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| 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 24267 | . 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 4920 | . . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ∪ 𝑢 = ∪ 𝑈) | 
| 4 | 3 | dmeqd 5916 | . . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom ∪ 𝑢 = dom ∪ 𝑈) | 
| 5 | 4 | opeq2d 4880 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 〈(Base‘ndx), dom ∪ 𝑢〉 = 〈(Base‘ndx), dom ∪ 𝑈〉) | 
| 6 | 2 | opeq2d 4880 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 〈(UnifSet‘ndx), 𝑢〉 = 〈(UnifSet‘ndx), 𝑈〉) | 
| 7 | 5, 6 | preq12d 4741 | . . 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 4880 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 〈(TopSet‘ndx), (unifTop‘𝑢)〉 = 〈(TopSet‘ndx), (unifTop‘𝑈)〉) | 
| 10 | 7, 9 | oveq12d 7449 | . 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 7466 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ({〈(Base‘ndx), dom ∪ 𝑈〉, 〈(UnifSet‘ndx), 𝑈〉} sSet 〈(TopSet‘ndx), (unifTop‘𝑈)〉) ∈ V) | |
| 13 | 1, 10, 11, 12 | fvmptd2 7024 | 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 1540 ∈ wcel 2108 Vcvv 3480 {cpr 4628 〈cop 4632 ∪ cuni 4907 dom cdm 5685 ran crn 5686 ‘cfv 6561 (class class class)co 7431 sSet csts 17200 ndxcnx 17230 Basecbs 17247 TopSetcts 17303 UnifSetcunif 17307 UnifOncust 24208 unifTopcutop 24239 toUnifSpctus 24264 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-tus 24267 | 
| This theorem is referenced by: tuslem 24275 tuslemOLD 24276 | 
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