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Theorem cvmsdisj 30952
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 2797 . . 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 30947 . . . . . . . . . 10 (𝑇 ∈ (𝑆𝑈) → (𝑈𝐽 ∧ (𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))))
43simp3d 1073 . . . . . . . . 9 (𝑇 ∈ (𝑆𝑈) → ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))
54simprd 479 . . . . . . . 8 (𝑇 ∈ (𝑆𝑈) → ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))
6 simpl 473 . . . . . . . . 9 ((∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))) → ∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅)
76ralimi 2952 . . . . . . . 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 4163 . . . . . . . . . 10 (𝑢 = 𝐴 → {𝑢} = {𝐴})
109difeq2d 3711 . . . . . . . . 9 (𝑢 = 𝐴 → (𝑇 ∖ {𝑢}) = (𝑇 ∖ {𝐴}))
11 ineq1 3790 . . . . . . . . . 10 (𝑢 = 𝐴 → (𝑢𝑣) = (𝐴𝑣))
1211eqeq1d 2628 . . . . . . . . 9 (𝑢 = 𝐴 → ((𝑢𝑣) = ∅ ↔ (𝐴𝑣) = ∅))
1310, 12raleqbidv 3146 . . . . . . . 8 (𝑢 = 𝐴 → (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ↔ ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅))
1413rspccva 3299 . . . . . . 7 ((∀𝑢𝑇𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ 𝐴𝑇) → ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅)
158, 14sylan 488 . . . . . 6 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅)
16 necom 2849 . . . . . . 7 (𝐴𝐵𝐵𝐴)
17 eldifsn 4292 . . . . . . . 8 (𝐵 ∈ (𝑇 ∖ {𝐴}) ↔ (𝐵𝑇𝐵𝐴))
1817biimpri 218 . . . . . . 7 ((𝐵𝑇𝐵𝐴) → 𝐵 ∈ (𝑇 ∖ {𝐴}))
1916, 18sylan2b 492 . . . . . 6 ((𝐵𝑇𝐴𝐵) → 𝐵 ∈ (𝑇 ∖ {𝐴}))
20 ineq2 3791 . . . . . . . 8 (𝑣 = 𝐵 → (𝐴𝑣) = (𝐴𝐵))
2120eqeq1d 2628 . . . . . . 7 (𝑣 = 𝐵 → ((𝐴𝑣) = ∅ ↔ (𝐴𝐵) = ∅))
2221rspccv 3297 . . . . . 6 (∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅ → (𝐵 ∈ (𝑇 ∖ {𝐴}) → (𝐴𝐵) = ∅))
2315, 19, 22syl2im 40 . . . . 5 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝐵𝑇𝐴𝐵) → (𝐴𝐵) = ∅))
2423expd 452 . . . 4 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐵𝑇 → (𝐴𝐵 → (𝐴𝐵) = ∅)))
25243impia 1258 . . 3 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝐵𝑇) → (𝐴𝐵 → (𝐴𝐵) = ∅))
261, 25syl5bir 233 . 2 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝐵𝑇) → (¬ 𝐴 = 𝐵 → (𝐴𝐵) = ∅))
2726orrd 393 1 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝐵𝑇) → (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wral 2912  {crab 2916  cdif 3557  cin 3559  wss 3560  c0 3896  𝒫 cpw 4135  {csn 4153   cuni 4407  cmpt 4678  ccnv 5078  cres 5081  cima 5082  cfv 5850  (class class class)co 6605  t crest 15997  Homeochmeo 21461
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608
This theorem is referenced by:  cvmscld  30955  cvmsss2  30956  cvmseu  30958
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