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Theorem cvmsdisj 33904
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 2945 . . 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 33899 . . . . . . . . . 10 (𝑇 ∈ (𝑆𝑈) → (𝑈𝐽 ∧ (𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))))
43simp3d 1145 . . . . . . . . 9 (𝑇 ∈ (𝑆𝑈) → ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))
54simprd 497 . . . . . . . 8 (𝑇 ∈ (𝑆𝑈) → ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))
6 simpl 484 . . . . . . . . 9 ((∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))) → ∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅)
76ralimi 3087 . . . . . . . 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 4601 . . . . . . . . . 10 (𝑢 = 𝐴 → {𝑢} = {𝐴})
109difeq2d 4087 . . . . . . . . 9 (𝑢 = 𝐴 → (𝑇 ∖ {𝑢}) = (𝑇 ∖ {𝐴}))
11 ineq1 4170 . . . . . . . . . 10 (𝑢 = 𝐴 → (𝑢𝑣) = (𝐴𝑣))
1211eqeq1d 2739 . . . . . . . . 9 (𝑢 = 𝐴 → ((𝑢𝑣) = ∅ ↔ (𝐴𝑣) = ∅))
1310, 12raleqbidv 3322 . . . . . . . 8 (𝑢 = 𝐴 → (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ↔ ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅))
1413rspccva 3583 . . . . . . 7 ((∀𝑢𝑇𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ 𝐴𝑇) → ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅)
158, 14sylan 581 . . . . . 6 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅)
16 necom 2998 . . . . . . 7 (𝐴𝐵𝐵𝐴)
17 eldifsn 4752 . . . . . . . 8 (𝐵 ∈ (𝑇 ∖ {𝐴}) ↔ (𝐵𝑇𝐵𝐴))
1817biimpri 227 . . . . . . 7 ((𝐵𝑇𝐵𝐴) → 𝐵 ∈ (𝑇 ∖ {𝐴}))
1916, 18sylan2b 595 . . . . . 6 ((𝐵𝑇𝐴𝐵) → 𝐵 ∈ (𝑇 ∖ {𝐴}))
20 ineq2 4171 . . . . . . . 8 (𝑣 = 𝐵 → (𝐴𝑣) = (𝐴𝐵))
2120eqeq1d 2739 . . . . . . 7 (𝑣 = 𝐵 → ((𝐴𝑣) = ∅ ↔ (𝐴𝐵) = ∅))
2221rspccv 3581 . . . . . 6 (∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅ → (𝐵 ∈ (𝑇 ∖ {𝐴}) → (𝐴𝐵) = ∅))
2315, 19, 22syl2im 40 . . . . 5 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝐵𝑇𝐴𝐵) → (𝐴𝐵) = ∅))
2423expd 417 . . . 4 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐵𝑇 → (𝐴𝐵 → (𝐴𝐵) = ∅)))
25243impia 1118 . . 3 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝐵𝑇) → (𝐴𝐵 → (𝐴𝐵) = ∅))
261, 25biimtrrid 242 . 2 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝐵𝑇) → (¬ 𝐴 = 𝐵 → (𝐴𝐵) = ∅))
2726orrd 862 1 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝐵𝑇) → (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2944  wral 3065  {crab 3410  cdif 3912  cin 3914  wss 3915  c0 4287  𝒫 cpw 4565  {csn 4591   cuni 4870  cmpt 5193  ccnv 5637  cres 5640  cima 5641  cfv 6501  (class class class)co 7362  t crest 17309  Homeochmeo 23120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365
This theorem is referenced by:  cvmscld  33907  cvmsss2  33908  cvmseu  33910
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