Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ( 𝑧 ∈ 𝑋 ↦ { 𝑤 ∈ 𝐽 ∣ 𝑧 ∈ 𝑤 } ) = ( 𝑧 ∈ 𝑋 ↦ { 𝑤 ∈ 𝐽 ∣ 𝑧 ∈ 𝑤 } ) |
2 |
1
|
isr0 |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → ( ( KQ ‘ 𝐽 ) ∈ Fre ↔ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜 ) → ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) ) ) ) |
3 |
2
|
biimpa |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ ( KQ ‘ 𝐽 ) ∈ Fre ) → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜 ) → ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) ) ) |
4 |
|
eleq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜 ) ) |
5 |
4
|
imbi1d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜 ) ↔ ( 𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜 ) ) ) |
6 |
5
|
ralbidv |
⊢ ( 𝑥 = 𝐴 → ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜 ) ↔ ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜 ) ) ) |
7 |
4
|
bibi1d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) ↔ ( 𝐴 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) ) ) |
8 |
7
|
ralbidv |
⊢ ( 𝑥 = 𝐴 → ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) ↔ ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) ) ) |
9 |
6 8
|
imbi12d |
⊢ ( 𝑥 = 𝐴 → ( ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜 ) → ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) ) ↔ ( ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜 ) → ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) ) ) ) |
10 |
|
eleq1 |
⊢ ( 𝑦 = 𝐵 → ( 𝑦 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜 ) ) |
11 |
10
|
imbi2d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜 ) ↔ ( 𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜 ) ) ) |
12 |
11
|
ralbidv |
⊢ ( 𝑦 = 𝐵 → ( ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜 ) ↔ ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜 ) ) ) |
13 |
10
|
bibi2d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) ↔ ( 𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜 ) ) ) |
14 |
13
|
ralbidv |
⊢ ( 𝑦 = 𝐵 → ( ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) ↔ ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜 ) ) ) |
15 |
12 14
|
imbi12d |
⊢ ( 𝑦 = 𝐵 → ( ( ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜 ) → ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) ) ↔ ( ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜 ) → ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜 ) ) ) ) |
16 |
9 15
|
rspc2v |
⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜 ) → ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) ) → ( ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜 ) → ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜 ) ) ) ) |
17 |
3 16
|
mpan9 |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ ( KQ ‘ 𝐽 ) ∈ Fre ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜 ) → ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜 ) ) ) |