Step | Hyp | Ref
| Expression |
1 | | tmsval.k |
. 2
⊢ 𝐾 = (toMetSp‘𝐷) |
2 | | df-tms 23828 |
. . 3
⊢ toMetSp =
(𝑑 ∈ ∪ ran ∞Met ↦ ({⟨(Base‘ndx), dom dom
𝑑⟩,
⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx),
(MetOpen‘𝑑)⟩)) |
3 | | dmeq 5904 |
. . . . . . . . 9
⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷) |
4 | 3 | dmeqd 5906 |
. . . . . . . 8
⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷) |
5 | | xmetf 23835 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
6 | 5 | fdmd 6729 |
. . . . . . . . . 10
⊢ (𝐷 ∈ (∞Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋)) |
7 | 6 | dmeqd 5906 |
. . . . . . . . 9
⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = dom (𝑋 × 𝑋)) |
8 | | dmxpid 5930 |
. . . . . . . . 9
⊢ dom
(𝑋 × 𝑋) = 𝑋 |
9 | 7, 8 | eqtrdi 2789 |
. . . . . . . 8
⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = 𝑋) |
10 | 4, 9 | sylan9eqr 2795 |
. . . . . . 7
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋) |
11 | 10 | opeq2d 4881 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ⟨(Base‘ndx), dom dom
𝑑⟩ =
⟨(Base‘ndx), 𝑋⟩) |
12 | | simpr 486 |
. . . . . . 7
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷) |
13 | 12 | opeq2d 4881 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ⟨(dist‘ndx), 𝑑⟩ = ⟨(dist‘ndx),
𝐷⟩) |
14 | 11, 13 | preq12d 4746 |
. . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → {⟨(Base‘ndx), dom dom
𝑑⟩,
⟨(dist‘ndx), 𝑑⟩} = {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx),
𝐷⟩}) |
15 | | tmsval.m |
. . . . 5
⊢ 𝑀 = {⟨(Base‘ndx),
𝑋⟩,
⟨(dist‘ndx), 𝐷⟩} |
16 | 14, 15 | eqtr4di 2791 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → {⟨(Base‘ndx), dom dom
𝑑⟩,
⟨(dist‘ndx), 𝑑⟩} = 𝑀) |
17 | 12 | fveq2d 6896 |
. . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (MetOpen‘𝑑) = (MetOpen‘𝐷)) |
18 | 17 | opeq2d 4881 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ⟨(TopSet‘ndx),
(MetOpen‘𝑑)⟩ =
⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩) |
19 | 16, 18 | oveq12d 7427 |
. . 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)⟩)) |
20 | | fvssunirn 6925 |
. . . 4
⊢
(∞Met‘𝑋)
⊆ ∪ ran ∞Met |
21 | 20 | sseli 3979 |
. . 3
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran
∞Met) |
22 | | ovexd 7444 |
. . 3
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑀 sSet ⟨(TopSet‘ndx),
(MetOpen‘𝐷)⟩)
∈ V) |
23 | 2, 19, 21, 22 | fvmptd2 7007 |
. 2
⊢ (𝐷 ∈ (∞Met‘𝑋) → (toMetSp‘𝐷) = (𝑀 sSet ⟨(TopSet‘ndx),
(MetOpen‘𝐷)⟩)) |
24 | 1, 23 | eqtrid 2785 |
1
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx),
(MetOpen‘𝐷)⟩)) |