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 ) ) |