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Theorem cvrfval 35878
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 3427 . 2 (𝐾𝐴𝐾 ∈ V)
2 cvrfval.c . . 3 𝐶 = ( ⋖ ‘𝐾)
3 fveq2 6496 . . . . . . . . 9 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4 cvrfval.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2826 . . . . . . . 8 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
65eleq2d 2845 . . . . . . 7 (𝑝 = 𝐾 → (𝑥 ∈ (Base‘𝑝) ↔ 𝑥𝐵))
75eleq2d 2845 . . . . . . 7 (𝑝 = 𝐾 → (𝑦 ∈ (Base‘𝑝) ↔ 𝑦𝐵))
86, 7anbi12d 621 . . . . . 6 (𝑝 = 𝐾 → ((𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝)) ↔ (𝑥𝐵𝑦𝐵)))
9 fveq2 6496 . . . . . . . 8 (𝑝 = 𝐾 → (lt‘𝑝) = (lt‘𝐾))
10 cvrfval.s . . . . . . . 8 < = (lt‘𝐾)
119, 10syl6eqr 2826 . . . . . . 7 (𝑝 = 𝐾 → (lt‘𝑝) = < )
1211breqd 4936 . . . . . 6 (𝑝 = 𝐾 → (𝑥(lt‘𝑝)𝑦𝑥 < 𝑦))
1311breqd 4936 . . . . . . . . 9 (𝑝 = 𝐾 → (𝑥(lt‘𝑝)𝑧𝑥 < 𝑧))
1411breqd 4936 . . . . . . . . 9 (𝑝 = 𝐾 → (𝑧(lt‘𝑝)𝑦𝑧 < 𝑦))
1513, 14anbi12d 621 . . . . . . . 8 (𝑝 = 𝐾 → ((𝑥(lt‘𝑝)𝑧𝑧(lt‘𝑝)𝑦) ↔ (𝑥 < 𝑧𝑧 < 𝑦)))
165, 15rexeqbidv 3336 . . . . . . 7 (𝑝 = 𝐾 → (∃𝑧 ∈ (Base‘𝑝)(𝑥(lt‘𝑝)𝑧𝑧(lt‘𝑝)𝑦) ↔ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))
1716notbid 310 . . . . . 6 (𝑝 = 𝐾 → (¬ ∃𝑧 ∈ (Base‘𝑝)(𝑥(lt‘𝑝)𝑧𝑧(lt‘𝑝)𝑦) ↔ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))
188, 12, 173anbi123d 1415 . . . . 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 4991 . . . 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 35876 . . . 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 1076 . . . . . 6 (((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)) ↔ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))))
2221opabbii 4992 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}
234fvexi 6510 . . . . . . 7 𝐵 ∈ V
2423, 23xpex 7291 . . . . . 6 (𝐵 × 𝐵) ∈ V
25 opabssxp 5489 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))} ⊆ (𝐵 × 𝐵)
2624, 25ssexi 5078 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))} ∈ V
2722, 26eqeltri 2856 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))} ∈ V
2819, 20, 27fvmpt 6593 . . 3 (𝐾 ∈ V → ( ⋖ ‘𝐾) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))})
292, 28syl5eq 2820 . 2 (𝐾 ∈ V → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))})
301, 29syl 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 387  w3a 1068   = wceq 1507  wcel 2050  wrex 3083  Vcvv 3409   class class class wbr 4925  {copab 4987   × cxp 5401  cfv 6185  Basecbs 16337  ltcplt 17421  ccvr 35872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-iota 6149  df-fun 6187  df-fv 6193  df-covers 35876
This theorem is referenced by:  cvrval  35879
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