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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftmoi | Structured version Visualization version GIF version |
Description: A lift of a continuous function from a connected and locally connected space over a covering map is unique when it exists. (Contributed by Mario Carneiro, 10-Mar-2015.) |
Ref | Expression |
---|---|
cvmliftmo.b | ⊢ 𝐵 = ∪ 𝐶 |
cvmliftmo.y | ⊢ 𝑌 = ∪ 𝐾 |
cvmliftmo.f | ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
cvmliftmo.k | ⊢ (𝜑 → 𝐾 ∈ Conn) |
cvmliftmo.l | ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn) |
cvmliftmo.o | ⊢ (𝜑 → 𝑂 ∈ 𝑌) |
cvmliftmoi.m | ⊢ (𝜑 → 𝑀 ∈ (𝐾 Cn 𝐶)) |
cvmliftmoi.n | ⊢ (𝜑 → 𝑁 ∈ (𝐾 Cn 𝐶)) |
cvmliftmoi.g | ⊢ (𝜑 → (𝐹 ∘ 𝑀) = (𝐹 ∘ 𝑁)) |
cvmliftmoi.p | ⊢ (𝜑 → (𝑀‘𝑂) = (𝑁‘𝑂)) |
Ref | Expression |
---|---|
cvmliftmoi | ⊢ (𝜑 → 𝑀 = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvmliftmo.b | . 2 ⊢ 𝐵 = ∪ 𝐶 | |
2 | cvmliftmo.y | . 2 ⊢ 𝑌 = ∪ 𝐾 | |
3 | cvmliftmo.f | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) | |
4 | cvmliftmo.k | . 2 ⊢ (𝜑 → 𝐾 ∈ Conn) | |
5 | cvmliftmo.l | . 2 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn) | |
6 | cvmliftmo.o | . 2 ⊢ (𝜑 → 𝑂 ∈ 𝑌) | |
7 | cvmliftmoi.m | . 2 ⊢ (𝜑 → 𝑀 ∈ (𝐾 Cn 𝐶)) | |
8 | cvmliftmoi.n | . 2 ⊢ (𝜑 → 𝑁 ∈ (𝐾 Cn 𝐶)) | |
9 | cvmliftmoi.g | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝑀) = (𝐹 ∘ 𝑁)) | |
10 | cvmliftmoi.p | . 2 ⊢ (𝜑 → (𝑀‘𝑂) = (𝑁‘𝑂)) | |
11 | eqid 2725 | . . 3 ⊢ (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) | |
12 | 11 | cvmscbv 35001 | . 2 ⊢ (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) = (𝑏 ∈ 𝐽 ↦ {𝑚 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑚 = (◡𝐹 “ 𝑏) ∧ ∀𝑟 ∈ 𝑚 (∀𝑤 ∈ (𝑚 ∖ {𝑟})(𝑟 ∩ 𝑤) = ∅ ∧ (𝐹 ↾ 𝑟) ∈ ((𝐶 ↾t 𝑟)Homeo(𝐽 ↾t 𝑏))))}) |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12 | cvmliftmolem2 35025 | 1 ⊢ (𝜑 → 𝑀 = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 {crab 3418 ∖ cdif 3941 ∩ cin 3943 ∅c0 4322 𝒫 cpw 4604 {csn 4630 ∪ cuni 4909 ↦ cmpt 5232 ◡ccnv 5677 ↾ cres 5680 “ cima 5681 ∘ ccom 5682 ‘cfv 6549 (class class class)co 7419 ↾t crest 17410 Cn ccn 23177 Conncconn 23364 𝑛-Locally cnlly 23418 Homeochmeo 23706 CovMap ccvm 34998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-map 8847 df-en 8965 df-fin 8968 df-fi 9441 df-rest 17412 df-topgen 17433 df-top 22845 df-topon 22862 df-bases 22898 df-cld 22972 df-nei 23051 df-cn 23180 df-conn 23365 df-nlly 23420 df-hmeo 23708 df-cvm 34999 |
This theorem is referenced by: cvmliftmo 35027 cvmliftphtlem 35060 |
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