| Step |
Hyp |
Ref |
Expression |
| 1 |
|
xmetres |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ∈ ( ∞Met ‘ ( 𝑋 ∩ 𝑌 ) ) ) |
| 2 |
|
iscfil2 |
⊢ ( ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ∈ ( ∞Met ‘ ( 𝑋 ∩ 𝑌 ) ) → ( 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ↔ ( 𝐹 ∈ ( Fil ‘ ( 𝑋 ∩ 𝑌 ) ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) 𝑣 ) < 𝑥 ) ) ) |
| 3 |
2
|
simplbda |
⊢ ( ( ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ∈ ( ∞Met ‘ ( 𝑋 ∩ 𝑌 ) ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) 𝑣 ) < 𝑥 ) |
| 4 |
1 3
|
sylan |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) 𝑣 ) < 𝑥 ) |
| 5 |
|
cfilfil |
⊢ ( ( ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ∈ ( ∞Met ‘ ( 𝑋 ∩ 𝑌 ) ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → 𝐹 ∈ ( Fil ‘ ( 𝑋 ∩ 𝑌 ) ) ) |
| 6 |
1 5
|
sylan |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → 𝐹 ∈ ( Fil ‘ ( 𝑋 ∩ 𝑌 ) ) ) |
| 7 |
|
filelss |
⊢ ( ( 𝐹 ∈ ( Fil ‘ ( 𝑋 ∩ 𝑌 ) ) ∧ 𝑦 ∈ 𝐹 ) → 𝑦 ⊆ ( 𝑋 ∩ 𝑌 ) ) |
| 8 |
6 7
|
sylan |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) ∧ 𝑦 ∈ 𝐹 ) → 𝑦 ⊆ ( 𝑋 ∩ 𝑌 ) ) |
| 9 |
|
inss2 |
⊢ ( 𝑋 ∩ 𝑌 ) ⊆ 𝑌 |
| 10 |
8 9
|
sstrdi |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) ∧ 𝑦 ∈ 𝐹 ) → 𝑦 ⊆ 𝑌 ) |
| 11 |
10
|
sselda |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) ∧ 𝑦 ∈ 𝐹 ) ∧ 𝑢 ∈ 𝑦 ) → 𝑢 ∈ 𝑌 ) |
| 12 |
10
|
sselda |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) ∧ 𝑦 ∈ 𝐹 ) ∧ 𝑣 ∈ 𝑦 ) → 𝑣 ∈ 𝑌 ) |
| 13 |
11 12
|
anim12dan |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) ∧ 𝑦 ∈ 𝐹 ) ∧ ( 𝑢 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦 ) ) → ( 𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌 ) ) |
| 14 |
|
ovres |
⊢ ( ( 𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌 ) → ( 𝑢 ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) 𝑣 ) = ( 𝑢 𝐷 𝑣 ) ) |
| 15 |
13 14
|
syl |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) ∧ 𝑦 ∈ 𝐹 ) ∧ ( 𝑢 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦 ) ) → ( 𝑢 ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) 𝑣 ) = ( 𝑢 𝐷 𝑣 ) ) |
| 16 |
15
|
breq1d |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) ∧ 𝑦 ∈ 𝐹 ) ∧ ( 𝑢 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦 ) ) → ( ( 𝑢 ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) 𝑣 ) < 𝑥 ↔ ( 𝑢 𝐷 𝑣 ) < 𝑥 ) ) |
| 17 |
16
|
2ralbidva |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) ∧ 𝑦 ∈ 𝐹 ) → ( ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) 𝑣 ) < 𝑥 ↔ ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 𝐷 𝑣 ) < 𝑥 ) ) |
| 18 |
17
|
rexbidva |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → ( ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) 𝑣 ) < 𝑥 ↔ ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 𝐷 𝑣 ) < 𝑥 ) ) |
| 19 |
18
|
ralbidv |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) 𝑣 ) < 𝑥 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 𝐷 𝑣 ) < 𝑥 ) ) |
| 20 |
4 19
|
mpbid |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 𝐷 𝑣 ) < 𝑥 ) |
| 21 |
|
filfbas |
⊢ ( 𝐹 ∈ ( Fil ‘ ( 𝑋 ∩ 𝑌 ) ) → 𝐹 ∈ ( fBas ‘ ( 𝑋 ∩ 𝑌 ) ) ) |
| 22 |
6 21
|
syl |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → 𝐹 ∈ ( fBas ‘ ( 𝑋 ∩ 𝑌 ) ) ) |
| 23 |
|
filsspw |
⊢ ( 𝐹 ∈ ( Fil ‘ ( 𝑋 ∩ 𝑌 ) ) → 𝐹 ⊆ 𝒫 ( 𝑋 ∩ 𝑌 ) ) |
| 24 |
6 23
|
syl |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → 𝐹 ⊆ 𝒫 ( 𝑋 ∩ 𝑌 ) ) |
| 25 |
|
inss1 |
⊢ ( 𝑋 ∩ 𝑌 ) ⊆ 𝑋 |
| 26 |
25
|
sspwi |
⊢ 𝒫 ( 𝑋 ∩ 𝑌 ) ⊆ 𝒫 𝑋 |
| 27 |
24 26
|
sstrdi |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → 𝐹 ⊆ 𝒫 𝑋 ) |
| 28 |
|
elfvdm |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝑋 ∈ dom ∞Met ) |
| 29 |
28
|
adantr |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → 𝑋 ∈ dom ∞Met ) |
| 30 |
|
fbasweak |
⊢ ( ( 𝐹 ∈ ( fBas ‘ ( 𝑋 ∩ 𝑌 ) ) ∧ 𝐹 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ dom ∞Met ) → 𝐹 ∈ ( fBas ‘ 𝑋 ) ) |
| 31 |
22 27 29 30
|
syl3anc |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → 𝐹 ∈ ( fBas ‘ 𝑋 ) ) |
| 32 |
|
fgcfil |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( fBas ‘ 𝑋 ) ) → ( ( 𝑋 filGen 𝐹 ) ∈ ( CauFil ‘ 𝐷 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 𝐷 𝑣 ) < 𝑥 ) ) |
| 33 |
31 32
|
syldan |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → ( ( 𝑋 filGen 𝐹 ) ∈ ( CauFil ‘ 𝐷 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 𝐷 𝑣 ) < 𝑥 ) ) |
| 34 |
20 33
|
mpbird |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → ( 𝑋 filGen 𝐹 ) ∈ ( CauFil ‘ 𝐷 ) ) |