Step |
Hyp |
Ref |
Expression |
1 |
|
ellimciota.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℂ ) |
2 |
|
ellimciota.a |
⊢ ( 𝜑 → 𝐴 ⊆ ℂ ) |
3 |
|
ellimciota.b |
⊢ ( 𝜑 → 𝐵 ∈ ( ( limPt ‘ 𝐾 ) ‘ 𝐴 ) ) |
4 |
|
ellimciota.4 |
⊢ ( 𝜑 → ( 𝐹 limℂ 𝐵 ) ≠ ∅ ) |
5 |
|
ellimciota.k |
⊢ 𝐾 = ( TopOpen ‘ ℂfld ) |
6 |
|
eleq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ ( 𝐹 limℂ 𝐵 ) ↔ 𝑦 ∈ ( 𝐹 limℂ 𝐵 ) ) ) |
7 |
6
|
cbviotavw |
⊢ ( ℩ 𝑥 𝑥 ∈ ( 𝐹 limℂ 𝐵 ) ) = ( ℩ 𝑦 𝑦 ∈ ( 𝐹 limℂ 𝐵 ) ) |
8 |
|
iotaex |
⊢ ( ℩ 𝑦 𝑦 ∈ ( 𝐹 limℂ 𝐵 ) ) ∈ V |
9 |
|
n0 |
⊢ ( ( 𝐹 limℂ 𝐵 ) ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ ( 𝐹 limℂ 𝐵 ) ) |
10 |
4 9
|
sylib |
⊢ ( 𝜑 → ∃ 𝑥 𝑥 ∈ ( 𝐹 limℂ 𝐵 ) ) |
11 |
1 2 3 5
|
limcmo |
⊢ ( 𝜑 → ∃* 𝑥 𝑥 ∈ ( 𝐹 limℂ 𝐵 ) ) |
12 |
|
df-eu |
⊢ ( ∃! 𝑥 𝑥 ∈ ( 𝐹 limℂ 𝐵 ) ↔ ( ∃ 𝑥 𝑥 ∈ ( 𝐹 limℂ 𝐵 ) ∧ ∃* 𝑥 𝑥 ∈ ( 𝐹 limℂ 𝐵 ) ) ) |
13 |
10 11 12
|
sylanbrc |
⊢ ( 𝜑 → ∃! 𝑥 𝑥 ∈ ( 𝐹 limℂ 𝐵 ) ) |
14 |
|
eleq1 |
⊢ ( 𝑥 = ( ℩ 𝑦 𝑦 ∈ ( 𝐹 limℂ 𝐵 ) ) → ( 𝑥 ∈ ( 𝐹 limℂ 𝐵 ) ↔ ( ℩ 𝑦 𝑦 ∈ ( 𝐹 limℂ 𝐵 ) ) ∈ ( 𝐹 limℂ 𝐵 ) ) ) |
15 |
14
|
iota2 |
⊢ ( ( ( ℩ 𝑦 𝑦 ∈ ( 𝐹 limℂ 𝐵 ) ) ∈ V ∧ ∃! 𝑥 𝑥 ∈ ( 𝐹 limℂ 𝐵 ) ) → ( ( ℩ 𝑦 𝑦 ∈ ( 𝐹 limℂ 𝐵 ) ) ∈ ( 𝐹 limℂ 𝐵 ) ↔ ( ℩ 𝑥 𝑥 ∈ ( 𝐹 limℂ 𝐵 ) ) = ( ℩ 𝑦 𝑦 ∈ ( 𝐹 limℂ 𝐵 ) ) ) ) |
16 |
8 13 15
|
sylancr |
⊢ ( 𝜑 → ( ( ℩ 𝑦 𝑦 ∈ ( 𝐹 limℂ 𝐵 ) ) ∈ ( 𝐹 limℂ 𝐵 ) ↔ ( ℩ 𝑥 𝑥 ∈ ( 𝐹 limℂ 𝐵 ) ) = ( ℩ 𝑦 𝑦 ∈ ( 𝐹 limℂ 𝐵 ) ) ) ) |
17 |
7 16
|
mpbiri |
⊢ ( 𝜑 → ( ℩ 𝑦 𝑦 ∈ ( 𝐹 limℂ 𝐵 ) ) ∈ ( 𝐹 limℂ 𝐵 ) ) |
18 |
7 17
|
eqeltrid |
⊢ ( 𝜑 → ( ℩ 𝑥 𝑥 ∈ ( 𝐹 limℂ 𝐵 ) ) ∈ ( 𝐹 limℂ 𝐵 ) ) |