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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 35265 | . . . . 5 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top) | |
| 2 | 1 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐶 ∈ Top) |
| 3 | eqid 2737 | . . . . 5 ⊢ ∪ 𝐶 = ∪ 𝐶 | |
| 4 | 3 | toptopon 22923 | . . . 4 ⊢ (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘∪ 𝐶)) |
| 5 | 2, 4 | sylib 218 | . . 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 35272 | . . . . . 6 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑇 ⊆ 𝐶) |
| 8 | 7 | 3ad2ant2 1135 | . . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝑇 ⊆ 𝐶) |
| 9 | simp3 1139 | . . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝑇) | |
| 10 | 8, 9 | sseldd 3984 | . . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝐶) |
| 11 | elssuni 4937 | . . . 4 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ⊆ ∪ 𝐶) | |
| 12 | 10, 11 | syl 17 | . . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ ∪ 𝐶) |
| 13 | resttopon 23169 | . . 3 ⊢ ((𝐶 ∈ (TopOn‘∪ 𝐶) ∧ 𝐴 ⊆ ∪ 𝐶) → (𝐶 ↾t 𝐴) ∈ (TopOn‘𝐴)) | |
| 14 | 5, 12, 13 | syl2anc 584 | . 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐶 ↾t 𝐴) ∈ (TopOn‘𝐴)) |
| 15 | cvmtop2 35266 | . . . . 5 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top) | |
| 16 | 15 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐽 ∈ Top) |
| 17 | eqid 2737 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 18 | 17 | toptopon 22923 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
| 19 | 16, 18 | sylib 218 | . . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
| 20 | 6 | cvmsrcl 35269 | . . . . 5 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑈 ∈ 𝐽) |
| 21 | 20 | 3ad2ant2 1135 | . . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝑈 ∈ 𝐽) |
| 22 | elssuni 4937 | . . . 4 ⊢ (𝑈 ∈ 𝐽 → 𝑈 ⊆ ∪ 𝐽) | |
| 23 | 21, 22 | syl 17 | . . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝑈 ⊆ ∪ 𝐽) |
| 24 | resttopon 23169 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑈 ⊆ ∪ 𝐽) → (𝐽 ↾t 𝑈) ∈ (TopOn‘𝑈)) | |
| 25 | 19, 23, 24 | syl2anc 584 | . 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐽 ↾t 𝑈) ∈ (TopOn‘𝑈)) |
| 26 | 6 | cvmshmeo 35276 | . . 3 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐹 ↾ 𝐴) ∈ ((𝐶 ↾t 𝐴)Homeo(𝐽 ↾t 𝑈))) |
| 27 | 26 | 3adant1 1131 | . 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐹 ↾ 𝐴) ∈ ((𝐶 ↾t 𝐴)Homeo(𝐽 ↾t 𝑈))) |
| 28 | hmeof1o2 23771 | . 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 1373 | 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 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 ∪ cuni 4907 ↦ cmpt 5225 ◡ccnv 5684 ↾ cres 5687 “ cima 5688 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ↾t crest 17465 Topctop 22899 TopOnctopon 22916 Homeochmeo 23761 CovMap ccvm 35260 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-map 8868 df-en 8986 df-fin 8989 df-fi 9451 df-rest 17467 df-topgen 17488 df-top 22900 df-topon 22917 df-bases 22953 df-cn 23235 df-hmeo 23763 df-cvm 35261 |
| This theorem is referenced by: cvmsss2 35279 cvmfolem 35284 cvmliftmolem1 35286 cvmliftlem6 35295 cvmliftlem9 35298 cvmlift2lem9a 35308 |
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