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Theorem brredunds 38154
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 4000 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
213adant3 1129 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑦𝐴𝐵))
3 ineq12 4201 . . . . 5 ((𝑥 = 𝐴𝑧 = 𝐶) → (𝑥𝑧) = (𝐴𝐶))
433adant2 1128 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑧) = (𝐴𝐶))
5 ineq12 4201 . . . . 5 ((𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑧) = (𝐵𝐶))
653adant1 1127 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑧) = (𝐵𝐶))
74, 6eqeq12d 2741 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑧) = (𝑦𝑧) ↔ (𝐴𝐶) = (𝐵𝐶)))
82, 7anbi12d 630 . 2 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑦 ∧ (𝑥𝑧) = (𝑦𝑧)) ↔ (𝐴𝐵 ∧ (𝐴𝐶) = (𝐵𝐶))))
9 df-redunds 38151 . 2 Redunds = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥𝑦 ∧ (𝑥𝑧) = (𝑦𝑧))}
108, 9brcnvrabga 37870 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 Redunds ⟨𝐵, 𝐶⟩ ↔ (𝐴𝐵 ∧ (𝐴𝐶) = (𝐵𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  cin 3938  wss 3939  cop 4630   class class class wbr 5143   Redunds credunds 37725
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-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-cnv 5680  df-oprab 7420  df-redunds 38151
This theorem is referenced by:  brredundsredund  38155
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