Theorem kqid

Description: The topological indistinguishability map is a continuous function into the Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015)

	Ref	Expression
Hypothesis	kqval.2	⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦 } )
Assertion	kqid	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐹 ∈ ( 𝐽 Cn ( KQ ‘ 𝐽 ) ) )

Proof

Step	Hyp	Ref	Expression
1		kqval.2	⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦 } )
2	1	kqffn	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐹 Fn 𝑋 )
3		qtopid	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 Fn 𝑋 ) → 𝐹 ∈ ( 𝐽 Cn ( 𝐽 qTop 𝐹 ) ) )
4	2 3	mpdan	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐹 ∈ ( 𝐽 Cn ( 𝐽 qTop 𝐹 ) ) )
5	1	kqval	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → ( KQ ‘ 𝐽 ) = ( 𝐽 qTop 𝐹 ) )
6	5	oveq2d	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → ( 𝐽 Cn ( KQ ‘ 𝐽 ) ) = ( 𝐽 Cn ( 𝐽 qTop 𝐹 ) ) )
7	4 6	eleqtrrd	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐹 ∈ ( 𝐽 Cn ( KQ ‘ 𝐽 ) ) )