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Theorem cvrfval 35050
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 3344 . 2 (𝐾𝐴𝐾 ∈ V)
2 cvrfval.c . . 3 𝐶 = ( ⋖ ‘𝐾)
3 fveq2 6344 . . . . . . . . 9 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4 cvrfval.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2804 . . . . . . . 8 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
65eleq2d 2817 . . . . . . 7 (𝑝 = 𝐾 → (𝑥 ∈ (Base‘𝑝) ↔ 𝑥𝐵))
75eleq2d 2817 . . . . . . 7 (𝑝 = 𝐾 → (𝑦 ∈ (Base‘𝑝) ↔ 𝑦𝐵))
86, 7anbi12d 749 . . . . . 6 (𝑝 = 𝐾 → ((𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝)) ↔ (𝑥𝐵𝑦𝐵)))
9 fveq2 6344 . . . . . . . 8 (𝑝 = 𝐾 → (lt‘𝑝) = (lt‘𝐾))
10 cvrfval.s . . . . . . . 8 < = (lt‘𝐾)
119, 10syl6eqr 2804 . . . . . . 7 (𝑝 = 𝐾 → (lt‘𝑝) = < )
1211breqd 4807 . . . . . 6 (𝑝 = 𝐾 → (𝑥(lt‘𝑝)𝑦𝑥 < 𝑦))
1311breqd 4807 . . . . . . . . 9 (𝑝 = 𝐾 → (𝑥(lt‘𝑝)𝑧𝑥 < 𝑧))
1411breqd 4807 . . . . . . . . 9 (𝑝 = 𝐾 → (𝑧(lt‘𝑝)𝑦𝑧 < 𝑦))
1513, 14anbi12d 749 . . . . . . . 8 (𝑝 = 𝐾 → ((𝑥(lt‘𝑝)𝑧𝑧(lt‘𝑝)𝑦) ↔ (𝑥 < 𝑧𝑧 < 𝑦)))
165, 15rexeqbidv 3284 . . . . . . 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 1540 . . . . 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 4860 . . . 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 35048 . . . 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 1081 . . . . . 6 (((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)) ↔ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))))
2221opabbii 4861 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}
23 fvex 6354 . . . . . . . 8 (Base‘𝐾) ∈ V
244, 23eqeltri 2827 . . . . . . 7 𝐵 ∈ V
2524, 24xpex 7119 . . . . . 6 (𝐵 × 𝐵) ∈ V
26 opabssxp 5342 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))} ⊆ (𝐵 × 𝐵)
2725, 26ssexi 4947 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))} ∈ V
2822, 27eqeltri 2827 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))} ∈ V
2919, 20, 28fvmpt 6436 . . 3 (𝐾 ∈ V → ( ⋖ ‘𝐾) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))})
302, 29syl5eq 2798 . 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 1072   = wceq 1624  wcel 2131  wrex 3043  Vcvv 3332   class class class wbr 4796  {copab 4856   × cxp 5256  cfv 6041  Basecbs 16051  ltcplt 17134  ccvr 35044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-iota 6004  df-fun 6043  df-fv 6049  df-covers 35048
This theorem is referenced by:  cvrval  35051
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