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Theorem cvmsf1o 35257
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 35245 . . . . 5 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
213ad2ant1 1132 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐶 ∈ Top)
3 eqid 2735 . . . . 5 𝐶 = 𝐶
43toptopon 22939 . . . 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 35252 . . . . . 6 (𝑇 ∈ (𝑆𝑈) → 𝑇𝐶)
873ad2ant2 1133 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇𝐶)
9 simp3 1137 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴𝑇)
108, 9sseldd 3996 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴𝐶)
11 elssuni 4942 . . . 4 (𝐴𝐶𝐴 𝐶)
1210, 11syl 17 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴 𝐶)
13 resttopon 23185 . . 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 35246 . . . . 5 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
16153ad2ant1 1132 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐽 ∈ Top)
17 eqid 2735 . . . . 5 𝐽 = 𝐽
1817toptopon 22939 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1916, 18sylib 218 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐽 ∈ (TopOn‘ 𝐽))
206cvmsrcl 35249 . . . . 5 (𝑇 ∈ (𝑆𝑈) → 𝑈𝐽)
21203ad2ant2 1133 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑈𝐽)
22 elssuni 4942 . . . 4 (𝑈𝐽𝑈 𝐽)
2321, 22syl 17 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑈 𝐽)
24 resttopon 23185 . . 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 35256 . . 3 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝐴) ∈ ((𝐶t 𝐴)Homeo(𝐽t 𝑈)))
27263adant1 1129 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝐴) ∈ ((𝐶t 𝐴)Homeo(𝐽t 𝑈)))
28 hmeof1o2 23787 . 2 (((𝐶t 𝐴) ∈ (TopOn‘𝐴) ∧ (𝐽t 𝑈) ∈ (TopOn‘𝑈) ∧ (𝐹𝐴) ∈ ((𝐶t 𝐴)Homeo(𝐽t 𝑈))) → (𝐹𝐴):𝐴1-1-onto𝑈)
2914, 25, 27, 28syl3anc 1370 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 1086   = wceq 1537  wcel 2106  wral 3059  {crab 3433  cdif 3960  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   cuni 4912  cmpt 5231  ccnv 5688  cres 5691  cima 5692  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  t crest 17467  Topctop 22915  TopOnctopon 22932  Homeochmeo 23777   CovMap ccvm 35240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-map 8867  df-en 8985  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cn 23251  df-hmeo 23779  df-cvm 35241
This theorem is referenced by:  cvmsss2  35259  cvmfolem  35264  cvmliftmolem1  35266  cvmliftlem6  35275  cvmliftlem9  35278  cvmlift2lem9a  35288
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