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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmsss | Structured version Visualization version GIF version |
Description: An even covering is a subset of the topology of the domain (i.e. a collection of open sets). (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
cvmcov.1 | ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) |
Ref | Expression |
---|---|
cvmsss | ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑇 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvmcov.1 | . . . 4 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) | |
2 | 1 | cvmsi 31756 | . . 3 ⊢ (𝑇 ∈ (𝑆‘𝑈) → (𝑈 ∈ 𝐽 ∧ (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))) |
3 | 2 | simp2d 1174 | . 2 ⊢ (𝑇 ∈ (𝑆‘𝑈) → (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅)) |
4 | 3 | simpld 489 | 1 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑇 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2969 ∀wral 3087 {crab 3091 ∖ cdif 3764 ∩ cin 3766 ⊆ wss 3767 ∅c0 4113 𝒫 cpw 4347 {csn 4366 ∪ cuni 4626 ↦ cmpt 4920 ◡ccnv 5309 ↾ cres 5312 “ cima 5313 ‘cfv 6099 (class class class)co 6876 ↾t crest 16393 Homeochmeo 21882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-sbc 3632 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fv 6107 df-ov 6879 |
This theorem is referenced by: cvmsf1o 31763 cvmscld 31764 cvmsss2 31765 cvmfolem 31770 cvmliftmolem1 31772 cvmliftmolem2 31773 cvmliftlem6 31781 cvmlift2lem9a 31794 cvmlift2lem9 31802 cvmlift3lem6 31815 |
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