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Theorem cvmseu 31001
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 1066 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝐴𝐵)
2 simpr3 1067 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → (𝐹𝐴) ∈ 𝑈)
3 cvmcn 30987 . . . . . . 7 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
43adantr 481 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝐹 ∈ (𝐶 Cn 𝐽))
5 cvmseu.1 . . . . . . 7 𝐵 = 𝐶
6 eqid 2621 . . . . . . 7 𝐽 = 𝐽
75, 6cnf 20973 . . . . . 6 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵 𝐽)
8 ffn 6007 . . . . . 6 (𝐹:𝐵 𝐽𝐹 Fn 𝐵)
9 elpreima 6298 . . . . . 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 956 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝐴 ∈ (𝐹𝑈))
12 simpr1 1065 . . . . 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 30994 . . . . 5 (𝑇 ∈ (𝑆𝑈) → 𝑇 = (𝐹𝑈))
1512, 14syl 17 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝑇 = (𝐹𝑈))
1611, 15eleqtrrd 2701 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → 𝐴 𝑇)
17 eluni2 4411 . . 3 (𝐴 𝑇 ↔ ∃𝑥𝑇 𝐴𝑥)
1816, 17sylib 208 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → ∃𝑥𝑇 𝐴𝑥)
19 inelcm 4009 . . . 4 ((𝐴𝑥𝐴𝑧) → (𝑥𝑧) ≠ ∅)
2013cvmsdisj 30995 . . . . . . . 8 ((𝑇 ∈ (𝑆𝑈) ∧ 𝑥𝑇𝑧𝑇) → (𝑥 = 𝑧 ∨ (𝑥𝑧) = ∅))
21203expb 1263 . . . . . . 7 ((𝑇 ∈ (𝑆𝑈) ∧ (𝑥𝑇𝑧𝑇)) → (𝑥 = 𝑧 ∨ (𝑥𝑧) = ∅))
2212, 21sylan 488 . . . . . 6 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) ∧ (𝑥𝑇𝑧𝑇)) → (𝑥 = 𝑧 ∨ (𝑥𝑧) = ∅))
2322ord 392 . . . . 5 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) ∧ (𝑥𝑇𝑧𝑇)) → (¬ 𝑥 = 𝑧 → (𝑥𝑧) = ∅))
2423necon1ad 2807 . . . 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 2966 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → ∀𝑥𝑇𝑧𝑇 ((𝐴𝑥𝐴𝑧) → 𝑥 = 𝑧))
27 eleq2 2687 . . 3 (𝑥 = 𝑧 → (𝐴𝑥𝐴𝑧))
2827reu4 3386 . 2 (∃!𝑥𝑇 𝐴𝑥 ↔ (∃𝑥𝑇 𝐴𝑥 ∧ ∀𝑥𝑇𝑧𝑇 ((𝐴𝑥𝐴𝑧) → 𝑥 = 𝑧)))
2918, 26, 28sylanbrc 697 1 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → ∃!𝑥𝑇 𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  ∃!wreu 2909  {crab 2911  cdif 3556  cin 3558  c0 3896  𝒫 cpw 4135  {csn 4153   cuni 4407  cmpt 4678  ccnv 5078  cres 5081  cima 5082   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  t crest 16013   Cn ccn 20951  Homeochmeo 21479   CovMap ccvm 30980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-top 20631  df-topon 20648  df-cn 20954  df-cvm 30981
This theorem is referenced by:  cvmsiota  31002
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