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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 22062 | . . 3 ⊢ toUnifSp = (𝑢 ∈ ∪ ran UnifOn ↦ ({〈(Base‘ndx), dom ∪ 𝑢〉, 〈(UnifSet‘ndx), 𝑢〉} sSet 〈(TopSet‘ndx), (unifTop‘𝑢)〉)) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → toUnifSp = (𝑢 ∈ ∪ ran UnifOn ↦ ({〈(Base‘ndx), dom ∪ 𝑢〉, 〈(UnifSet‘ndx), 𝑢〉} sSet 〈(TopSet‘ndx), (unifTop‘𝑢)〉))) |
3 | simpr 477 | . . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈) | |
4 | 3 | unieqd 4446 | . . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ∪ 𝑢 = ∪ 𝑈) |
5 | 4 | dmeqd 5326 | . . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom ∪ 𝑢 = dom ∪ 𝑈) |
6 | 5 | opeq2d 4409 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 〈(Base‘ndx), dom ∪ 𝑢〉 = 〈(Base‘ndx), dom ∪ 𝑈〉) |
7 | 3 | opeq2d 4409 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 〈(UnifSet‘ndx), 𝑢〉 = 〈(UnifSet‘ndx), 𝑈〉) |
8 | 6, 7 | preq12d 4276 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → {〈(Base‘ndx), dom ∪ 𝑢〉, 〈(UnifSet‘ndx), 𝑢〉} = {〈(Base‘ndx), dom ∪ 𝑈〉, 〈(UnifSet‘ndx), 𝑈〉}) |
9 | 3 | fveq2d 6195 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → (unifTop‘𝑢) = (unifTop‘𝑈)) |
10 | 9 | opeq2d 4409 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 〈(TopSet‘ndx), (unifTop‘𝑢)〉 = 〈(TopSet‘ndx), (unifTop‘𝑈)〉) |
11 | 8, 10 | oveq12d 6668 | . 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)〉)) |
12 | elrnust 22028 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn) | |
13 | ovexd 6680 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ({〈(Base‘ndx), dom ∪ 𝑈〉, 〈(UnifSet‘ndx), 𝑈〉} sSet 〈(TopSet‘ndx), (unifTop‘𝑈)〉) ∈ V) | |
14 | 2, 11, 12, 13 | fvmptd 6288 | 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 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {cpr 4179 〈cop 4183 ∪ cuni 4436 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 ‘cfv 5888 (class class class)co 6650 ndxcnx 15854 sSet csts 15855 Basecbs 15857 TopSetcts 15947 UnifSetcunif 15951 UnifOncust 22003 unifTopcutop 22034 toUnifSpctus 22059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 df-ov 6653 df-ust 22004 df-tus 22062 |
This theorem is referenced by: tuslem 22071 |
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