Theorem cvmseu 32527

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 1191	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → 𝐴 ∈ 𝐵)
2		simpr3 1192	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → (𝐹‘𝐴) ∈ 𝑈)
3		cvmcn 32513	. . . . . . 7 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
4	3	adantr 483	. . . . . 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 2824	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
7	5, 6	cnf 21857	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶∪ 𝐽)
8		ffn 6517	. . . . . 6 ⊢ (𝐹:𝐵⟶∪ 𝐽 → 𝐹 Fn 𝐵)
9		elpreima 6831	. . . . . 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 1190	. . . . 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 32520	. . . . 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 2919	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → 𝐴 ∈ ∪ 𝑇)
17		eluni2 4845	. . 3 ⊢ (𝐴 ∈ ∪ 𝑇 ↔ ∃𝑥 ∈ 𝑇 𝐴 ∈ 𝑥)
18	16, 17	sylib 220	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → ∃𝑥 ∈ 𝑇 𝐴 ∈ 𝑥)
19		inelcm 4417	. . . 4 ⊢ ((𝐴 ∈ 𝑥 ∧ 𝐴 ∈ 𝑧) → (𝑥 ∩ 𝑧) ≠ ∅)
20	13	cvmsdisj 32521	. . . . . . . 8 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) → (𝑥 = 𝑧 ∨ (𝑥 ∩ 𝑧) = ∅))
21	20	3expb 1116	. . . . . . 7 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ (𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (𝑥 = 𝑧 ∨ (𝑥 ∩ 𝑧) = ∅))
22	12, 21	sylan 582	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) ∧ (𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (𝑥 = 𝑧 ∨ (𝑥 ∩ 𝑧) = ∅))
23	22	ord 860	. . . . 5 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) ∧ (𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (¬ 𝑥 = 𝑧 → (𝑥 ∩ 𝑧) = ∅))
24	23	necon1ad 3036	. . . 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 3194	. 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 2899	. . 3 ⊢ (𝑥 = 𝑧 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑧))
28	27	reu4 3725	. 2 ⊢ (∃!𝑥 ∈ 𝑇 𝐴 ∈ 𝑥 ↔ (∃𝑥 ∈ 𝑇 𝐴 ∈ 𝑥 ∧ ∀𝑥 ∈ 𝑇 ∀𝑧 ∈ 𝑇 ((𝐴 ∈ 𝑥 ∧ 𝐴 ∈ 𝑧) → 𝑥 = 𝑧)))
29	18, 26, 28	sylanbrc 585	1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → ∃!𝑥 ∈ 𝑇 𝐴 ∈ 𝑥)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113   ≠ wne 3019  ∀wral 3141  ∃wrex 3142  ∃!wreu 3143  {crab 3145   ∖ cdif 3936   ∩ cin 3938  ∅c0 4294  𝒫 cpw 4542  {csn 4570  ∪ cuni 4841   ↦ cmpt 5149  ^◡ccnv 5557   ↾ cres 5560   “ cima 5561   Fn wfn 6353  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159   ↾_t crest 16697   Cn ccn 21835  Homeochmeo 22364   CovMap ccvm 32506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-map 8411  df-top 21505  df-topon 21522  df-cn 21838  df-cvm 32507

This theorem is referenced by:  cvmsiota  32528