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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmsf1o | Structured version Visualization version GIF version |
Description: 𝐹, localized to an element of an even covering of 𝑈, is a bijection. (Contributed by Mario Carneiro, 14-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 |
---|---|
cvmsf1o | ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvmtop1 35003 | . . . . 5 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top) | |
2 | 1 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐶 ∈ Top) |
3 | eqid 2725 | . . . . 5 ⊢ ∪ 𝐶 = ∪ 𝐶 | |
4 | 3 | toptopon 22868 | . . . 4 ⊢ (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘∪ 𝐶)) |
5 | 2, 4 | sylib 217 | . . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐶 ∈ (TopOn‘∪ 𝐶)) |
6 | cvmcov.1 | . . . . . . 7 ⊢ 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))))}) | |
7 | 6 | cvmsss 35010 | . . . . . 6 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑇 ⊆ 𝐶) |
8 | 7 | 3ad2ant2 1131 | . . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝑇 ⊆ 𝐶) |
9 | simp3 1135 | . . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝑇) | |
10 | 8, 9 | sseldd 3977 | . . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝐶) |
11 | elssuni 4941 | . . . 4 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ⊆ ∪ 𝐶) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ ∪ 𝐶) |
13 | resttopon 23114 | . . 3 ⊢ ((𝐶 ∈ (TopOn‘∪ 𝐶) ∧ 𝐴 ⊆ ∪ 𝐶) → (𝐶 ↾t 𝐴) ∈ (TopOn‘𝐴)) | |
14 | 5, 12, 13 | syl2anc 582 | . 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐶 ↾t 𝐴) ∈ (TopOn‘𝐴)) |
15 | cvmtop2 35004 | . . . . 5 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top) | |
16 | 15 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐽 ∈ Top) |
17 | eqid 2725 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
18 | 17 | toptopon 22868 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
19 | 16, 18 | sylib 217 | . . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
20 | 6 | cvmsrcl 35007 | . . . . 5 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑈 ∈ 𝐽) |
21 | 20 | 3ad2ant2 1131 | . . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝑈 ∈ 𝐽) |
22 | elssuni 4941 | . . . 4 ⊢ (𝑈 ∈ 𝐽 → 𝑈 ⊆ ∪ 𝐽) | |
23 | 21, 22 | syl 17 | . . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝑈 ⊆ ∪ 𝐽) |
24 | resttopon 23114 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑈 ⊆ ∪ 𝐽) → (𝐽 ↾t 𝑈) ∈ (TopOn‘𝑈)) | |
25 | 19, 23, 24 | syl2anc 582 | . 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐽 ↾t 𝑈) ∈ (TopOn‘𝑈)) |
26 | 6 | cvmshmeo 35014 | . . 3 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐹 ↾ 𝐴) ∈ ((𝐶 ↾t 𝐴)Homeo(𝐽 ↾t 𝑈))) |
27 | 26 | 3adant1 1127 | . 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐹 ↾ 𝐴) ∈ ((𝐶 ↾t 𝐴)Homeo(𝐽 ↾t 𝑈))) |
28 | hmeof1o2 23716 | . 2 ⊢ (((𝐶 ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ (𝐽 ↾t 𝑈) ∈ (TopOn‘𝑈) ∧ (𝐹 ↾ 𝐴) ∈ ((𝐶 ↾t 𝐴)Homeo(𝐽 ↾t 𝑈))) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝑈) | |
29 | 14, 25, 27, 28 | syl3anc 1368 | 1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3050 {crab 3418 ∖ cdif 3941 ∩ cin 3943 ⊆ wss 3944 ∅c0 4322 𝒫 cpw 4604 {csn 4630 ∪ cuni 4909 ↦ cmpt 5232 ◡ccnv 5677 ↾ cres 5680 “ cima 5681 –1-1-onto→wf1o 6548 ‘cfv 6549 (class class class)co 7419 ↾t crest 17410 Topctop 22844 TopOnctopon 22861 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-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-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-cn 23180 df-hmeo 23708 df-cvm 34999 |
This theorem is referenced by: cvmsss2 35017 cvmfolem 35022 cvmliftmolem1 35024 cvmliftlem6 35033 cvmliftlem9 35036 cvmlift2lem9a 35046 |
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