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Theorem brredunds 38627
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 4011 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
213adant3 1133 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑦𝐴𝐵))
3 ineq12 4215 . . . . 5 ((𝑥 = 𝐴𝑧 = 𝐶) → (𝑥𝑧) = (𝐴𝐶))
433adant2 1132 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑧) = (𝐴𝐶))
5 ineq12 4215 . . . . 5 ((𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑧) = (𝐵𝐶))
653adant1 1131 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑧) = (𝐵𝐶))
74, 6eqeq12d 2753 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑧) = (𝑦𝑧) ↔ (𝐴𝐶) = (𝐵𝐶)))
82, 7anbi12d 632 . 2 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑦 ∧ (𝑥𝑧) = (𝑦𝑧)) ↔ (𝐴𝐵 ∧ (𝐴𝐶) = (𝐵𝐶))))
9 df-redunds 38624 . 2 Redunds = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥𝑦 ∧ (𝑥𝑧) = (𝑦𝑧))}
108, 9brcnvrabga 38343 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 Redunds ⟨𝐵, 𝐶⟩ ↔ (𝐴𝐵 ∧ (𝐴𝐶) = (𝐵𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  cin 3950  wss 3951  cop 4632   class class class wbr 5143   Redunds credunds 38202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-oprab 7435  df-redunds 38624
This theorem is referenced by:  brredundsredund  38628
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