theorem kqfval

Description: Value of the function appearing in df-kq . (Contributed by Mario Carneiro, 25-Aug-2015)

	Ref	Expression
Hypothesis	kqval.2	$$⊢F=\left(x\in X⟼\left\{y\in J|x\in y\right\}\right)$$
Assertion	kqfval	$$⊢\left(J\in V\wedge A\in X\right)\to F\left(A\right)=\left\{y\in J|A\in y\right\}$$

Proof

Step	Hyp	Ref	Expression
1		kqval.2	$$⊢F=\left(x\in X⟼\left\{y\in J|x\in y\right\}\right)$$
2		id	$$⊢A\in X\to A\in X$$
3		rabexg	$$⊢J\in V\to \left\{y\in J|A\in y\right\}\in \mathrm{V}$$
4		eleq1	$$⊢x=A\to \left(x\in y\leftrightarrow A\in y\right)$$
5	4	rabbidv	$$⊢x=A\to \left\{y\in J|x\in y\right\}=\left\{y\in J|A\in y\right\}$$
6	5 1	fvmptg	$$⊢\left(A\in X\wedge \left\{y\in J|A\in y\right\}\in \mathrm{V}\right)\to F\left(A\right)=\left\{y\in J|A\in y\right\}$$
7	2 3 6	syl2anr	$$⊢\left(J\in V\wedge A\in X\right)\to F\left(A\right)=\left\{y\in J|A\in y\right\}$$