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 2823 | . . 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 32507 | . 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 32531 | 1 ⊢ (𝜑 → 𝑀 = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 ∖ cdif 3935 ∩ cin 3937 ∅c0 4293 𝒫 cpw 4541 {csn 4569 ∪ cuni 4840 ↦ cmpt 5148 ◡ccnv 5556 ↾ cres 5559 “ cima 5560 ∘ ccom 5561 ‘cfv 6357 (class class class)co 7158 ↾t crest 16696 Cn ccn 21834 Conncconn 22021 𝑛-Locally cnlly 22075 Homeochmeo 22363 CovMap ccvm 32504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-fin 8515 df-fi 8877 df-rest 16698 df-topgen 16719 df-top 21504 df-topon 21521 df-bases 21556 df-cld 21629 df-nei 21708 df-cn 21837 df-conn 22022 df-nlly 22077 df-hmeo 22365 df-cvm 32505 |
This theorem is referenced by: cvmliftmo 32533 cvmliftphtlem 32566 |
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