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Theorem cvrfval 36561
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 3459 . 2 (𝐾𝐴𝐾 ∈ V)
2 cvrfval.c . . 3 𝐶 = ( ⋖ ‘𝐾)
3 fveq2 6645 . . . . . . . . 9 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4 cvrfval.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
53, 4eqtr4di 2851 . . . . . . . 8 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
65eleq2d 2875 . . . . . . 7 (𝑝 = 𝐾 → (𝑥 ∈ (Base‘𝑝) ↔ 𝑥𝐵))
75eleq2d 2875 . . . . . . 7 (𝑝 = 𝐾 → (𝑦 ∈ (Base‘𝑝) ↔ 𝑦𝐵))
86, 7anbi12d 633 . . . . . 6 (𝑝 = 𝐾 → ((𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝)) ↔ (𝑥𝐵𝑦𝐵)))
9 fveq2 6645 . . . . . . . 8 (𝑝 = 𝐾 → (lt‘𝑝) = (lt‘𝐾))
10 cvrfval.s . . . . . . . 8 < = (lt‘𝐾)
119, 10eqtr4di 2851 . . . . . . 7 (𝑝 = 𝐾 → (lt‘𝑝) = < )
1211breqd 5041 . . . . . 6 (𝑝 = 𝐾 → (𝑥(lt‘𝑝)𝑦𝑥 < 𝑦))
1311breqd 5041 . . . . . . . . 9 (𝑝 = 𝐾 → (𝑥(lt‘𝑝)𝑧𝑥 < 𝑧))
1411breqd 5041 . . . . . . . . 9 (𝑝 = 𝐾 → (𝑧(lt‘𝑝)𝑦𝑧 < 𝑦))
1513, 14anbi12d 633 . . . . . . . 8 (𝑝 = 𝐾 → ((𝑥(lt‘𝑝)𝑧𝑧(lt‘𝑝)𝑦) ↔ (𝑥 < 𝑧𝑧 < 𝑦)))
165, 15rexeqbidv 3355 . . . . . . 7 (𝑝 = 𝐾 → (∃𝑧 ∈ (Base‘𝑝)(𝑥(lt‘𝑝)𝑧𝑧(lt‘𝑝)𝑦) ↔ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))
1716notbid 321 . . . . . 6 (𝑝 = 𝐾 → (¬ ∃𝑧 ∈ (Base‘𝑝)(𝑥(lt‘𝑝)𝑧𝑧(lt‘𝑝)𝑦) ↔ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))
188, 12, 173anbi123d 1433 . . . . 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 5096 . . . 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 36559 . . . 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 1092 . . . . . 6 (((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)) ↔ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))))
2221opabbii 5097 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}
234fvexi 6659 . . . . . . 7 𝐵 ∈ V
2423, 23xpex 7456 . . . . . 6 (𝐵 × 𝐵) ∈ V
25 opabssxp 5607 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))} ⊆ (𝐵 × 𝐵)
2624, 25ssexi 5190 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))} ∈ V
2722, 26eqeltri 2886 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))} ∈ V
2819, 20, 27fvmpt 6745 . . 3 (𝐾 ∈ V → ( ⋖ ‘𝐾) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))})
292, 28syl5eq 2845 . 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 399  w3a 1084   = wceq 1538  wcel 2111  wrex 3107  Vcvv 3441   class class class wbr 5030  {copab 5092   × cxp 5517  cfv 6324  Basecbs 16475  ltcplt 17543  ccvr 36555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-covers 36559
This theorem is referenced by:  cvrval  36562
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