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 31800
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 1254 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝐴𝐵)
2 simpr3 1256 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → (𝐹𝐴) ∈ 𝑈)
3 cvmcn 31786 . . . . . . 7 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
43adantr 474 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝐹 ∈ (𝐶 Cn 𝐽))
5 cvmseu.1 . . . . . . 7 𝐵 = 𝐶
6 eqid 2825 . . . . . . 7 𝐽 = 𝐽
75, 6cnf 21428 . . . . . 6 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵 𝐽)
8 ffn 6282 . . . . . 6 (𝐹:𝐵 𝐽𝐹 Fn 𝐵)
9 elpreima 6591 . . . . . 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 704 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝐴 ∈ (𝐹𝑈))
12 simpr1 1252 . . . . 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 31793 . . . . 5 (𝑇 ∈ (𝑆𝑈) → 𝑇 = (𝐹𝑈))
1512, 14syl 17 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝑇 = (𝐹𝑈))
1611, 15eleqtrrd 2909 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝐴 𝑇)
17 eluni2 4664 . . 3 (𝐴 𝑇 ↔ ∃𝑥𝑇 𝐴𝑥)
1816, 17sylib 210 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → ∃𝑥𝑇 𝐴𝑥)
19 inelcm 4258 . . . 4 ((𝐴𝑥𝐴𝑧) → (𝑥𝑧) ≠ ∅)
2013cvmsdisj 31794 . . . . . . . 8 ((𝑇 ∈ (𝑆𝑈) ∧ 𝑥𝑇𝑧𝑇) → (𝑥 = 𝑧 ∨ (𝑥𝑧) = ∅))
21203expb 1153 . . . . . . 7 ((𝑇 ∈ (𝑆𝑈) ∧ (𝑥𝑇𝑧𝑇)) → (𝑥 = 𝑧 ∨ (𝑥𝑧) = ∅))
2212, 21sylan 575 . . . . . 6 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) ∧ (𝑥𝑇𝑧𝑇)) → (𝑥 = 𝑧 ∨ (𝑥𝑧) = ∅))
2322ord 895 . . . . 5 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) ∧ (𝑥𝑇𝑧𝑇)) → (¬ 𝑥 = 𝑧 → (𝑥𝑧) = ∅))
2423necon1ad 3016 . . . 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 3180 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → ∀𝑥𝑇𝑧𝑇 ((𝐴𝑥𝐴𝑧) → 𝑥 = 𝑧))
27 eleq2w 2890 . . 3 (𝑥 = 𝑧 → (𝐴𝑥𝐴𝑧))
2827reu4 3625 . 2 (∃!𝑥𝑇 𝐴𝑥 ↔ (∃𝑥𝑇 𝐴𝑥 ∧ ∀𝑥𝑇𝑧𝑇 ((𝐴𝑥𝐴𝑧) → 𝑥 = 𝑧)))
2918, 26, 28sylanbrc 578 1 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → ∃!𝑥𝑇 𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wo 878  w3a 1111   = wceq 1656  wcel 2164  wne 2999  wral 3117  wrex 3118  ∃!wreu 3119  {crab 3121  cdif 3795  cin 3797  c0 4146  𝒫 cpw 4380  {csn 4399   cuni 4660  cmpt 4954  ccnv 5345  cres 5348  cima 5349   Fn wfn 6122  wf 6123  cfv 6127  (class class class)co 6910  t crest 16441   Cn ccn 21406  Homeochmeo 21934   CovMap ccvm 31779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-map 8129  df-top 21076  df-topon 21093  df-cn 21409  df-cvm 31780
This theorem is referenced by:  cvmsiota  31801
  Copyright terms: Public domain W3C validator