Theorem brcnvrabga 37211

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 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))
brcnvrabga.2	⊢ 𝑅 = ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}

Assertion

Ref	Expression
brcnvrabga	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑅⟨𝐵, 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

Step	Hyp	Ref	Expression
1		relcnv 6104	. . . 4 ⊢ Rel ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
2		brcnvrabga.2	. . . . 5 ⊢ 𝑅 = ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
3	2	releqi 5778	. . . 4 ⊢ (Rel 𝑅 ↔ Rel ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑})
4	1, 3	mpbir 230	. . 3 ⊢ Rel 𝑅
5	4	relbrcnv 6107	. 2 ⊢ (⟨𝐵, 𝐶⟩◡𝑅𝐴 ↔ 𝐴𝑅⟨𝐵, 𝐶⟩)
6		brrabga.1	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))
7	6	3coml 1128	. . . 4 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
8	2	cnveqi 5875	. . . . 5 ⊢ ◡𝑅 = ◡◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
9		reloprab 7468	. . . . . 6 ⊢ Rel {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
10		dfrel2 6189	. . . . . 6 ⊢ (Rel {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} ↔ ◡◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑})
11	9, 10	mpbi 229	. . . . 5 ⊢ ◡◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
12	8, 11	eqtri 2761	. . . 4 ⊢ ◡𝑅 = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
13	7, 12	brrabga 37210	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑉) → (⟨𝐵, 𝐶⟩◡𝑅𝐴 ↔ 𝜓))
14	13	3comr 1126	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩◡𝑅𝐴 ↔ 𝜓))
15	5, 14	bitr3id 285	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑅⟨𝐵, 𝐶⟩ ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ⟨cop 4635   class class class wbr 5149  ^◡ccnv 5676  Rel wrel 5682  {coprab 7410

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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

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 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-oprab 7413

This theorem is referenced by:  brredunds  37496