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Metamath Proof Explorer

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 ) )