Theorem brrabga 35613

Description: The law of concretion for operation class abstraction. (Contributed by Peter Mazsa, 24-Oct-2022.)

Hypotheses

Ref	Expression
brrabga.1	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))
brrabga.2	⊢ 𝑅 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Assertion

Ref	Expression
brrabga	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐴, 𝐵⟩𝑅𝐶 ↔ 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 brrabga

Step	Hyp	Ref	Expression
1		df-br 5067	. . 3 ⊢ (⟨𝐴, 𝐵⟩𝑅𝐶 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ 𝑅)
2		brrabga.2	. . . 4 ⊢ 𝑅 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
3	2	eleq2i 2904	. . 3 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ 𝑅 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
4	1, 3	bitri 277	. 2 ⊢ (⟨𝐴, 𝐵⟩𝑅𝐶 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
5		brrabga.1	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))
6	5	eloprabga 7261	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓))
7	4, 6	syl5bb 285	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐴, 𝐵⟩𝑅𝐶 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ⟨cop 4573   class class class wbr 5066  {coprab 7157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-oprab 7160

This theorem is referenced by:  brcnvrabga  35614