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Theorem cvmsiota 34892
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 34891 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → ∃!𝑥𝑇 𝐴𝑥)
5 riotacl2 7397 . . . 4 (∃!𝑥𝑇 𝐴𝑥 → (𝑥𝑇 𝐴𝑥) ∈ {𝑥𝑇𝐴𝑥})
64, 5syl 17 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → (𝑥𝑇 𝐴𝑥) ∈ {𝑥𝑇𝐴𝑥})
71, 6eqeltrid 2832 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝑊 ∈ {𝑥𝑇𝐴𝑥})
8 eleq2 2817 . . 3 (𝑣 = 𝑊 → (𝐴𝑣𝐴𝑊))
9 eleq2 2817 . . . 4 (𝑥 = 𝑣 → (𝐴𝑥𝐴𝑣))
109cbvrabv 3439 . . 3 {𝑥𝑇𝐴𝑥} = {𝑣𝑇𝐴𝑣}
118, 10elrab2 3685 . 2 (𝑊 ∈ {𝑥𝑇𝐴𝑥} ↔ (𝑊𝑇𝐴𝑊))
127, 11sylib 217 1 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → (𝑊𝑇𝐴𝑊))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3057  ∃!wreu 3370  {crab 3428  cdif 3944  cin 3946  c0 4324  𝒫 cpw 4604  {csn 4630   cuni 4910  cmpt 5233  ccnv 5679  cres 5682  cima 5683  cfv 6551  crio 7379  (class class class)co 7424  t crest 17407  Homeochmeo 23675   CovMap ccvm 34870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-map 8851  df-top 22814  df-topon 22831  df-cn 23149  df-cvm 34871
This theorem is referenced by:  cvmopnlem  34893  cvmliftmolem2  34897  cvmliftlem6  34905  cvmliftlem8  34907  cvmliftlem9  34908  cvmlift2lem9  34926  cvmlift3lem6  34939  cvmlift3lem7  34940
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