Theorem brcnvrabga 35759

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 5934	. . . 4 ⊢ Rel ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
2		brcnvrabga.2	. . . . 5 ⊢ 𝑅 = ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
3	2	releqi 5616	. . . 4 ⊢ (Rel 𝑅 ↔ Rel ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑})
4	1, 3	mpbir 234	. . 3 ⊢ Rel 𝑅
5	4	relbrcnv 5937	. 2 ⊢ (⟨𝐵, 𝐶⟩◡𝑅𝐴 ↔ 𝐴𝑅⟨𝐵, 𝐶⟩)
6		brrabga.1	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))
7	6	3coml 1124	. . . 4 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
8	2	cnveqi 5709	. . . . 5 ⊢ ◡𝑅 = ◡◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
9		reloprab 7192	. . . . . 6 ⊢ Rel {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
10		dfrel2 6013	. . . . . 6 ⊢ (Rel {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} ↔ ◡◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑})
11	9, 10	mpbi 233	. . . . 5 ⊢ ◡◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
12	8, 11	eqtri 2821	. . . 4 ⊢ ◡𝑅 = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}
13	7, 12	brrabga 35758	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑉) → (⟨𝐵, 𝐶⟩◡𝑅𝐴 ↔ 𝜓))
14	13	3comr 1122	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩◡𝑅𝐴 ↔ 𝜓))
15	5, 14	bitr3id 288	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑅⟨𝐵, 𝐶⟩ ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ⟨cop 4531   class class class wbr 5030  ^◡ccnv 5518  Rel wrel 5524  {coprab 7136

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-oprab 7139

This theorem is referenced by:  brredunds  36021