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Theorem cvmsf1o 32751
 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 32739 . . . . 5 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
213ad2ant1 1131 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐶 ∈ Top)
3 eqid 2759 . . . . 5 𝐶 = 𝐶
43toptopon 21618 . . . 4 (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘ 𝐶))
52, 4sylib 221 . . 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 32746 . . . . . 6 (𝑇 ∈ (𝑆𝑈) → 𝑇𝐶)
873ad2ant2 1132 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇𝐶)
9 simp3 1136 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴𝑇)
108, 9sseldd 3894 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴𝐶)
11 elssuni 4831 . . . 4 (𝐴𝐶𝐴 𝐶)
1210, 11syl 17 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴 𝐶)
13 resttopon 21862 . . 3 ((𝐶 ∈ (TopOn‘ 𝐶) ∧ 𝐴 𝐶) → (𝐶t 𝐴) ∈ (TopOn‘𝐴))
145, 12, 13syl2anc 588 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐶t 𝐴) ∈ (TopOn‘𝐴))
15 cvmtop2 32740 . . . . 5 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
16153ad2ant1 1131 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐽 ∈ Top)
17 eqid 2759 . . . . 5 𝐽 = 𝐽
1817toptopon 21618 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1916, 18sylib 221 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐽 ∈ (TopOn‘ 𝐽))
206cvmsrcl 32743 . . . . 5 (𝑇 ∈ (𝑆𝑈) → 𝑈𝐽)
21203ad2ant2 1132 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑈𝐽)
22 elssuni 4831 . . . 4 (𝑈𝐽𝑈 𝐽)
2321, 22syl 17 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑈 𝐽)
24 resttopon 21862 . . 3 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝑈 𝐽) → (𝐽t 𝑈) ∈ (TopOn‘𝑈))
2519, 23, 24syl2anc 588 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐽t 𝑈) ∈ (TopOn‘𝑈))
266cvmshmeo 32750 . . 3 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝐴) ∈ ((𝐶t 𝐴)Homeo(𝐽t 𝑈)))
27263adant1 1128 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝐴) ∈ ((𝐶t 𝐴)Homeo(𝐽t 𝑈)))
28 hmeof1o2 22464 . 2 (((𝐶t 𝐴) ∈ (TopOn‘𝐴) ∧ (𝐽t 𝑈) ∈ (TopOn‘𝑈) ∧ (𝐹𝐴) ∈ ((𝐶t 𝐴)Homeo(𝐽t 𝑈))) → (𝐹𝐴):𝐴1-1-onto𝑈)
2914, 25, 27, 28syl3anc 1369 1 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝐴):𝐴1-1-onto𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3071  {crab 3075   ∖ cdif 3856   ∩ cin 3858   ⊆ wss 3859  ∅c0 4226  𝒫 cpw 4495  {csn 4523  ∪ cuni 4799   ↦ cmpt 5113  ◡ccnv 5524   ↾ cres 5527   “ cima 5528  –1-1-onto→wf1o 6335  ‘cfv 6336  (class class class)co 7151   ↾t crest 16753  Topctop 21594  TopOnctopon 21611  Homeochmeo 22454   CovMap ccvm 32734 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7581  df-1st 7694  df-2nd 7695  df-map 8419  df-en 8529  df-fin 8532  df-fi 8909  df-rest 16755  df-topgen 16776  df-top 21595  df-topon 21612  df-bases 21647  df-cn 21928  df-hmeo 22456  df-cvm 32735 This theorem is referenced by:  cvmsss2  32753  cvmfolem  32758  cvmliftmolem1  32760  cvmliftlem6  32769  cvmliftlem9  32772  cvmlift2lem9a  32782
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