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Theorem cvmsrcl 34553
Description: Reverse closure for an even covering. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypothesis
Ref Expression
cvmcov.1 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
Assertion
Ref Expression
cvmsrcl (𝑇 ∈ (𝑆𝑈) → 𝑈𝐽)
Distinct variable groups:   𝑘,𝑠,𝑢,𝑣,𝐶   𝑘,𝐹,𝑠,𝑢,𝑣   𝑘,𝐽,𝑠,𝑢,𝑣   𝑈,𝑘,𝑠,𝑢,𝑣   𝑇,𝑠,𝑢,𝑣
Allowed substitution hints:   𝑆(𝑣,𝑢,𝑘,𝑠)   𝑇(𝑘)

Proof of Theorem cvmsrcl
StepHypRef Expression
1 cvmcov.1 . . 3 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
21dmmptss 6239 . 2 dom 𝑆𝐽
3 elfvdm 6927 . 2 (𝑇 ∈ (𝑆𝑈) → 𝑈 ∈ dom 𝑆)
42, 3sselid 3979 1 (𝑇 ∈ (𝑆𝑈) → 𝑈𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  wral 3059  {crab 3430  cdif 3944  cin 3946  c0 4321  𝒫 cpw 4601  {csn 4627   cuni 4907  cmpt 5230  ccnv 5674  dom cdm 5675  cres 5677  cima 5678  cfv 6542  (class class class)co 7411  t crest 17370  Homeochmeo 23477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fv 6550
This theorem is referenced by:  cvmsi  34554  cvmsf1o  34561  cvmsss2  34563  cvmopnlem  34567  cvmliftlem8  34581  cvmlift2lem9  34600  cvmlift2lem10  34601  cvmlift3lem6  34613  cvmlift3lem8  34615
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