Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmseu Structured version   Visualization version   GIF version

Theorem cvmseu 34563
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 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
cvmseu.1 𝐵 = 𝐶
Assertion
Ref Expression
cvmseu ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → ∃!𝑥𝑇 𝐴𝑥)
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.
StepHypRef Expression
1 simpr2 1193 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝐴𝐵)
2 simpr3 1194 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → (𝐹𝐴) ∈ 𝑈)
3 cvmcn 34549 . . . . . . 7 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
43adantr 479 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝐹 ∈ (𝐶 Cn 𝐽))
5 cvmseu.1 . . . . . . 7 𝐵 = 𝐶
6 eqid 2730 . . . . . . 7 𝐽 = 𝐽
75, 6cnf 22972 . . . . . 6 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵 𝐽)
8 ffn 6718 . . . . . 6 (𝐹:𝐵 𝐽𝐹 Fn 𝐵)
9 elpreima 7060 . . . . . 6 (𝐹 Fn 𝐵 → (𝐴 ∈ (𝐹𝑈) ↔ (𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)))
104, 7, 8, 94syl 19 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → (𝐴 ∈ (𝐹𝑈) ↔ (𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)))
111, 2, 10mpbir2and 709 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝐴 ∈ (𝐹𝑈))
12 simpr1 1192 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝑇 ∈ (𝑆𝑈))
13 cvmcov.1 . . . . . 6 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
1413cvmsuni 34556 . . . . 5 (𝑇 ∈ (𝑆𝑈) → 𝑇 = (𝐹𝑈))
1512, 14syl 17 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝑇 = (𝐹𝑈))
1611, 15eleqtrrd 2834 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝐴 𝑇)
17 eluni2 4913 . . 3 (𝐴 𝑇 ↔ ∃𝑥𝑇 𝐴𝑥)
1816, 17sylib 217 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → ∃𝑥𝑇 𝐴𝑥)
19 inelcm 4465 . . . 4 ((𝐴𝑥𝐴𝑧) → (𝑥𝑧) ≠ ∅)
2013cvmsdisj 34557 . . . . . . . 8 ((𝑇 ∈ (𝑆𝑈) ∧ 𝑥𝑇𝑧𝑇) → (𝑥 = 𝑧 ∨ (𝑥𝑧) = ∅))
21203expb 1118 . . . . . . 7 ((𝑇 ∈ (𝑆𝑈) ∧ (𝑥𝑇𝑧𝑇)) → (𝑥 = 𝑧 ∨ (𝑥𝑧) = ∅))
2212, 21sylan 578 . . . . . 6 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) ∧ (𝑥𝑇𝑧𝑇)) → (𝑥 = 𝑧 ∨ (𝑥𝑧) = ∅))
2322ord 860 . . . . 5 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) ∧ (𝑥𝑇𝑧𝑇)) → (¬ 𝑥 = 𝑧 → (𝑥𝑧) = ∅))
2423necon1ad 2955 . . . 4 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) ∧ (𝑥𝑇𝑧𝑇)) → ((𝑥𝑧) ≠ ∅ → 𝑥 = 𝑧))
2519, 24syl5 34 . . 3 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) ∧ (𝑥𝑇𝑧𝑇)) → ((𝐴𝑥𝐴𝑧) → 𝑥 = 𝑧))
2625ralrimivva 3198 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → ∀𝑥𝑇𝑧𝑇 ((𝐴𝑥𝐴𝑧) → 𝑥 = 𝑧))
27 eleq2w 2815 . . 3 (𝑥 = 𝑧 → (𝐴𝑥𝐴𝑧))
2827reu4 3728 . 2 (∃!𝑥𝑇 𝐴𝑥 ↔ (∃𝑥𝑇 𝐴𝑥 ∧ ∀𝑥𝑇𝑧𝑇 ((𝐴𝑥𝐴𝑧) → 𝑥 = 𝑧)))
2918, 26, 28sylanbrc 581 1 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → ∃!𝑥𝑇 𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 843  w3a 1085   = wceq 1539  wcel 2104  wne 2938  wral 3059  wrex 3068  ∃!wreu 3372  {crab 3430  cdif 3946  cin 3948  c0 4323  𝒫 cpw 4603  {csn 4629   cuni 4909  cmpt 5232  ccnv 5676  cres 5679  cima 5680   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7413  t crest 17372   Cn ccn 22950  Homeochmeo 23479   CovMap ccvm 34542
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7416  df-oprab 7417  df-mpo 7418  df-map 8826  df-top 22618  df-topon 22635  df-cn 22953  df-cvm 34543
This theorem is referenced by:  cvmsiota  34564
  Copyright terms: Public domain W3C validator