Theorem cvmsdisj 35312

Description: An even covering of 𝑈 is a disjoint union. (Contributed by Mario Carneiro, 13-Feb-2015.)

Hypothesis

Ref	Expression
cvmcov.1	⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})

Assertion

Ref	Expression
cvmsdisj	⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 = 𝐵 ∨ (𝐴 ∩ 𝐵) = ∅))

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

Step	Hyp	Ref	Expression
1		df-ne 2929	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		cvmcov.1	. . . . . . . . . . 11 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})
3	2	cvmsi 35307	. . . . . . . . . 10 ⊢ (𝑇 ∈ (𝑆‘𝑈) → (𝑈 ∈ 𝐽 ∧ (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))))
4	3	simp3d 1144	. . . . . . . . 9 ⊢ (𝑇 ∈ (𝑆‘𝑈) → (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))
5	4	simprd 495	. . . . . . . 8 ⊢ (𝑇 ∈ (𝑆‘𝑈) → ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))
6		simpl 482	. . . . . . . . 9 ⊢ ((∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))) → ∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅)
7	6	ralimi 3069	. . . . . . . 8 ⊢ (∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))) → ∀𝑢 ∈ 𝑇 ∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅)
8	5, 7	syl 17	. . . . . . 7 ⊢ (𝑇 ∈ (𝑆‘𝑈) → ∀𝑢 ∈ 𝑇 ∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅)
9		sneq 4586	. . . . . . . . . 10 ⊢ (𝑢 = 𝐴 → {𝑢} = {𝐴})
10	9	difeq2d 4076	. . . . . . . . 9 ⊢ (𝑢 = 𝐴 → (𝑇 ∖ {𝑢}) = (𝑇 ∖ {𝐴}))
11		ineq1 4163	. . . . . . . . . 10 ⊢ (𝑢 = 𝐴 → (𝑢 ∩ 𝑣) = (𝐴 ∩ 𝑣))
12	11	eqeq1d 2733	. . . . . . . . 9 ⊢ (𝑢 = 𝐴 → ((𝑢 ∩ 𝑣) = ∅ ↔ (𝐴 ∩ 𝑣) = ∅))
13	10, 12	raleqbidv 3312	. . . . . . . 8 ⊢ (𝑢 = 𝐴 → (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ↔ ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑣) = ∅))
14	13	rspccva 3576	. . . . . . 7 ⊢ ((∀𝑢 ∈ 𝑇 ∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ 𝐴 ∈ 𝑇) → ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑣) = ∅)
15	8, 14	sylan 580	. . . . . 6 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑣) = ∅)
16		necom 2981	. . . . . . 7 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
17		eldifsn 4738	. . . . . . . 8 ⊢ (𝐵 ∈ (𝑇 ∖ {𝐴}) ↔ (𝐵 ∈ 𝑇 ∧ 𝐵 ≠ 𝐴))
18	17	biimpri 228	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑇 ∧ 𝐵 ≠ 𝐴) → 𝐵 ∈ (𝑇 ∖ {𝐴}))
19	16, 18	sylan2b 594	. . . . . 6 ⊢ ((𝐵 ∈ 𝑇 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ (𝑇 ∖ {𝐴}))
20		ineq2 4164	. . . . . . . 8 ⊢ (𝑣 = 𝐵 → (𝐴 ∩ 𝑣) = (𝐴 ∩ 𝐵))
21	20	eqeq1d 2733	. . . . . . 7 ⊢ (𝑣 = 𝐵 → ((𝐴 ∩ 𝑣) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅))
22	21	rspccv 3574	. . . . . 6 ⊢ (∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑣) = ∅ → (𝐵 ∈ (𝑇 ∖ {𝐴}) → (𝐴 ∩ 𝐵) = ∅))
23	15, 19, 22	syl2im 40	. . . . 5 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ((𝐵 ∈ 𝑇 ∧ 𝐴 ≠ 𝐵) → (𝐴 ∩ 𝐵) = ∅))
24	23	expd 415	. . . 4 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐵 ∈ 𝑇 → (𝐴 ≠ 𝐵 → (𝐴 ∩ 𝐵) = ∅)))
25	24	3impia 1117	. . 3 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 ≠ 𝐵 → (𝐴 ∩ 𝐵) = ∅))
26	1, 25	biimtrrid 243	. 2 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (¬ 𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = ∅))
27	26	orrd 863	1 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 = 𝐵 ∨ (𝐴 ∩ 𝐵) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  {crab 3395   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4550  {csn 4576  ∪ cuni 4859   ↦ cmpt 5172  ^◡ccnv 5615   ↾ cres 5618   “ cima 5619  ‘cfv 6481  (class class class)co 7346   ↾_t crest 17324  Homeochmeo 23669

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349

This theorem is referenced by:  cvmscld  35315  cvmsss2  35316  cvmseu  35318