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Theorem cvmsiota 31591
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 31590 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → ∃!𝑥𝑇 𝐴𝑥)
5 riotacl2 6766 . . . 4 (∃!𝑥𝑇 𝐴𝑥 → (𝑥𝑇 𝐴𝑥) ∈ {𝑥𝑇𝐴𝑥})
64, 5syl 17 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → (𝑥𝑇 𝐴𝑥) ∈ {𝑥𝑇𝐴𝑥})
71, 6syl5eqel 2853 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝑊 ∈ {𝑥𝑇𝐴𝑥})
8 eleq2 2838 . . 3 (𝑣 = 𝑊 → (𝐴𝑣𝐴𝑊))
9 eleq2 2838 . . . 4 (𝑥 = 𝑣 → (𝐴𝑥𝐴𝑣))
109cbvrabv 3348 . . 3 {𝑥𝑇𝐴𝑥} = {𝑣𝑇𝐴𝑣}
118, 10elrab2 3516 . 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 382  w3a 1070   = wceq 1630  wcel 2144  wral 3060  ∃!wreu 3062  {crab 3064  cdif 3718  cin 3720  c0 4061  𝒫 cpw 4295  {csn 4314   cuni 4572  cmpt 4861  ccnv 5248  cres 5251  cima 5252  cfv 6031  crio 6752  (class class class)co 6792  t crest 16288  Homeochmeo 21776   CovMap ccvm 31569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-map 8010  df-top 20918  df-topon 20935  df-cn 21251  df-cvm 31570
This theorem is referenced by:  cvmopnlem  31592  cvmliftmolem2  31596  cvmliftlem6  31604  cvmliftlem8  31606  cvmliftlem9  31607  cvmlift2lem9  31625  cvmlift3lem6  31638  cvmlift3lem7  31639
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