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 = 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 6122	. . . 4 ⊢ Rel ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
2		brcnvrabga.2	. . . . 5 ⊢ 𝑅 = ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
3	2	releqi 5787	. . . 4 ⊢ (Rel 𝑅 ↔ Rel ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑})
4	1, 3	mpbir 231	. . 3 ⊢ Rel 𝑅
5	4	relbrcnv 6125	. 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 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})
11	9, 10	mpbi 230	. . . . 5 ⊢ ◡◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
12	8, 11	eqtri 2765	. . . 4 ⊢ ◡𝑅 = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
13	7, 12	brrabga 38342	. . 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 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