Theorem brintclab 13786

Description: Two ways to express a binary relation which is the intersection of a class. (Contributed by RP, 4-Apr-2020.)

Assertion

Ref	Expression
brintclab	⊢ (𝐴∩ {𝑥 ∣ 𝜑}𝐵 ↔ ∀𝑥(𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem brintclab

Step	Hyp	Ref	Expression
1		df-br 4686	. 2 ⊢ (𝐴∩ {𝑥 ∣ 𝜑}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∩ {𝑥 ∣ 𝜑})
2		opex 4962	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
3	2	elintab 4519	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑥))
4	1, 3	bitri 264	1 ⊢ (𝐴∩ {𝑥 ∣ 𝜑}𝐵 ↔ ∀𝑥(𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1521   ∈ wcel 2030  {cab 2637  ⟨cop 4216  ∩ cint 4507   class class class wbr 4685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-int 4508  df-br 4686

This theorem is referenced by:  brtrclfv  13787