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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmsuni | Structured version Visualization version GIF version |
Description: An even covering of 𝑈 has union equal to the preimage of 𝑈 by 𝐹. (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 |
---|---|
cvmsuni | ⊢ (𝑇 ∈ (𝑆‘𝑈) → ∪ 𝑇 = (◡𝐹 “ 𝑈)) |
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 30501 | . . 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 | simp3d 1068 | . 2 ⊢ (𝑇 ∈ (𝑆‘𝑈) → (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))) |
4 | 3 | simpld 474 | 1 ⊢ (𝑇 ∈ (𝑆‘𝑈) → ∪ 𝑇 = (◡𝐹 “ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 ∪ cuni 4372 ↦ cmpt 4643 ◡ccnv 5037 ↾ cres 5040 “ cima 5041 ‘cfv 5804 (class class class)co 6549 ↾t crest 15904 Homeochmeo 21366 |
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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
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-eu 2462 df-mo 2463 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-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 |
This theorem is referenced by: cvmscld 30509 cvmsss2 30510 cvmseu 30512 cvmliftmolem1 30517 cvmlift2lem9 30547 cvmlift3lem6 30560 |
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