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Theorem brredunds 37117
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 3976 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
213adant3 1133 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑦𝐴𝐵))
3 ineq12 4172 . . . . 5 ((𝑥 = 𝐴𝑧 = 𝐶) → (𝑥𝑧) = (𝐴𝐶))
433adant2 1132 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑧) = (𝐴𝐶))
5 ineq12 4172 . . . . 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 37114 . 2 Redunds = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥𝑦 ∧ (𝑥𝑧) = (𝑦𝑧))}
108, 9brcnvrabga 36832 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 Redunds ⟨𝐵, 𝐶⟩ ↔ (𝐴𝐵 ∧ (𝐴𝐶) = (𝐵𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  cin 3914  wss 3915  cop 4597   class class class wbr 5110   Redunds credunds 36683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646  df-oprab 7366  df-redunds 37114
This theorem is referenced by:  brredundsredund  37118
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