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Theorem cvmsf1o 35015
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 35003 . . . . 5 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
213ad2ant1 1130 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐶 ∈ Top)
3 eqid 2725 . . . . 5 𝐶 = 𝐶
43toptopon 22868 . . . 4 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘ 𝐶))
52, 4sylib 217 . . 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 35010 . . . . . 6 (𝑇 ∈ (𝑆𝑈) → 𝑇𝐶)
873ad2ant2 1131 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇𝐶)
9 simp3 1135 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴𝑇)
108, 9sseldd 3977 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴𝐶)
11 elssuni 4941 . . . 4 (𝐴𝐶𝐴 𝐶)
1210, 11syl 17 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴 𝐶)
13 resttopon 23114 . . 3 ((𝐶 ∈ (TopOn‘ 𝐶) ∧ 𝐴 𝐶) → (𝐶t 𝐴) ∈ (TopOn‘𝐴))
145, 12, 13syl2anc 582 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐶t 𝐴) ∈ (TopOn‘𝐴))
15 cvmtop2 35004 . . . . 5 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
16153ad2ant1 1130 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐽 ∈ Top)
17 eqid 2725 . . . . 5 𝐽 = 𝐽
1817toptopon 22868 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1916, 18sylib 217 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐽 ∈ (TopOn‘ 𝐽))
206cvmsrcl 35007 . . . . 5 (𝑇 ∈ (𝑆𝑈) → 𝑈𝐽)
21203ad2ant2 1131 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑈𝐽)
22 elssuni 4941 . . . 4 (𝑈𝐽𝑈 𝐽)
2321, 22syl 17 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑈 𝐽)
24 resttopon 23114 . . 3 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝑈 𝐽) → (𝐽t 𝑈) ∈ (TopOn‘𝑈))
2519, 23, 24syl2anc 582 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐽t 𝑈) ∈ (TopOn‘𝑈))
266cvmshmeo 35014 . . 3 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝐴) ∈ ((𝐶t 𝐴)Homeo(𝐽t 𝑈)))
27263adant1 1127 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝐴) ∈ ((𝐶t 𝐴)Homeo(𝐽t 𝑈)))
28 hmeof1o2 23716 . 2 (((𝐶t 𝐴) ∈ (TopOn‘𝐴) ∧ (𝐽t 𝑈) ∈ (TopOn‘𝑈) ∧ (𝐹𝐴) ∈ ((𝐶t 𝐴)Homeo(𝐽t 𝑈))) → (𝐹𝐴):𝐴1-1-onto𝑈)
2914, 25, 27, 28syl3anc 1368 1 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝐴):𝐴1-1-onto𝑈)
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-ontowf1o 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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