Theorem cvmseu 35260

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 1194	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → 𝐴 ∈ 𝐵)
2		simpr3 1195	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → (𝐹‘𝐴) ∈ 𝑈)
3		cvmcn 35246	. . . . . . 7 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
4	3	adantr 480	. . . . . 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 2734	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
7	5, 6	cnf 23269	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶∪ 𝐽)
8		ffn 6736	. . . . . 6 ⊢ (𝐹:𝐵⟶∪ 𝐽 → 𝐹 Fn 𝐵)
9		elpreima 7077	. . . . . 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 713	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → 𝐴 ∈ (◡𝐹 “ 𝑈))
12		simpr1 1193	. . . . 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 35253	. . . . 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 2841	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → 𝐴 ∈ ∪ 𝑇)
17		eluni2 4915	. . 3 ⊢ (𝐴 ∈ ∪ 𝑇 ↔ ∃𝑥 ∈ 𝑇 𝐴 ∈ 𝑥)
18	16, 17	sylib 218	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → ∃𝑥 ∈ 𝑇 𝐴 ∈ 𝑥)
19		inelcm 4470	. . . 4 ⊢ ((𝐴 ∈ 𝑥 ∧ 𝐴 ∈ 𝑧) → (𝑥 ∩ 𝑧) ≠ ∅)
20	13	cvmsdisj 35254	. . . . . . . 8 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) → (𝑥 = 𝑧 ∨ (𝑥 ∩ 𝑧) = ∅))
21	20	3expb 1119	. . . . . . 7 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ (𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (𝑥 = 𝑧 ∨ (𝑥 ∩ 𝑧) = ∅))
22	12, 21	sylan 580	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) ∧ (𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (𝑥 = 𝑧 ∨ (𝑥 ∩ 𝑧) = ∅))
23	22	ord 864	. . . . 5 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) ∧ (𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (¬ 𝑥 = 𝑧 → (𝑥 ∩ 𝑧) = ∅))
24	23	necon1ad 2954	. . . 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 3199	. 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 2822	. . 3 ⊢ (𝑥 = 𝑧 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑧))
28	27	reu4 3739	. 2 ⊢ (∃!𝑥 ∈ 𝑇 𝐴 ∈ 𝑥 ↔ (∃𝑥 ∈ 𝑇 𝐴 ∈ 𝑥 ∧ ∀𝑥 ∈ 𝑇 ∀𝑧 ∈ 𝑇 ((𝐴 ∈ 𝑥 ∧ 𝐴 ∈ 𝑧) → 𝑥 = 𝑧)))
29	18, 26, 28	sylanbrc 583	1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → ∃!𝑥 ∈ 𝑇 𝐴 ∈ 𝑥)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  ∃!wreu 3375  {crab 3432   ∖ cdif 3959   ∩ cin 3961  ∅c0 4338  𝒫 cpw 4604  {csn 4630  ∪ cuni 4911   ↦ cmpt 5230  ^◡ccnv 5687   ↾ cres 5690   “ cima 5691   Fn wfn 6557  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430   ↾_t crest 17466   Cn ccn 23247  Homeochmeo 23776   CovMap ccvm 35239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-top 22915  df-topon 22932  df-cn 23250  df-cvm 35240

This theorem is referenced by:  cvmsiota  35261