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Theorem cvmsiota 35262
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 35261 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → ∃!𝑥𝑇 𝐴𝑥)
5 riotacl2 7404 . . . 4 (∃!𝑥𝑇 𝐴𝑥 → (𝑥𝑇 𝐴𝑥) ∈ {𝑥𝑇𝐴𝑥})
64, 5syl 17 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → (𝑥𝑇 𝐴𝑥) ∈ {𝑥𝑇𝐴𝑥})
71, 6eqeltrid 2843 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝑊 ∈ {𝑥𝑇𝐴𝑥})
8 eleq2 2828 . . 3 (𝑣 = 𝑊 → (𝐴𝑣𝐴𝑊))
9 eleq2 2828 . . . 4 (𝑥 = 𝑣 → (𝐴𝑥𝐴𝑣))
109cbvrabv 3444 . . 3 {𝑥𝑇𝐴𝑥} = {𝑣𝑇𝐴𝑣}
118, 10elrab2 3698 . 2 (𝑊 ∈ {𝑥𝑇𝐴𝑥} ↔ (𝑊𝑇𝐴𝑊))
127, 11sylib 218 1 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → (𝑊𝑇𝐴𝑊))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  ∃!wreu 3376  {crab 3433  cdif 3960  cin 3962  c0 4339  𝒫 cpw 4605  {csn 4631   cuni 4912  cmpt 5231  ccnv 5688  cres 5691  cima 5692  cfv 6563  crio 7387  (class class class)co 7431  t crest 17467  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-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-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-rmo 3378  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-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  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-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-top 22916  df-topon 22933  df-cn 23251  df-cvm 35241
This theorem is referenced by:  cvmopnlem  35263  cvmliftmolem2  35267  cvmliftlem6  35275  cvmliftlem8  35277  cvmliftlem9  35278  cvmlift2lem9  35296  cvmlift3lem6  35309  cvmlift3lem7  35310
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