Theorem cvmseu 33238

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 33224	. . . . . . 7 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
4	3	adantr 481	. . . . . 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 2738	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
7	5, 6	cnf 22397	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶∪ 𝐽)
8		ffn 6600	. . . . . 6 ⊢ (𝐹:𝐵⟶∪ 𝐽 → 𝐹 Fn 𝐵)
9		elpreima 6935	. . . . . 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 710	. . . 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 33231	. . . . 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 2842	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → 𝐴 ∈ ∪ 𝑇)
17		eluni2 4843	. . 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 4398	. . . 4 ⊢ ((𝐴 ∈ 𝑥 ∧ 𝐴 ∈ 𝑧) → (𝑥 ∩ 𝑧) ≠ ∅)
20	13	cvmsdisj 33232	. . . . . . . 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 861	. . . . 5 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) ∧ (𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (¬ 𝑥 = 𝑧 → (𝑥 ∩ 𝑧) = ∅))
24	23	necon1ad 2960	. . . 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 3123	. 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 3666	. 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 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  ∃!wreu 3066  {crab 3068   ∖ cdif 3884   ∩ cin 3886  ∅c0 4256  𝒫 cpw 4533  {csn 4561  ∪ cuni 4839   ↦ cmpt 5157  ^◡ccnv 5588   ↾ cres 5591   “ cima 5592   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↾_t crest 17131   Cn ccn 22375  Homeochmeo 22904   CovMap ccvm 33217

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-top 22043  df-topon 22060  df-cn 22378  df-cvm 33218

This theorem is referenced by:  cvmsiota  33239