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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 23692 | . 2 ⊢ toUnifSp = (𝑢 ∈ ∪ ran UnifOn ↦ ({〈(Base‘ndx), dom ∪ 𝑢〉, 〈(UnifSet‘ndx), 𝑢〉} sSet 〈(TopSet‘ndx), (unifTop‘𝑢)〉)) | |
2 | simpr 485 | . . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈) | |
3 | 2 | unieqd 4915 | . . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ∪ 𝑢 = ∪ 𝑈) |
4 | 3 | dmeqd 5897 | . . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom ∪ 𝑢 = dom ∪ 𝑈) |
5 | 4 | opeq2d 4873 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 〈(Base‘ndx), dom ∪ 𝑢〉 = 〈(Base‘ndx), dom ∪ 𝑈〉) |
6 | 2 | opeq2d 4873 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 〈(UnifSet‘ndx), 𝑢〉 = 〈(UnifSet‘ndx), 𝑈〉) |
7 | 5, 6 | preq12d 4738 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → {〈(Base‘ndx), dom ∪ 𝑢〉, 〈(UnifSet‘ndx), 𝑢〉} = {〈(Base‘ndx), dom ∪ 𝑈〉, 〈(UnifSet‘ndx), 𝑈〉}) |
8 | 2 | fveq2d 6882 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → (unifTop‘𝑢) = (unifTop‘𝑈)) |
9 | 8 | opeq2d 4873 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 〈(TopSet‘ndx), (unifTop‘𝑢)〉 = 〈(TopSet‘ndx), (unifTop‘𝑈)〉) |
10 | 7, 9 | oveq12d 7411 | . 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 6910 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn) | |
12 | ovexd 7428 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ({〈(Base‘ndx), dom ∪ 𝑈〉, 〈(UnifSet‘ndx), 𝑈〉} sSet 〈(TopSet‘ndx), (unifTop‘𝑈)〉) ∈ V) | |
13 | 1, 10, 11, 12 | fvmptd2 6992 | 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 396 = wceq 1541 ∈ wcel 2106 Vcvv 3473 {cpr 4624 〈cop 4628 ∪ cuni 4901 dom cdm 5669 ran crn 5670 ‘cfv 6532 (class class class)co 7393 sSet csts 17078 ndxcnx 17108 Basecbs 17126 TopSetcts 17185 UnifSetcunif 17189 UnifOncust 23633 unifTopcutop 23664 toUnifSpctus 23689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6484 df-fun 6534 df-fv 6540 df-ov 7396 df-tus 23692 |
This theorem is referenced by: tuslem 23700 tuslemOLD 23701 |
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