Theorem cvmseu 35017

Description: Every element in ∪ 𝑇 is a member of a unique element of 𝑇. (Contributed by Mario Carneiro, 14-Feb-2015.)

Hypotheses

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))))})
cvmseu.1	⊢ 𝐵 = ∪ 𝐶

Assertion

Ref	Expression
cvmseu	⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → ∃!𝑥 ∈ 𝑇 𝐴 ∈ 𝑥)

Distinct variable groups:   𝑘,𝑠,𝑢,𝑣,𝑥,𝐶   𝑘,𝐹,𝑠,𝑢,𝑣,𝑥   𝑘,𝐽,𝑠,𝑢,𝑣,𝑥   𝑥,𝑆   𝑈,𝑘,𝑠,𝑢,𝑣,𝑥   𝑇,𝑠,𝑢,𝑣,𝑥   𝑢,𝐴,𝑣,𝑥   𝑣,𝐵,𝑥

Allowed substitution hints:   𝐴(𝑘,𝑠)   𝐵(𝑢,𝑘,𝑠)   𝑆(𝑣,𝑢,𝑘,𝑠)   𝑇(𝑘)

Proof of Theorem cvmseu

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr2 1192	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → 𝐴 ∈ 𝐵)
2		simpr3 1193	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → (𝐹‘𝐴) ∈ 𝑈)
3		cvmcn 35003	. . . . . . 7 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
4	3	adantr 479	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → 𝐹 ∈ (𝐶 Cn 𝐽))
5		cvmseu.1	. . . . . . 7 ⊢ 𝐵 = ∪ 𝐶
6		eqid 2725	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
7	5, 6	cnf 23194	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶∪ 𝐽)
8		ffn 6723	. . . . . 6 ⊢ (𝐹:𝐵⟶∪ 𝐽 → 𝐹 Fn 𝐵)
9		elpreima 7066	. . . . . 6 ⊢ (𝐹 Fn 𝐵 → (𝐴 ∈ (◡𝐹 “ 𝑈) ↔ (𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)))
10	4, 7, 8, 9	4syl 19	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → (𝐴 ∈ (◡𝐹 “ 𝑈) ↔ (𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)))
11	1, 2, 10	mpbir2and 711	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → 𝐴 ∈ (◡𝐹 “ 𝑈))
12		simpr1 1191	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → 𝑇 ∈ (𝑆‘𝑈))
13		cvmcov.1	. . . . . 6 ⊢ 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))))})
14	13	cvmsuni 35010	. . . . 5 ⊢ (𝑇 ∈ (𝑆‘𝑈) → ∪ 𝑇 = (◡𝐹 “ 𝑈))
15	12, 14	syl 17	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → ∪ 𝑇 = (◡𝐹 “ 𝑈))
16	11, 15	eleqtrrd 2828	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → 𝐴 ∈ ∪ 𝑇)
17		eluni2 4913	. . 3 ⊢ (𝐴 ∈ ∪ 𝑇 ↔ ∃𝑥 ∈ 𝑇 𝐴 ∈ 𝑥)
18	16, 17	sylib 217	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → ∃𝑥 ∈ 𝑇 𝐴 ∈ 𝑥)
19		inelcm 4466	. . . 4 ⊢ ((𝐴 ∈ 𝑥 ∧ 𝐴 ∈ 𝑧) → (𝑥 ∩ 𝑧) ≠ ∅)
20	13	cvmsdisj 35011	. . . . . . . 8 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) → (𝑥 = 𝑧 ∨ (𝑥 ∩ 𝑧) = ∅))
21	20	3expb 1117	. . . . . . 7 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ (𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (𝑥 = 𝑧 ∨ (𝑥 ∩ 𝑧) = ∅))
22	12, 21	sylan 578	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) ∧ (𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (𝑥 = 𝑧 ∨ (𝑥 ∩ 𝑧) = ∅))
23	22	ord 862	. . . . 5 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) ∧ (𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (¬ 𝑥 = 𝑧 → (𝑥 ∩ 𝑧) = ∅))
24	23	necon1ad 2946	. . . 4 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) ∧ (𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → ((𝑥 ∩ 𝑧) ≠ ∅ → 𝑥 = 𝑧))
25	19, 24	syl5 34	. . 3 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) ∧ (𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → ((𝐴 ∈ 𝑥 ∧ 𝐴 ∈ 𝑧) → 𝑥 = 𝑧))
26	25	ralrimivva 3190	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → ∀𝑥 ∈ 𝑇 ∀𝑧 ∈ 𝑇 ((𝐴 ∈ 𝑥 ∧ 𝐴 ∈ 𝑧) → 𝑥 = 𝑧))
27		eleq2w 2809	. . 3 ⊢ (𝑥 = 𝑧 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑧))
28	27	reu4 3723	. 2 ⊢ (∃!𝑥 ∈ 𝑇 𝐴 ∈ 𝑥 ↔ (∃𝑥 ∈ 𝑇 𝐴 ∈ 𝑥 ∧ ∀𝑥 ∈ 𝑇 ∀𝑧 ∈ 𝑇 ((𝐴 ∈ 𝑥 ∧ 𝐴 ∈ 𝑧) → 𝑥 = 𝑧)))
29	18, 26, 28	sylanbrc 581	1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → ∃!𝑥 ∈ 𝑇 𝐴 ∈ 𝑥)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2929  ∀wral 3050  ∃wrex 3059  ∃!wreu 3361  {crab 3418   ∖ cdif 3941   ∩ cin 3943  ∅c0 4322  𝒫 cpw 4604  {csn 4630  ∪ cuni 4909   ↦ cmpt 5232  ^◡ccnv 5677   ↾ cres 5680   “ cima 5681   Fn wfn 6544  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419   ↾_t crest 17405   Cn ccn 23172  Homeochmeo 23701   CovMap ccvm 34996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8847  df-top 22840  df-topon 22857  df-cn 23175  df-cvm 34997

This theorem is referenced by:  cvmsiota  35018