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Theorem brcnvrabga 38343
Description: The law of concretion for the converse of operation class abstraction. (Contributed by Peter Mazsa, 25-Oct-2022.)
Hypotheses
Ref Expression
brrabga.1 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
brcnvrabga.2 𝑅 = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
Assertion
Ref Expression
brcnvrabga ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑅𝐵, 𝐶⟩ ↔ 𝜓))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜓,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝑅(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦,𝑧)

Proof of Theorem brcnvrabga
StepHypRef Expression
1 relcnv 6122 . . . 4 Rel {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
2 brcnvrabga.2 . . . . 5 𝑅 = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
32releqi 5787 . . . 4 (Rel 𝑅 ↔ Rel {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑})
41, 3mpbir 231 . . 3 Rel 𝑅
54relbrcnv 6125 . 2 (⟨𝐵, 𝐶𝑅𝐴𝐴𝑅𝐵, 𝐶⟩)
6 brrabga.1 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
763coml 1128 . . . 4 ((𝑦 = 𝐵𝑧 = 𝐶𝑥 = 𝐴) → (𝜑𝜓))
82cnveqi 5885 . . . . 5 𝑅 = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
9 reloprab 7492 . . . . . 6 Rel {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
10 dfrel2 6209 . . . . . 6 (Rel {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} ↔ {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑})
119, 10mpbi 230 . . . . 5 {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
128, 11eqtri 2765 . . . 4 𝑅 = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
137, 12brrabga 38342 . . 3 ((𝐵𝑊𝐶𝑋𝐴𝑉) → (⟨𝐵, 𝐶𝑅𝐴𝜓))
14133comr 1126 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶𝑅𝐴𝜓))
155, 14bitr3id 285 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑅𝐵, 𝐶⟩ ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   = wceq 1540  wcel 2108  cop 4632   class class class wbr 5143  ccnv 5684  Rel wrel 5690  {coprab 7432
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-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
This theorem is referenced by:  brredunds  38627
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