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Theorem brcnvrabga 35586
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 5960 . . . 4 Rel {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
2 brcnvrabga.2 . . . . 5 𝑅 = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
32releqi 5645 . . . 4 (Rel 𝑅 ↔ Rel {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑})
41, 3mpbir 233 . . 3 Rel 𝑅
54relbrcnv 5963 . 2 (⟨𝐵, 𝐶𝑅𝐴𝐴𝑅𝐵, 𝐶⟩)
6 brrabga.1 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
763coml 1121 . . . 4 ((𝑦 = 𝐵𝑧 = 𝐶𝑥 = 𝐴) → (𝜑𝜓))
82cnveqi 5738 . . . . 5 𝑅 = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
9 reloprab 7205 . . . . . 6 Rel {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
10 dfrel2 6039 . . . . . 6 (Rel {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} ↔ {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑})
119, 10mpbi 232 . . . . 5 {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
128, 11eqtri 2842 . . . 4 𝑅 = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
137, 12brrabga 35585 . . 3 ((𝐵𝑊𝐶𝑋𝐴𝑉) → (⟨𝐵, 𝐶𝑅𝐴𝜓))
14133comr 1119 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶𝑅𝐴𝜓))
155, 14syl5bbr 287 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑅𝐵, 𝐶⟩ ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  w3a 1081   = wceq 1530  wcel 2107  cop 4565   class class class wbr 5057  ccnv 5547  Rel wrel 5553  {coprab 7149
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-oprab 7152
This theorem is referenced by:  brredunds  35848
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