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 |
|
pi1val.o |
⊢ 𝑂 = ( 𝐽 Ω1 𝑌 ) |
5 |
|
df-pi1 |
⊢ π1 = ( 𝑗 ∈ Top , 𝑦 ∈ ∪ 𝑗 ↦ ( ( 𝑗 Ω1 𝑦 ) /s ( ≃ph ‘ 𝑗 ) ) ) |
6 |
5
|
a1i |
⊢ ( 𝜑 → π1 = ( 𝑗 ∈ Top , 𝑦 ∈ ∪ 𝑗 ↦ ( ( 𝑗 Ω1 𝑦 ) /s ( ≃ph ‘ 𝑗 ) ) ) ) |
7 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → 𝑗 = 𝐽 ) |
8 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → 𝑦 = 𝑌 ) |
9 |
7 8
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → ( 𝑗 Ω1 𝑦 ) = ( 𝐽 Ω1 𝑌 ) ) |
10 |
9 4
|
eqtr4di |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → ( 𝑗 Ω1 𝑦 ) = 𝑂 ) |
11 |
7
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → ( ≃ph ‘ 𝑗 ) = ( ≃ph ‘ 𝐽 ) ) |
12 |
10 11
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → ( ( 𝑗 Ω1 𝑦 ) /s ( ≃ph ‘ 𝑗 ) ) = ( 𝑂 /s ( ≃ph ‘ 𝐽 ) ) ) |
13 |
|
unieq |
⊢ ( 𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽 ) |
14 |
13
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 = 𝐽 ) → ∪ 𝑗 = ∪ 𝐽 ) |
15 |
|
toponuni |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 ) |
16 |
2 15
|
syl |
⊢ ( 𝜑 → 𝑋 = ∪ 𝐽 ) |
17 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 = 𝐽 ) → 𝑋 = ∪ 𝐽 ) |
18 |
14 17
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑗 = 𝐽 ) → ∪ 𝑗 = 𝑋 ) |
19 |
|
topontop |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ Top ) |
20 |
2 19
|
syl |
⊢ ( 𝜑 → 𝐽 ∈ Top ) |
21 |
|
ovexd |
⊢ ( 𝜑 → ( 𝑂 /s ( ≃ph ‘ 𝐽 ) ) ∈ V ) |
22 |
6 12 18 20 3 21
|
ovmpodx |
⊢ ( 𝜑 → ( 𝐽 π1 𝑌 ) = ( 𝑂 /s ( ≃ph ‘ 𝐽 ) ) ) |
23 |
1 22
|
eqtrid |
⊢ ( 𝜑 → 𝐺 = ( 𝑂 /s ( ≃ph ‘ 𝐽 ) ) ) |