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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 6191 | . . . 4 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
2 | 1 | fveq2d 6195 | . . 3 ⊢ (𝑤 = 𝑊 → (Fil‘(Base‘𝑤)) = (Fil‘(Base‘𝑊))) |
3 | fveq2 6191 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (UnifSt‘𝑤) = (UnifSt‘𝑊)) | |
4 | 3 | fveq2d 6195 | . . . . 5 ⊢ (𝑤 = 𝑊 → (CauFilu‘(UnifSt‘𝑤)) = (CauFilu‘(UnifSt‘𝑊))) |
5 | 4 | eleq2d 2687 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) ↔ 𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)))) |
6 | fveq2 6191 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊)) | |
7 | 6 | oveq1d 6665 | . . . . 5 ⊢ (𝑤 = 𝑊 → ((TopOpen‘𝑤) fLim 𝑐) = ((TopOpen‘𝑊) fLim 𝑐)) |
8 | 7 | neeq1d 2853 | . . . 4 ⊢ (𝑤 = 𝑊 → (((TopOpen‘𝑤) fLim 𝑐) ≠ ∅ ↔ ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)) |
9 | 5, 8 | imbi12d 334 | . . 3 ⊢ (𝑤 = 𝑊 → ((𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅) ↔ (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))) |
10 | 2, 9 | raleqbidv 3152 | . 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 22102 | . 2 ⊢ CUnifSp = {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)} | |
12 | 10, 11 | elrab2 3366 | 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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∅c0 3915 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 TopOpenctopn 16082 Filcfil 21649 fLim cflim 21738 UnifStcuss 22057 UnifSpcusp 22058 CauFiluccfilu 22090 CUnifSpccusp 22101 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
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-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-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-cusp 22102 |
This theorem is referenced by: cuspusp 22104 cuspcvg 22105 iscusp2 22106 cmetcusp 23150 |
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