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Theorem cvmsf1o 35277
Description: 𝐹, localized to an element of an even covering of 𝑈, is a bijection. (Contributed by Mario Carneiro, 14-Feb-2015.)
Hypothesis
Ref Expression
cvmcov.1 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
Assertion
Ref Expression
cvmsf1o ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝐴):𝐴1-1-onto𝑈)
Distinct variable groups:   𝑘,𝑠,𝑢,𝑣,𝐶   𝑘,𝐹,𝑠,𝑢,𝑣   𝑘,𝐽,𝑠,𝑢,𝑣   𝑈,𝑘,𝑠,𝑢,𝑣   𝑇,𝑠,𝑢,𝑣   𝑢,𝐴,𝑣
Allowed substitution hints:   𝐴(𝑘,𝑠)   𝑆(𝑣,𝑢,𝑘,𝑠)   𝑇(𝑘)

Proof of Theorem cvmsf1o
StepHypRef Expression
1 cvmtop1 35265 . . . . 5 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
213ad2ant1 1134 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐶 ∈ Top)
3 eqid 2737 . . . . 5 𝐶 = 𝐶
43toptopon 22923 . . . 4 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘ 𝐶))
52, 4sylib 218 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐶 ∈ (TopOn‘ 𝐶))
6 cvmcov.1 . . . . . . 7 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
76cvmsss 35272 . . . . . 6 (𝑇 ∈ (𝑆𝑈) → 𝑇𝐶)
873ad2ant2 1135 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇𝐶)
9 simp3 1139 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴𝑇)
108, 9sseldd 3984 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴𝐶)
11 elssuni 4937 . . . 4 (𝐴𝐶𝐴 𝐶)
1210, 11syl 17 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴 𝐶)
13 resttopon 23169 . . 3 ((𝐶 ∈ (TopOn‘ 𝐶) ∧ 𝐴 𝐶) → (𝐶t 𝐴) ∈ (TopOn‘𝐴))
145, 12, 13syl2anc 584 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐶t 𝐴) ∈ (TopOn‘𝐴))
15 cvmtop2 35266 . . . . 5 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
16153ad2ant1 1134 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐽 ∈ Top)
17 eqid 2737 . . . . 5 𝐽 = 𝐽
1817toptopon 22923 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1916, 18sylib 218 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐽 ∈ (TopOn‘ 𝐽))
206cvmsrcl 35269 . . . . 5 (𝑇 ∈ (𝑆𝑈) → 𝑈𝐽)
21203ad2ant2 1135 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑈𝐽)
22 elssuni 4937 . . . 4 (𝑈𝐽𝑈 𝐽)
2321, 22syl 17 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑈 𝐽)
24 resttopon 23169 . . 3 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝑈 𝐽) → (𝐽t 𝑈) ∈ (TopOn‘𝑈))
2519, 23, 24syl2anc 584 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐽t 𝑈) ∈ (TopOn‘𝑈))
266cvmshmeo 35276 . . 3 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝐴) ∈ ((𝐶t 𝐴)Homeo(𝐽t 𝑈)))
27263adant1 1131 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝐴) ∈ ((𝐶t 𝐴)Homeo(𝐽t 𝑈)))
28 hmeof1o2 23771 . 2 (((𝐶t 𝐴) ∈ (TopOn‘𝐴) ∧ (𝐽t 𝑈) ∈ (TopOn‘𝑈) ∧ (𝐹𝐴) ∈ ((𝐶t 𝐴)Homeo(𝐽t 𝑈))) → (𝐹𝐴):𝐴1-1-onto𝑈)
2914, 25, 27, 28syl3anc 1373 1 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝐴):𝐴1-1-onto𝑈)
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-ontowf1o 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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