Metamath Proof Explorer


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 = ( 𝑥𝑋 ↦ { 𝑦𝐽𝑥𝑦 } )
Assertion kqid ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐹 ∈ ( 𝐽 Cn ( KQ ‘ 𝐽 ) ) )

Proof

Step Hyp Ref Expression
1 kqval.2 𝐹 = ( 𝑥𝑋 ↦ { 𝑦𝐽𝑥𝑦 } )
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 ) ) )