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Theorem brredunds 36476
Description: Binary relation on the class of all redundant sets. (Contributed by Peter Mazsa, 25-Oct-2022.)
Assertion
Ref Expression
brredunds ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 Redunds ⟨𝐵, 𝐶⟩ ↔ (𝐴𝐵 ∧ (𝐴𝐶) = (𝐵𝐶))))

Proof of Theorem brredunds
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq12 3928 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
213adant3 1134 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑦𝐴𝐵))
3 ineq12 4122 . . . . 5 ((𝑥 = 𝐴𝑧 = 𝐶) → (𝑥𝑧) = (𝐴𝐶))
433adant2 1133 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑧) = (𝐴𝐶))
5 ineq12 4122 . . . . 5 ((𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑧) = (𝐵𝐶))
653adant1 1132 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑧) = (𝐵𝐶))
74, 6eqeq12d 2753 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑧) = (𝑦𝑧) ↔ (𝐴𝐶) = (𝐵𝐶)))
82, 7anbi12d 634 . 2 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑦 ∧ (𝑥𝑧) = (𝑦𝑧)) ↔ (𝐴𝐵 ∧ (𝐴𝐶) = (𝐵𝐶))))
9 df-redunds 36473 . 2 Redunds = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥𝑦 ∧ (𝑥𝑧) = (𝑦𝑧))}
108, 9brcnvrabga 36214 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 Redunds ⟨𝐵, 𝐶⟩ ↔ (𝐴𝐵 ∧ (𝐴𝐶) = (𝐵𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  cin 3865  wss 3866  cop 4547   class class class wbr 5053   Redunds credunds 36090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-cnv 5559  df-oprab 7217  df-redunds 36473
This theorem is referenced by:  brredundsredund  36477
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