Step |
Hyp |
Ref |
Expression |
1 |
|
is2ndc |
⊢ ( 𝐽 ∈ 2ndω ↔ ∃ 𝑥 ∈ TopBases ( 𝑥 ≼ ω ∧ ( topGen ‘ 𝑥 ) = 𝐽 ) ) |
2 |
|
simplr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases ) ∧ 𝑥 ≼ ω ) → 𝑥 ∈ TopBases ) |
3 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases ) ∧ 𝑥 ≼ ω ) → 𝐴 ∈ 𝑉 ) |
4 |
|
tgrest |
⊢ ( ( 𝑥 ∈ TopBases ∧ 𝐴 ∈ 𝑉 ) → ( topGen ‘ ( 𝑥 ↾t 𝐴 ) ) = ( ( topGen ‘ 𝑥 ) ↾t 𝐴 ) ) |
5 |
2 3 4
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases ) ∧ 𝑥 ≼ ω ) → ( topGen ‘ ( 𝑥 ↾t 𝐴 ) ) = ( ( topGen ‘ 𝑥 ) ↾t 𝐴 ) ) |
6 |
|
restbas |
⊢ ( 𝑥 ∈ TopBases → ( 𝑥 ↾t 𝐴 ) ∈ TopBases ) |
7 |
6
|
ad2antlr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases ) ∧ 𝑥 ≼ ω ) → ( 𝑥 ↾t 𝐴 ) ∈ TopBases ) |
8 |
|
restval |
⊢ ( ( 𝑥 ∈ TopBases ∧ 𝐴 ∈ 𝑉 ) → ( 𝑥 ↾t 𝐴 ) = ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ 𝐴 ) ) ) |
9 |
2 3 8
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases ) ∧ 𝑥 ≼ ω ) → ( 𝑥 ↾t 𝐴 ) = ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ 𝐴 ) ) ) |
10 |
|
1stcrestlem |
⊢ ( 𝑥 ≼ ω → ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ 𝐴 ) ) ≼ ω ) |
11 |
10
|
adantl |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases ) ∧ 𝑥 ≼ ω ) → ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ 𝐴 ) ) ≼ ω ) |
12 |
9 11
|
eqbrtrd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases ) ∧ 𝑥 ≼ ω ) → ( 𝑥 ↾t 𝐴 ) ≼ ω ) |
13 |
|
2ndci |
⊢ ( ( ( 𝑥 ↾t 𝐴 ) ∈ TopBases ∧ ( 𝑥 ↾t 𝐴 ) ≼ ω ) → ( topGen ‘ ( 𝑥 ↾t 𝐴 ) ) ∈ 2ndω ) |
14 |
7 12 13
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases ) ∧ 𝑥 ≼ ω ) → ( topGen ‘ ( 𝑥 ↾t 𝐴 ) ) ∈ 2ndω ) |
15 |
5 14
|
eqeltrrd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases ) ∧ 𝑥 ≼ ω ) → ( ( topGen ‘ 𝑥 ) ↾t 𝐴 ) ∈ 2ndω ) |
16 |
|
oveq1 |
⊢ ( ( topGen ‘ 𝑥 ) = 𝐽 → ( ( topGen ‘ 𝑥 ) ↾t 𝐴 ) = ( 𝐽 ↾t 𝐴 ) ) |
17 |
16
|
eleq1d |
⊢ ( ( topGen ‘ 𝑥 ) = 𝐽 → ( ( ( topGen ‘ 𝑥 ) ↾t 𝐴 ) ∈ 2ndω ↔ ( 𝐽 ↾t 𝐴 ) ∈ 2ndω ) ) |
18 |
15 17
|
syl5ibcom |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases ) ∧ 𝑥 ≼ ω ) → ( ( topGen ‘ 𝑥 ) = 𝐽 → ( 𝐽 ↾t 𝐴 ) ∈ 2ndω ) ) |
19 |
18
|
expimpd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ TopBases ) → ( ( 𝑥 ≼ ω ∧ ( topGen ‘ 𝑥 ) = 𝐽 ) → ( 𝐽 ↾t 𝐴 ) ∈ 2ndω ) ) |
20 |
19
|
rexlimdva |
⊢ ( 𝐴 ∈ 𝑉 → ( ∃ 𝑥 ∈ TopBases ( 𝑥 ≼ ω ∧ ( topGen ‘ 𝑥 ) = 𝐽 ) → ( 𝐽 ↾t 𝐴 ) ∈ 2ndω ) ) |
21 |
1 20
|
syl5bi |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝐽 ∈ 2ndω → ( 𝐽 ↾t 𝐴 ) ∈ 2ndω ) ) |
22 |
21
|
impcom |
⊢ ( ( 𝐽 ∈ 2ndω ∧ 𝐴 ∈ 𝑉 ) → ( 𝐽 ↾t 𝐴 ) ∈ 2ndω ) |