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Mirrors > Home > MPE Home > Th. List > tusval | Structured version Visualization version GIF version |
Description: The value of the uniform space mapping function. (Contributed by Thierry Arnoux, 5-Dec-2017.) |
Ref | Expression |
---|---|
tusval | ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({〈(Base‘ndx), dom ∪ 𝑈〉, 〈(UnifSet‘ndx), 𝑈〉} sSet 〈(TopSet‘ndx), (unifTop‘𝑈)〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tus 24282 | . 2 ⊢ toUnifSp = (𝑢 ∈ ∪ ran UnifOn ↦ ({〈(Base‘ndx), dom ∪ 𝑢〉, 〈(UnifSet‘ndx), 𝑢〉} sSet 〈(TopSet‘ndx), (unifTop‘𝑢)〉)) | |
2 | simpr 484 | . . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈) | |
3 | 2 | unieqd 4924 | . . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ∪ 𝑢 = ∪ 𝑈) |
4 | 3 | dmeqd 5918 | . . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom ∪ 𝑢 = dom ∪ 𝑈) |
5 | 4 | opeq2d 4884 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 〈(Base‘ndx), dom ∪ 𝑢〉 = 〈(Base‘ndx), dom ∪ 𝑈〉) |
6 | 2 | opeq2d 4884 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 〈(UnifSet‘ndx), 𝑢〉 = 〈(UnifSet‘ndx), 𝑈〉) |
7 | 5, 6 | preq12d 4745 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → {〈(Base‘ndx), dom ∪ 𝑢〉, 〈(UnifSet‘ndx), 𝑢〉} = {〈(Base‘ndx), dom ∪ 𝑈〉, 〈(UnifSet‘ndx), 𝑈〉}) |
8 | 2 | fveq2d 6910 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → (unifTop‘𝑢) = (unifTop‘𝑈)) |
9 | 8 | opeq2d 4884 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 〈(TopSet‘ndx), (unifTop‘𝑢)〉 = 〈(TopSet‘ndx), (unifTop‘𝑈)〉) |
10 | 7, 9 | oveq12d 7448 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ({〈(Base‘ndx), dom ∪ 𝑢〉, 〈(UnifSet‘ndx), 𝑢〉} sSet 〈(TopSet‘ndx), (unifTop‘𝑢)〉) = ({〈(Base‘ndx), dom ∪ 𝑈〉, 〈(UnifSet‘ndx), 𝑈〉} sSet 〈(TopSet‘ndx), (unifTop‘𝑈)〉)) |
11 | elfvunirn 6938 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn) | |
12 | ovexd 7465 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ({〈(Base‘ndx), dom ∪ 𝑈〉, 〈(UnifSet‘ndx), 𝑈〉} sSet 〈(TopSet‘ndx), (unifTop‘𝑈)〉) ∈ V) | |
13 | 1, 10, 11, 12 | fvmptd2 7023 | 1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({〈(Base‘ndx), dom ∪ 𝑈〉, 〈(UnifSet‘ndx), 𝑈〉} sSet 〈(TopSet‘ndx), (unifTop‘𝑈)〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 {cpr 4632 〈cop 4636 ∪ cuni 4911 dom cdm 5688 ran crn 5689 ‘cfv 6562 (class class class)co 7430 sSet csts 17196 ndxcnx 17226 Basecbs 17244 TopSetcts 17303 UnifSetcunif 17307 UnifOncust 24223 unifTopcutop 24254 toUnifSpctus 24279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-tus 24282 |
This theorem is referenced by: tuslem 24290 tuslemOLD 24291 |
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