Step |
Hyp |
Ref |
Expression |
1 |
|
pi1val.g |
⊢ 𝐺 = ( 𝐽 π1 𝑌 ) |
2 |
|
pi1val.1 |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
3 |
|
pi1val.2 |
⊢ ( 𝜑 → 𝑌 ∈ 𝑋 ) |
4 |
|
pi1bas2.b |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐺 ) ) |
5 |
|
pi1bas3.r |
⊢ 𝑅 = ( ( ≃ph ‘ 𝐽 ) ∩ ( ∪ 𝐵 × ∪ 𝐵 ) ) |
6 |
1 2 3 4
|
pi1bas2 |
⊢ ( 𝜑 → 𝐵 = ( ∪ 𝐵 / ( ≃ph ‘ 𝐽 ) ) ) |
7 |
|
eqid |
⊢ ( 𝐽 Ω1 𝑌 ) = ( 𝐽 Ω1 𝑌 ) |
8 |
|
eqidd |
⊢ ( 𝜑 → ( Base ‘ ( 𝐽 Ω1 𝑌 ) ) = ( Base ‘ ( 𝐽 Ω1 𝑌 ) ) ) |
9 |
1 2 3 7 4 8
|
pi1buni |
⊢ ( 𝜑 → ∪ 𝐵 = ( Base ‘ ( 𝐽 Ω1 𝑌 ) ) ) |
10 |
1 2 3 7 4 9
|
pi1blem |
⊢ ( 𝜑 → ( ( ( ≃ph ‘ 𝐽 ) “ ∪ 𝐵 ) ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ⊆ ( II Cn 𝐽 ) ) ) |
11 |
10
|
simpld |
⊢ ( 𝜑 → ( ( ≃ph ‘ 𝐽 ) “ ∪ 𝐵 ) ⊆ ∪ 𝐵 ) |
12 |
|
qsinxp |
⊢ ( ( ( ≃ph ‘ 𝐽 ) “ ∪ 𝐵 ) ⊆ ∪ 𝐵 → ( ∪ 𝐵 / ( ≃ph ‘ 𝐽 ) ) = ( ∪ 𝐵 / ( ( ≃ph ‘ 𝐽 ) ∩ ( ∪ 𝐵 × ∪ 𝐵 ) ) ) ) |
13 |
11 12
|
syl |
⊢ ( 𝜑 → ( ∪ 𝐵 / ( ≃ph ‘ 𝐽 ) ) = ( ∪ 𝐵 / ( ( ≃ph ‘ 𝐽 ) ∩ ( ∪ 𝐵 × ∪ 𝐵 ) ) ) ) |
14 |
6 13
|
eqtrd |
⊢ ( 𝜑 → 𝐵 = ( ∪ 𝐵 / ( ( ≃ph ‘ 𝐽 ) ∩ ( ∪ 𝐵 × ∪ 𝐵 ) ) ) ) |
15 |
|
qseq2 |
⊢ ( 𝑅 = ( ( ≃ph ‘ 𝐽 ) ∩ ( ∪ 𝐵 × ∪ 𝐵 ) ) → ( ∪ 𝐵 / 𝑅 ) = ( ∪ 𝐵 / ( ( ≃ph ‘ 𝐽 ) ∩ ( ∪ 𝐵 × ∪ 𝐵 ) ) ) ) |
16 |
5 15
|
ax-mp |
⊢ ( ∪ 𝐵 / 𝑅 ) = ( ∪ 𝐵 / ( ( ≃ph ‘ 𝐽 ) ∩ ( ∪ 𝐵 × ∪ 𝐵 ) ) ) |
17 |
14 16
|
eqtr4di |
⊢ ( 𝜑 → 𝐵 = ( ∪ 𝐵 / 𝑅 ) ) |