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Theorem brcnvrabga 36244
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 5987 . . . 4 Rel {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
2 brcnvrabga.2 . . . . 5 𝑅 = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
32releqi 5664 . . . 4 (Rel 𝑅 ↔ Rel {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑})
41, 3mpbir 234 . . 3 Rel 𝑅
54relbrcnv 5990 . 2 (⟨𝐵, 𝐶𝑅𝐴𝐴𝑅𝐵, 𝐶⟩)
6 brrabga.1 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
763coml 1129 . . . 4 ((𝑦 = 𝐵𝑧 = 𝐶𝑥 = 𝐴) → (𝜑𝜓))
82cnveqi 5758 . . . . 5 𝑅 = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
9 reloprab 7289 . . . . . 6 Rel {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
10 dfrel2 6067 . . . . . 6 (Rel {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} ↔ {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑})
119, 10mpbi 233 . . . . 5 {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
128, 11eqtri 2766 . . . 4 𝑅 = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
137, 12brrabga 36243 . . 3 ((𝐵𝑊𝐶𝑋𝐴𝑉) → (⟨𝐵, 𝐶𝑅𝐴𝜓))
14133comr 1127 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶𝑅𝐴𝜓))
155, 14bitr3id 288 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑅𝐵, 𝐶⟩ ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1089   = wceq 1543  wcel 2111  cop 4562   class class class wbr 5068  ccnv 5565  Rel wrel 5571  {coprab 7233
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 2113  ax-9 2121  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337
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 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-br 5069  df-opab 5131  df-xp 5572  df-rel 5573  df-cnv 5574  df-oprab 7236
This theorem is referenced by:  brredunds  36506
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