Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmsdisj Structured version   Visualization version   GIF version

Theorem cvmsdisj 34560
Description: An even covering of 𝑈 is a disjoint union. (Contributed by Mario Carneiro, 13-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
cvmsdisj ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝐵𝑇) → (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))
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   𝑢,𝐴,𝑣   𝑣,𝐵
Allowed substitution hints:   𝐴(𝑘,𝑠)   𝐵(𝑢,𝑘,𝑠)   𝑆(𝑣,𝑢,𝑘,𝑠)   𝑇(𝑘)

Proof of Theorem cvmsdisj
StepHypRef Expression
1 df-ne 2940 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 cvmcov.1 . . . . . . . . . . 11 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
32cvmsi 34555 . . . . . . . . . 10 (𝑇 ∈ (𝑆𝑈) → (𝑈𝐽 ∧ (𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))))
43simp3d 1143 . . . . . . . . 9 (𝑇 ∈ (𝑆𝑈) → ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))
54simprd 495 . . . . . . . 8 (𝑇 ∈ (𝑆𝑈) → ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))
6 simpl 482 . . . . . . . . 9 ((∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))) → ∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅)
76ralimi 3082 . . . . . . . 8 (∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))) → ∀𝑢𝑇𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅)
85, 7syl 17 . . . . . . 7 (𝑇 ∈ (𝑆𝑈) → ∀𝑢𝑇𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅)
9 sneq 4638 . . . . . . . . . 10 (𝑢 = 𝐴 → {𝑢} = {𝐴})
109difeq2d 4122 . . . . . . . . 9 (𝑢 = 𝐴 → (𝑇 ∖ {𝑢}) = (𝑇 ∖ {𝐴}))
11 ineq1 4205 . . . . . . . . . 10 (𝑢 = 𝐴 → (𝑢𝑣) = (𝐴𝑣))
1211eqeq1d 2733 . . . . . . . . 9 (𝑢 = 𝐴 → ((𝑢𝑣) = ∅ ↔ (𝐴𝑣) = ∅))
1310, 12raleqbidv 3341 . . . . . . . 8 (𝑢 = 𝐴 → (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ↔ ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅))
1413rspccva 3611 . . . . . . 7 ((∀𝑢𝑇𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ 𝐴𝑇) → ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅)
158, 14sylan 579 . . . . . 6 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅)
16 necom 2993 . . . . . . 7 (𝐴𝐵𝐵𝐴)
17 eldifsn 4790 . . . . . . . 8 (𝐵 ∈ (𝑇 ∖ {𝐴}) ↔ (𝐵𝑇𝐵𝐴))
1817biimpri 227 . . . . . . 7 ((𝐵𝑇𝐵𝐴) → 𝐵 ∈ (𝑇 ∖ {𝐴}))
1916, 18sylan2b 593 . . . . . 6 ((𝐵𝑇𝐴𝐵) → 𝐵 ∈ (𝑇 ∖ {𝐴}))
20 ineq2 4206 . . . . . . . 8 (𝑣 = 𝐵 → (𝐴𝑣) = (𝐴𝐵))
2120eqeq1d 2733 . . . . . . 7 (𝑣 = 𝐵 → ((𝐴𝑣) = ∅ ↔ (𝐴𝐵) = ∅))
2221rspccv 3609 . . . . . 6 (∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅ → (𝐵 ∈ (𝑇 ∖ {𝐴}) → (𝐴𝐵) = ∅))
2315, 19, 22syl2im 40 . . . . 5 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝐵𝑇𝐴𝐵) → (𝐴𝐵) = ∅))
2423expd 415 . . . 4 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐵𝑇 → (𝐴𝐵 → (𝐴𝐵) = ∅)))
25243impia 1116 . . 3 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝐵𝑇) → (𝐴𝐵 → (𝐴𝐵) = ∅))
261, 25biimtrrid 242 . 2 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝐵𝑇) → (¬ 𝐴 = 𝐵 → (𝐴𝐵) = ∅))
2726orrd 860 1 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝐵𝑇) → (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 844  w3a 1086   = wceq 1540  wcel 2105  wne 2939  wral 3060  {crab 3431  cdif 3945  cin 3947  wss 3948  c0 4322  𝒫 cpw 4602  {csn 4628   cuni 4908  cmpt 5231  ccnv 5675  cres 5678  cima 5679  cfv 6543  (class class class)co 7412  t crest 17371  Homeochmeo 23478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415
This theorem is referenced by:  cvmscld  34563  cvmsss2  34564  cvmseu  34566
  Copyright terms: Public domain W3C validator