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Mirrors > Home > MPE Home > Th. List > iscusp | Structured version Visualization version GIF version |
Description: The predicate "𝑊 is a complete uniform space." (Contributed by Thierry Arnoux, 3-Dec-2017.) |
Ref | Expression |
---|---|
iscusp | ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6103 | . . . 4 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
2 | 1 | fveq2d 6107 | . . 3 ⊢ (𝑤 = 𝑊 → (Fil‘(Base‘𝑤)) = (Fil‘(Base‘𝑊))) |
3 | fveq2 6103 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (UnifSt‘𝑤) = (UnifSt‘𝑊)) | |
4 | 3 | fveq2d 6107 | . . . . 5 ⊢ (𝑤 = 𝑊 → (CauFilu‘(UnifSt‘𝑤)) = (CauFilu‘(UnifSt‘𝑊))) |
5 | 4 | eleq2d 2673 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) ↔ 𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)))) |
6 | fveq2 6103 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊)) | |
7 | 6 | oveq1d 6564 | . . . . 5 ⊢ (𝑤 = 𝑊 → ((TopOpen‘𝑤) fLim 𝑐) = ((TopOpen‘𝑊) fLim 𝑐)) |
8 | 7 | neeq1d 2841 | . . . 4 ⊢ (𝑤 = 𝑊 → (((TopOpen‘𝑤) fLim 𝑐) ≠ ∅ ↔ ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)) |
9 | 5, 8 | imbi12d 333 | . . 3 ⊢ (𝑤 = 𝑊 → ((𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅) ↔ (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))) |
10 | 2, 9 | raleqbidv 3129 | . 2 ⊢ (𝑤 = 𝑊 → (∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅) ↔ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))) |
11 | df-cusp 21912 | . 2 ⊢ CUnifSp = {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)} | |
12 | 10, 11 | elrab2 3333 | 1 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 TopOpenctopn 15905 Filcfil 21459 fLim cflim 21548 UnifStcuss 21867 UnifSpcusp 21868 CauFiluccfilu 21900 CUnifSpccusp 21911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-cusp 21912 |
This theorem is referenced by: cuspusp 21914 cuspcvg 21915 iscusp2 21916 cmetcusp 22958 |
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