Step |
Hyp |
Ref |
Expression |
1 |
|
fveq2 |
⊢ ( 𝑗 = 𝐽 → ( cls ‘ 𝑗 ) = ( cls ‘ 𝐽 ) ) |
2 |
1
|
fveq1d |
⊢ ( 𝑗 = 𝐽 → ( ( cls ‘ 𝑗 ) ‘ 𝑧 ) = ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) |
3 |
2
|
sseq1d |
⊢ ( 𝑗 = 𝐽 → ( ( ( cls ‘ 𝑗 ) ‘ 𝑧 ) ⊆ 𝑥 ↔ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑥 ) ) |
4 |
3
|
anbi2d |
⊢ ( 𝑗 = 𝐽 → ( ( 𝑦 ∈ 𝑧 ∧ ( ( cls ‘ 𝑗 ) ‘ 𝑧 ) ⊆ 𝑥 ) ↔ ( 𝑦 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑥 ) ) ) |
5 |
4
|
rexeqbi1dv |
⊢ ( 𝑗 = 𝐽 → ( ∃ 𝑧 ∈ 𝑗 ( 𝑦 ∈ 𝑧 ∧ ( ( cls ‘ 𝑗 ) ‘ 𝑧 ) ⊆ 𝑥 ) ↔ ∃ 𝑧 ∈ 𝐽 ( 𝑦 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑥 ) ) ) |
6 |
5
|
ralbidv |
⊢ ( 𝑗 = 𝐽 → ( ∀ 𝑦 ∈ 𝑥 ∃ 𝑧 ∈ 𝑗 ( 𝑦 ∈ 𝑧 ∧ ( ( cls ‘ 𝑗 ) ‘ 𝑧 ) ⊆ 𝑥 ) ↔ ∀ 𝑦 ∈ 𝑥 ∃ 𝑧 ∈ 𝐽 ( 𝑦 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑥 ) ) ) |
7 |
6
|
raleqbi1dv |
⊢ ( 𝑗 = 𝐽 → ( ∀ 𝑥 ∈ 𝑗 ∀ 𝑦 ∈ 𝑥 ∃ 𝑧 ∈ 𝑗 ( 𝑦 ∈ 𝑧 ∧ ( ( cls ‘ 𝑗 ) ‘ 𝑧 ) ⊆ 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐽 ∀ 𝑦 ∈ 𝑥 ∃ 𝑧 ∈ 𝐽 ( 𝑦 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑥 ) ) ) |
8 |
|
df-reg |
⊢ Reg = { 𝑗 ∈ Top ∣ ∀ 𝑥 ∈ 𝑗 ∀ 𝑦 ∈ 𝑥 ∃ 𝑧 ∈ 𝑗 ( 𝑦 ∈ 𝑧 ∧ ( ( cls ‘ 𝑗 ) ‘ 𝑧 ) ⊆ 𝑥 ) } |
9 |
7 8
|
elrab2 |
⊢ ( 𝐽 ∈ Reg ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ 𝐽 ∀ 𝑦 ∈ 𝑥 ∃ 𝑧 ∈ 𝐽 ( 𝑦 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑥 ) ) ) |