Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvrfval Structured version   Visualization version   GIF version

Theorem cvrfval 34873
Description: Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
cvrfval.b 𝐵 = (Base‘𝐾)
cvrfval.s < = (lt‘𝐾)
cvrfval.c 𝐶 = ( ⋖ ‘𝐾)
Assertion
Ref Expression
cvrfval (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))})
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝐾,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦,𝑧)   < (𝑥,𝑦,𝑧)

Proof of Theorem cvrfval
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 elex 3243 . 2 (𝐾𝐴𝐾 ∈ V)
2 cvrfval.c . . 3 𝐶 = ( ⋖ ‘𝐾)
3 fveq2 6229 . . . . . . . . 9 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4 cvrfval.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2703 . . . . . . . 8 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
65eleq2d 2716 . . . . . . 7 (𝑝 = 𝐾 → (𝑥 ∈ (Base‘𝑝) ↔ 𝑥𝐵))
75eleq2d 2716 . . . . . . 7 (𝑝 = 𝐾 → (𝑦 ∈ (Base‘𝑝) ↔ 𝑦𝐵))
86, 7anbi12d 747 . . . . . 6 (𝑝 = 𝐾 → ((𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝)) ↔ (𝑥𝐵𝑦𝐵)))
9 fveq2 6229 . . . . . . . 8 (𝑝 = 𝐾 → (lt‘𝑝) = (lt‘𝐾))
10 cvrfval.s . . . . . . . 8 < = (lt‘𝐾)
119, 10syl6eqr 2703 . . . . . . 7 (𝑝 = 𝐾 → (lt‘𝑝) = < )
1211breqd 4696 . . . . . 6 (𝑝 = 𝐾 → (𝑥(lt‘𝑝)𝑦𝑥 < 𝑦))
1311breqd 4696 . . . . . . . . 9 (𝑝 = 𝐾 → (𝑥(lt‘𝑝)𝑧𝑥 < 𝑧))
1411breqd 4696 . . . . . . . . 9 (𝑝 = 𝐾 → (𝑧(lt‘𝑝)𝑦𝑧 < 𝑦))
1513, 14anbi12d 747 . . . . . . . 8 (𝑝 = 𝐾 → ((𝑥(lt‘𝑝)𝑧𝑧(lt‘𝑝)𝑦) ↔ (𝑥 < 𝑧𝑧 < 𝑦)))
165, 15rexeqbidv 3183 . . . . . . 7 (𝑝 = 𝐾 → (∃𝑧 ∈ (Base‘𝑝)(𝑥(lt‘𝑝)𝑧𝑧(lt‘𝑝)𝑦) ↔ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))
1716notbid 307 . . . . . 6 (𝑝 = 𝐾 → (¬ ∃𝑧 ∈ (Base‘𝑝)(𝑥(lt‘𝑝)𝑧𝑧(lt‘𝑝)𝑦) ↔ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))
188, 12, 173anbi123d 1439 . . . . 5 (𝑝 = 𝐾 → (((𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝)) ∧ 𝑥(lt‘𝑝)𝑦 ∧ ¬ ∃𝑧 ∈ (Base‘𝑝)(𝑥(lt‘𝑝)𝑧𝑧(lt‘𝑝)𝑦)) ↔ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))))
1918opabbidv 4749 . . . 4 (𝑝 = 𝐾 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝)) ∧ 𝑥(lt‘𝑝)𝑦 ∧ ¬ ∃𝑧 ∈ (Base‘𝑝)(𝑥(lt‘𝑝)𝑧𝑧(lt‘𝑝)𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))})
20 df-covers 34871 . . . 4 ⋖ = (𝑝 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝)) ∧ 𝑥(lt‘𝑝)𝑦 ∧ ¬ ∃𝑧 ∈ (Base‘𝑝)(𝑥(lt‘𝑝)𝑧𝑧(lt‘𝑝)𝑦))})
21 3anass 1059 . . . . . 6 (((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)) ↔ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))))
2221opabbii 4750 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}
23 fvex 6239 . . . . . . . 8 (Base‘𝐾) ∈ V
244, 23eqeltri 2726 . . . . . . 7 𝐵 ∈ V
2524, 24xpex 7004 . . . . . 6 (𝐵 × 𝐵) ∈ V
26 opabssxp 5227 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))} ⊆ (𝐵 × 𝐵)
2725, 26ssexi 4836 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))} ∈ V
2822, 27eqeltri 2726 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))} ∈ V
2919, 20, 28fvmpt 6321 . . 3 (𝐾 ∈ V → ( ⋖ ‘𝐾) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))})
302, 29syl5eq 2697 . 2 (𝐾 ∈ V → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))})
311, 30syl 17 1 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942  Vcvv 3231   class class class wbr 4685  {copab 4745   × cxp 5141  cfv 5926  Basecbs 15904  ltcplt 16988  ccvr 34867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-covers 34871
This theorem is referenced by:  cvrval  34874
  Copyright terms: Public domain W3C validator