Theorem brcnvrabga 38512

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 6062	. . . 4 ⊢ Rel ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
2		brcnvrabga.2	. . . . 5 ⊢ 𝑅 = ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
3	2	releqi 5726	. . . 4 ⊢ (Rel 𝑅 ↔ Rel ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑})
4	1, 3	mpbir 231	. . 3 ⊢ Rel 𝑅
5	4	relbrcnv 6065	. 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 5822	. . . . 5 ⊢ ◡𝑅 = ◡◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
9		reloprab 7417	. . . . . 6 ⊢ Rel {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
10		dfrel2 6146	. . . . . 6 ⊢ (Rel {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} ↔ ◡◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑})
11	9, 10	mpbi 230	. . . . 5 ⊢ ◡◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
12	8, 11	eqtri 2758	. . . 4 ⊢ ◡𝑅 = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
13	7, 12	brrabga 38511	. . 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 206   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ⟨cop 4585   class class class wbr 5097  ^◡ccnv 5622  Rel wrel 5628  {coprab 7359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5629  df-rel 5630  df-cnv 5631  df-oprab 7362

This theorem is referenced by:  brredunds  38880