Step | Hyp | Ref
| Expression |
1 | | tmsval.k |
. 2
⊢ 𝐾 = (toMetSp‘𝐷) |
2 | | df-tms 21937 |
. . . 4
⊢ toMetSp =
(𝑑 ∈ ∪ ran ∞Met ↦ ({〈(Base‘ndx), dom dom
𝑑〉,
〈(dist‘ndx), 𝑑〉} sSet 〈(TopSet‘ndx),
(MetOpen‘𝑑)〉)) |
3 | 2 | a1i 11 |
. . 3
⊢ (𝐷 ∈ (∞Met‘𝑋) → toMetSp = (𝑑 ∈ ∪ ran ∞Met ↦ ({〈(Base‘ndx), dom dom
𝑑〉,
〈(dist‘ndx), 𝑑〉} sSet 〈(TopSet‘ndx),
(MetOpen‘𝑑)〉))) |
4 | | dmeq 5246 |
. . . . . . . . 9
⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷) |
5 | 4 | dmeqd 5248 |
. . . . . . . 8
⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷) |
6 | | xmetf 21944 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
7 | | fdm 5964 |
. . . . . . . . . . 11
⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom
𝐷 = (𝑋 × 𝑋)) |
8 | 6, 7 | syl 17 |
. . . . . . . . . 10
⊢ (𝐷 ∈ (∞Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋)) |
9 | 8 | dmeqd 5248 |
. . . . . . . . 9
⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = dom (𝑋 × 𝑋)) |
10 | | dmxpid 5266 |
. . . . . . . . 9
⊢ dom
(𝑋 × 𝑋) = 𝑋 |
11 | 9, 10 | syl6eq 2660 |
. . . . . . . 8
⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = 𝑋) |
12 | 5, 11 | sylan9eqr 2666 |
. . . . . . 7
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋) |
13 | 12 | opeq2d 4347 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → 〈(Base‘ndx), dom dom
𝑑〉 =
〈(Base‘ndx), 𝑋〉) |
14 | | simpr 476 |
. . . . . . 7
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷) |
15 | 14 | opeq2d 4347 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → 〈(dist‘ndx), 𝑑〉 = 〈(dist‘ndx),
𝐷〉) |
16 | 13, 15 | preq12d 4220 |
. . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → {〈(Base‘ndx), dom dom
𝑑〉,
〈(dist‘ndx), 𝑑〉} = {〈(Base‘ndx), 𝑋〉, 〈(dist‘ndx),
𝐷〉}) |
17 | | tmsval.m |
. . . . 5
⊢ 𝑀 = {〈(Base‘ndx),
𝑋〉,
〈(dist‘ndx), 𝐷〉} |
18 | 16, 17 | syl6eqr 2662 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → {〈(Base‘ndx), dom dom
𝑑〉,
〈(dist‘ndx), 𝑑〉} = 𝑀) |
19 | 14 | fveq2d 6107 |
. . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (MetOpen‘𝑑) = (MetOpen‘𝐷)) |
20 | 19 | opeq2d 4347 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → 〈(TopSet‘ndx),
(MetOpen‘𝑑)〉 =
〈(TopSet‘ndx), (MetOpen‘𝐷)〉) |
21 | 18, 20 | oveq12d 6567 |
. . 3
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ({〈(Base‘ndx), dom dom
𝑑〉,
〈(dist‘ndx), 𝑑〉} sSet 〈(TopSet‘ndx),
(MetOpen‘𝑑)〉) =
(𝑀 sSet
〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) |
22 | | fvssunirn 6127 |
. . . 4
⊢
(∞Met‘𝑋)
⊆ ∪ ran ∞Met |
23 | 22 | sseli 3564 |
. . 3
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran
∞Met) |
24 | | ovex 6577 |
. . . 4
⊢ (𝑀 sSet 〈(TopSet‘ndx),
(MetOpen‘𝐷)〉)
∈ V |
25 | 24 | a1i 11 |
. . 3
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑀 sSet 〈(TopSet‘ndx),
(MetOpen‘𝐷)〉)
∈ V) |
26 | 3, 21, 23, 25 | fvmptd 6197 |
. 2
⊢ (𝐷 ∈ (∞Met‘𝑋) → (toMetSp‘𝐷) = (𝑀 sSet 〈(TopSet‘ndx),
(MetOpen‘𝐷)〉)) |
27 | 1, 26 | syl5eq 2656 |
1
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx),
(MetOpen‘𝐷)〉)) |