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Theorem cvmsf1o 32121
 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 32109 . . . . 5 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
213ad2ant1 1126 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐶 ∈ Top)
3 eqid 2794 . . . . 5 𝐶 = 𝐶
43toptopon 21209 . . . 4 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘ 𝐶))
52, 4sylib 219 . . 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 32116 . . . . . 6 (𝑇 ∈ (𝑆𝑈) → 𝑇𝐶)
873ad2ant2 1127 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇𝐶)
9 simp3 1131 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴𝑇)
108, 9sseldd 3892 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴𝐶)
11 elssuni 4776 . . . 4 (𝐴𝐶𝐴 𝐶)
1210, 11syl 17 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴 𝐶)
13 resttopon 21453 . . 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 32110 . . . . 5 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
16153ad2ant1 1126 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐽 ∈ Top)
17 eqid 2794 . . . . 5 𝐽 = 𝐽
1817toptopon 21209 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1916, 18sylib 219 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐽 ∈ (TopOn‘ 𝐽))
206cvmsrcl 32113 . . . . 5 (𝑇 ∈ (𝑆𝑈) → 𝑈𝐽)
21203ad2ant2 1127 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑈𝐽)
22 elssuni 4776 . . . 4 (𝑈𝐽𝑈 𝐽)
2321, 22syl 17 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑈 𝐽)
24 resttopon 21453 . . 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 32120 . . 3 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝐴) ∈ ((𝐶t 𝐴)Homeo(𝐽t 𝑈)))
27263adant1 1123 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝐴) ∈ ((𝐶t 𝐴)Homeo(𝐽t 𝑈)))
28 hmeof1o2 22055 . 2 (((𝐶t 𝐴) ∈ (TopOn‘𝐴) ∧ (𝐽t 𝑈) ∈ (TopOn‘𝑈) ∧ (𝐹𝐴) ∈ ((𝐶t 𝐴)Homeo(𝐽t 𝑈))) → (𝐹𝐴):𝐴1-1-onto𝑈)
2914, 25, 27, 28syl3anc 1364 1 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝐴):𝐴1-1-onto𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2080  ∀wral 3104  {crab 3108   ∖ cdif 3858   ∩ cin 3860   ⊆ wss 3861  ∅c0 4213  𝒫 cpw 4455  {csn 4474  ∪ cuni 4747   ↦ cmpt 5043  ◡ccnv 5445   ↾ cres 5448   “ cima 5449  –1-1-onto→wf1o 6227  ‘cfv 6228  (class class class)co 7019   ↾t crest 16523  Topctop 21185  TopOnctopon 21202  Homeochmeo 22045   CovMap ccvm 32104 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-rep 5084  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-reu 3111  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-pss 3878  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-tp 4479  df-op 4481  df-uni 4748  df-int 4785  df-iun 4829  df-br 4965  df-opab 5027  df-mpt 5044  df-tr 5067  df-id 5351  df-eprel 5356  df-po 5365  df-so 5366  df-fr 5405  df-we 5407  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-pred 6026  df-ord 6072  df-on 6073  df-lim 6074  df-suc 6075  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-ov 7022  df-oprab 7023  df-mpo 7024  df-om 7440  df-1st 7548  df-2nd 7549  df-wrecs 7801  df-recs 7863  df-rdg 7901  df-oadd 7960  df-er 8142  df-map 8261  df-en 8361  df-fin 8364  df-fi 8724  df-rest 16525  df-topgen 16546  df-top 21186  df-topon 21203  df-bases 21238  df-cn 21519  df-hmeo 22047  df-cvm 32105 This theorem is referenced by:  cvmsss2  32123  cvmfolem  32128  cvmliftmolem1  32130  cvmliftlem6  32139  cvmliftlem9  32142  cvmlift2lem9a  32152
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