Theorem kqtopon

Description: The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015)

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

Proof

Step	Hyp	Ref	Expression
1		kqval.2	⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦 } )
2	1	kqval	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → ( KQ ‘ 𝐽 ) = ( 𝐽 qTop 𝐹 ) )
3	1	kqffn	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐹 Fn 𝑋 )
4		dffn4	⊢ ( 𝐹 Fn 𝑋 ↔ 𝐹 : 𝑋 –onto→ ran 𝐹 )
5	3 4	sylib	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐹 : 𝑋 –onto→ ran 𝐹 )
6		qtoptopon	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 : 𝑋 –onto→ ran 𝐹 ) → ( 𝐽 qTop 𝐹 ) ∈ ( TopOn ‘ ran 𝐹 ) )
7	5 6	mpdan	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → ( 𝐽 qTop 𝐹 ) ∈ ( TopOn ‘ ran 𝐹 ) )
8	2 7	eqeltrd	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → ( KQ ‘ 𝐽 ) ∈ ( TopOn ‘ ran 𝐹 ) )