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Theorem cvrfval 38650
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 3487 . 2 (𝐾𝐴𝐾 ∈ V)
2 cvrfval.c . . 3 𝐶 = ( ⋖ ‘𝐾)
3 fveq2 6884 . . . . . . . . 9 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4 cvrfval.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
53, 4eqtr4di 2784 . . . . . . . 8 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
65eleq2d 2813 . . . . . . 7 (𝑝 = 𝐾 → (𝑥 ∈ (Base‘𝑝) ↔ 𝑥𝐵))
75eleq2d 2813 . . . . . . 7 (𝑝 = 𝐾 → (𝑦 ∈ (Base‘𝑝) ↔ 𝑦𝐵))
86, 7anbi12d 630 . . . . . 6 (𝑝 = 𝐾 → ((𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝)) ↔ (𝑥𝐵𝑦𝐵)))
9 fveq2 6884 . . . . . . . 8 (𝑝 = 𝐾 → (lt‘𝑝) = (lt‘𝐾))
10 cvrfval.s . . . . . . . 8 < = (lt‘𝐾)
119, 10eqtr4di 2784 . . . . . . 7 (𝑝 = 𝐾 → (lt‘𝑝) = < )
1211breqd 5152 . . . . . 6 (𝑝 = 𝐾 → (𝑥(lt‘𝑝)𝑦𝑥 < 𝑦))
1311breqd 5152 . . . . . . . . 9 (𝑝 = 𝐾 → (𝑥(lt‘𝑝)𝑧𝑥 < 𝑧))
1411breqd 5152 . . . . . . . . 9 (𝑝 = 𝐾 → (𝑧(lt‘𝑝)𝑦𝑧 < 𝑦))
1513, 14anbi12d 630 . . . . . . . 8 (𝑝 = 𝐾 → ((𝑥(lt‘𝑝)𝑧𝑧(lt‘𝑝)𝑦) ↔ (𝑥 < 𝑧𝑧 < 𝑦)))
165, 15rexeqbidv 3337 . . . . . . 7 (𝑝 = 𝐾 → (∃𝑧 ∈ (Base‘𝑝)(𝑥(lt‘𝑝)𝑧𝑧(lt‘𝑝)𝑦) ↔ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))
1716notbid 318 . . . . . 6 (𝑝 = 𝐾 → (¬ ∃𝑧 ∈ (Base‘𝑝)(𝑥(lt‘𝑝)𝑧𝑧(lt‘𝑝)𝑦) ↔ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))
188, 12, 173anbi123d 1432 . . . . 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 5207 . . . 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 38648 . . . 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 5208 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))}
234fvexi 6898 . . . . . . 7 𝐵 ∈ V
2423, 23xpex 7736 . . . . . 6 (𝐵 × 𝐵) ∈ V
25 opabssxp 5761 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))} ⊆ (𝐵 × 𝐵)
2624, 25ssexi 5315 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦)))} ∈ V
2722, 26eqeltri 2823 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))} ∈ V
2819, 20, 27fvmpt 6991 . . 3 (𝐾 ∈ V → ( ⋖ ‘𝐾) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))})
292, 28eqtrid 2778 . 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 395  w3a 1084   = wceq 1533  wcel 2098  wrex 3064  Vcvv 3468   class class class wbr 5141  {copab 5203   × cxp 5667  cfv 6536  Basecbs 17150  ltcplt 18270  ccvr 38644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-covers 38648
This theorem is referenced by:  cvrval  38651
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