Step |
Hyp |
Ref |
Expression |
1 |
|
pibt1.19 |
⊢ 𝐶 = { 𝑥 ∈ Top ∣ ∀ 𝑦 ∈ 𝒫 𝑥 ( ( ∪ 𝑥 = ∪ 𝑦 ∧ 𝑦 ≼ ω ) → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) ∪ 𝑥 = ∪ 𝑧 ) } |
2 |
|
pm3.41 |
⊢ ( ( ∪ 𝐽 = ∪ 𝑦 → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) ∪ 𝐽 = ∪ 𝑧 ) → ( ( ∪ 𝐽 = ∪ 𝑦 ∧ 𝑦 ≼ ω ) → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) ∪ 𝐽 = ∪ 𝑧 ) ) |
3 |
2
|
ralimi |
⊢ ( ∀ 𝑦 ∈ 𝒫 𝐽 ( ∪ 𝐽 = ∪ 𝑦 → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) ∪ 𝐽 = ∪ 𝑧 ) → ∀ 𝑦 ∈ 𝒫 𝐽 ( ( ∪ 𝐽 = ∪ 𝑦 ∧ 𝑦 ≼ ω ) → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) ∪ 𝐽 = ∪ 𝑧 ) ) |
4 |
3
|
anim2i |
⊢ ( ( 𝐽 ∈ Top ∧ ∀ 𝑦 ∈ 𝒫 𝐽 ( ∪ 𝐽 = ∪ 𝑦 → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) ∪ 𝐽 = ∪ 𝑧 ) ) → ( 𝐽 ∈ Top ∧ ∀ 𝑦 ∈ 𝒫 𝐽 ( ( ∪ 𝐽 = ∪ 𝑦 ∧ 𝑦 ≼ ω ) → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) ∪ 𝐽 = ∪ 𝑧 ) ) ) |
5 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
6 |
5
|
pibp16 |
⊢ ( 𝐽 ∈ Comp ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑦 ∈ 𝒫 𝐽 ( ∪ 𝐽 = ∪ 𝑦 → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) ∪ 𝐽 = ∪ 𝑧 ) ) ) |
7 |
5 1
|
pibp19 |
⊢ ( 𝐽 ∈ 𝐶 ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑦 ∈ 𝒫 𝐽 ( ( ∪ 𝐽 = ∪ 𝑦 ∧ 𝑦 ≼ ω ) → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) ∪ 𝐽 = ∪ 𝑧 ) ) ) |
8 |
4 6 7
|
3imtr4i |
⊢ ( 𝐽 ∈ Comp → 𝐽 ∈ 𝐶 ) |