Step |
Hyp |
Ref |
Expression |
1 |
|
is1stc.1 |
⊢ 𝑋 = ∪ 𝐽 |
2 |
|
unieq |
⊢ ( 𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽 ) |
3 |
2 1
|
eqtr4di |
⊢ ( 𝑗 = 𝐽 → ∪ 𝑗 = 𝑋 ) |
4 |
|
pweq |
⊢ ( 𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝐽 ) |
5 |
|
raleq |
⊢ ( 𝑗 = 𝐽 → ( ∀ 𝑧 ∈ 𝑗 ( 𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ ( 𝑦 ∩ 𝒫 𝑧 ) ) ↔ ∀ 𝑧 ∈ 𝐽 ( 𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ ( 𝑦 ∩ 𝒫 𝑧 ) ) ) ) |
6 |
5
|
anbi2d |
⊢ ( 𝑗 = 𝐽 → ( ( 𝑦 ≼ ω ∧ ∀ 𝑧 ∈ 𝑗 ( 𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ ( 𝑦 ∩ 𝒫 𝑧 ) ) ) ↔ ( 𝑦 ≼ ω ∧ ∀ 𝑧 ∈ 𝐽 ( 𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ ( 𝑦 ∩ 𝒫 𝑧 ) ) ) ) ) |
7 |
4 6
|
rexeqbidv |
⊢ ( 𝑗 = 𝐽 → ( ∃ 𝑦 ∈ 𝒫 𝑗 ( 𝑦 ≼ ω ∧ ∀ 𝑧 ∈ 𝑗 ( 𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ ( 𝑦 ∩ 𝒫 𝑧 ) ) ) ↔ ∃ 𝑦 ∈ 𝒫 𝐽 ( 𝑦 ≼ ω ∧ ∀ 𝑧 ∈ 𝐽 ( 𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ ( 𝑦 ∩ 𝒫 𝑧 ) ) ) ) ) |
8 |
3 7
|
raleqbidv |
⊢ ( 𝑗 = 𝐽 → ( ∀ 𝑥 ∈ ∪ 𝑗 ∃ 𝑦 ∈ 𝒫 𝑗 ( 𝑦 ≼ ω ∧ ∀ 𝑧 ∈ 𝑗 ( 𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ ( 𝑦 ∩ 𝒫 𝑧 ) ) ) ↔ ∀ 𝑥 ∈ 𝑋 ∃ 𝑦 ∈ 𝒫 𝐽 ( 𝑦 ≼ ω ∧ ∀ 𝑧 ∈ 𝐽 ( 𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ ( 𝑦 ∩ 𝒫 𝑧 ) ) ) ) ) |
9 |
|
df-1stc |
⊢ 1stω = { 𝑗 ∈ Top ∣ ∀ 𝑥 ∈ ∪ 𝑗 ∃ 𝑦 ∈ 𝒫 𝑗 ( 𝑦 ≼ ω ∧ ∀ 𝑧 ∈ 𝑗 ( 𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ ( 𝑦 ∩ 𝒫 𝑧 ) ) ) } |
10 |
8 9
|
elrab2 |
⊢ ( 𝐽 ∈ 1stω ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ 𝑋 ∃ 𝑦 ∈ 𝒫 𝐽 ( 𝑦 ≼ ω ∧ ∀ 𝑧 ∈ 𝐽 ( 𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ ( 𝑦 ∩ 𝒫 𝑧 ) ) ) ) ) |