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Theorem cvmseu 34267
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 1196 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝐴𝐵)
2 simpr3 1197 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → (𝐹𝐴) ∈ 𝑈)
3 cvmcn 34253 . . . . . . 7 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
43adantr 482 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝐹 ∈ (𝐶 Cn 𝐽))
5 cvmseu.1 . . . . . . 7 𝐵 = 𝐶
6 eqid 2733 . . . . . . 7 𝐽 = 𝐽
75, 6cnf 22750 . . . . . 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 712 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝐴 ∈ (𝐹𝑈))
12 simpr1 1195 . . . . 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 34260 . . . . 5 (𝑇 ∈ (𝑆𝑈) → 𝑇 = (𝐹𝑈))
1512, 14syl 17 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝑇 = (𝐹𝑈))
1611, 15eleqtrrd 2837 . . 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 34261 . . . . . . . 8 ((𝑇 ∈ (𝑆𝑈) ∧ 𝑥𝑇𝑧𝑇) → (𝑥 = 𝑧 ∨ (𝑥𝑧) = ∅))
21203expb 1121 . . . . . . 7 ((𝑇 ∈ (𝑆𝑈) ∧ (𝑥𝑇𝑧𝑇)) → (𝑥 = 𝑧 ∨ (𝑥𝑧) = ∅))
2212, 21sylan 581 . . . . . 6 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) ∧ (𝑥𝑇𝑧𝑇)) → (𝑥 = 𝑧 ∨ (𝑥𝑧) = ∅))
2322ord 863 . . . . 5 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) ∧ (𝑥𝑇𝑧𝑇)) → (¬ 𝑥 = 𝑧 → (𝑥𝑧) = ∅))
2423necon1ad 2958 . . . 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 3201 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → ∀𝑥𝑇𝑧𝑇 ((𝐴𝑥𝐴𝑧) → 𝑥 = 𝑧))
27 eleq2w 2818 . . 3 (𝑥 = 𝑧 → (𝐴𝑥𝐴𝑧))
2827reu4 3728 . 2 (∃!𝑥𝑇 𝐴𝑥 ↔ (∃𝑥𝑇 𝐴𝑥 ∧ ∀𝑥𝑇𝑧𝑇 ((𝐴𝑥𝐴𝑧) → 𝑥 = 𝑧)))
2918, 26, 28sylanbrc 584 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 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wral 3062  wrex 3071  ∃!wreu 3375  {crab 3433  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 7409  t crest 17366   Cn ccn 22728  Homeochmeo 23257   CovMap ccvm 34246
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  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 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-top 22396  df-topon 22413  df-cn 22731  df-cvm 34247
This theorem is referenced by:  cvmsiota  34268
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