Theorem brintclab 13590

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 4584	. 2 ⊢ (𝐴∩ {𝑥 ∣ 𝜑}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∩ {𝑥 ∣ 𝜑})
2		opex 4859	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
3	2	elintab 4422	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑥))
4	1, 3	bitri 263	1 ⊢ (𝐴∩ {𝑥 ∣ 𝜑}𝐵 ↔ ∀𝑥(𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   ∈ wcel 1977  {cab 2596  ⟨cop 4131  ∩ cint 4410   class class class wbr 4583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-int 4411  df-br 4584

This theorem is referenced by:  brtrclfv  13591