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Theorem cvmsiota 31020
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 31019 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → ∃!𝑥𝑇 𝐴𝑥)
5 riotacl2 6589 . . . 4 (∃!𝑥𝑇 𝐴𝑥 → (𝑥𝑇 𝐴𝑥) ∈ {𝑥𝑇𝐴𝑥})
64, 5syl 17 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → (𝑥𝑇 𝐴𝑥) ∈ {𝑥𝑇𝐴𝑥})
71, 6syl5eqel 2702 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝑊 ∈ {𝑥𝑇𝐴𝑥})
8 eleq2 2687 . . 3 (𝑣 = 𝑊 → (𝐴𝑣𝐴𝑊))
9 eleq2 2687 . . . 4 (𝑥 = 𝑣 → (𝐴𝑥𝐴𝑣))
109cbvrabv 3189 . . 3 {𝑥𝑇𝐴𝑥} = {𝑣𝑇𝐴𝑣}
118, 10elrab2 3353 . 2 (𝑊 ∈ {𝑥𝑇𝐴𝑥} ↔ (𝑊𝑇𝐴𝑊))
127, 11sylib 208 1 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → (𝑊𝑇𝐴𝑊))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  ∃!wreu 2910  {crab 2912  cdif 3557  cin 3559  c0 3897  𝒫 cpw 4136  {csn 4155   cuni 4409  cmpt 4683  ccnv 5083  cres 5086  cima 5087  cfv 5857  crio 6575  (class class class)co 6615  t crest 16021  Homeochmeo 21496   CovMap ccvm 30998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-top 20639  df-topon 20656  df-cn 20971  df-cvm 30999
This theorem is referenced by:  cvmopnlem  31021  cvmliftmolem2  31025  cvmliftlem6  31033  cvmliftlem8  31035  cvmliftlem9  31036  cvmlift2lem9  31054  cvmlift3lem6  31067  cvmlift3lem7  31068
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