Theorem kqfeq

Description: Two points in the Kolmogorov quotient are equal iff the original points are topologically indistinguishable. (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	kqfeq	$$⊢\left(J\in V\wedge A\in X\wedge B\in X\right)\to \left(F\left(A\right)=F\left(B\right)\leftrightarrow \forall y\in J\left(A\in y\leftrightarrow B\in y\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		kqval.2	$$⊢F=\left(x\in X⟼\left\{y\in J|x\in y\right\}\right)$$
2	1	kqfval	$$⊢\left(J\in V\wedge A\in X\right)\to F\left(A\right)=\left\{y\in J|A\in y\right\}$$
3	2	3adant3	$$⊢\left(J\in V\wedge A\in X\wedge B\in X\right)\to F\left(A\right)=\left\{y\in J|A\in y\right\}$$
4	1	kqfval	$$⊢\left(J\in V\wedge B\in X\right)\to F\left(B\right)=\left\{y\in J|B\in y\right\}$$
5	4	3adant2	$$⊢\left(J\in V\wedge A\in X\wedge B\in X\right)\to F\left(B\right)=\left\{y\in J|B\in y\right\}$$
6	3 5	eqeq12d	$$⊢\left(J\in V\wedge A\in X\wedge B\in X\right)\to \left(F\left(A\right)=F\left(B\right)\leftrightarrow \left\{y\in J|A\in y\right\}=\left\{y\in J|B\in y\right\}\right)$$
7		rabbi	$$⊢\forall y\in J\left(A\in y\leftrightarrow B\in y\right)\leftrightarrow \left\{y\in J|A\in y\right\}=\left\{y\in J|B\in y\right\}$$
8	6 7	bitr4di	$$⊢\left(J\in V\wedge A\in X\wedge B\in X\right)\to \left(F\left(A\right)=F\left(B\right)\leftrightarrow \forall y\in J\left(A\in y\leftrightarrow B\in y\right)\right)$$