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Theorem cvmsiota 32703
 Description: Identify the unique element of 𝑇 containing 𝐴. (Contributed by Mario Carneiro, 14-Feb-2015.)
Hypotheses
Ref Expression
cvmcov.1 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
cvmseu.1 𝐵 = 𝐶
cvmsiota.2 𝑊 = (𝑥𝑇 𝐴𝑥)
Assertion
Ref Expression
cvmsiota ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → (𝑊𝑇𝐴𝑊))
Distinct variable groups:   𝑘,𝑠,𝑢,𝑣,𝑥,𝐶   𝑘,𝐹,𝑠,𝑢,𝑣,𝑥   𝑘,𝐽,𝑠,𝑢,𝑣,𝑥   𝑥,𝑆   𝑈,𝑘,𝑠,𝑢,𝑣,𝑥   𝑇,𝑠,𝑢,𝑣,𝑥   𝑣,𝑊   𝑢,𝐴,𝑣,𝑥   𝑣,𝐵,𝑥
Allowed substitution hints:   𝐴(𝑘,𝑠)   𝐵(𝑢,𝑘,𝑠)   𝑆(𝑣,𝑢,𝑘,𝑠)   𝑇(𝑘)   𝑊(𝑥,𝑢,𝑘,𝑠)

Proof of Theorem cvmsiota
StepHypRef Expression
1 cvmsiota.2 . . 3 𝑊 = (𝑥𝑇 𝐴𝑥)
2 cvmcov.1 . . . . 5 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
3 cvmseu.1 . . . . 5 𝐵 = 𝐶
42, 3cvmseu 32702 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → ∃!𝑥𝑇 𝐴𝑥)
5 riotacl2 7119 . . . 4 (∃!𝑥𝑇 𝐴𝑥 → (𝑥𝑇 𝐴𝑥) ∈ {𝑥𝑇𝐴𝑥})
64, 5syl 17 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → (𝑥𝑇 𝐴𝑥) ∈ {𝑥𝑇𝐴𝑥})
71, 6eqeltrid 2894 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝑊 ∈ {𝑥𝑇𝐴𝑥})
8 eleq2 2878 . . 3 (𝑣 = 𝑊 → (𝐴𝑣𝐴𝑊))
9 eleq2 2878 . . . 4 (𝑥 = 𝑣 → (𝐴𝑥𝐴𝑣))
109cbvrabv 3440 . . 3 {𝑥𝑇𝐴𝑥} = {𝑣𝑇𝐴𝑣}
118, 10elrab2 3633 . 2 (𝑊 ∈ {𝑥𝑇𝐴𝑥} ↔ (𝑊𝑇𝐴𝑊))
127, 11sylib 221 1 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → (𝑊𝑇𝐴𝑊))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃!wreu 3108  {crab 3110   ∖ cdif 3880   ∩ cin 3882  ∅c0 4246  𝒫 cpw 4500  {csn 4528  ∪ cuni 4804   ↦ cmpt 5114  ◡ccnv 5522   ↾ cres 5525   “ cima 5526  ‘cfv 6332  ℩crio 7102  (class class class)co 7145   ↾t crest 16706  Homeochmeo 22399   CovMap ccvm 32681 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8409  df-top 21540  df-topon 21557  df-cn 21873  df-cvm 32682 This theorem is referenced by:  cvmopnlem  32704  cvmliftmolem2  32708  cvmliftlem6  32716  cvmliftlem8  32718  cvmliftlem9  32719  cvmlift2lem9  32737  cvmlift3lem6  32750  cvmlift3lem7  32751
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