Theorem brintclab 13742

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 4654	. 2 ⊢ (𝐴∩ {𝑥 ∣ 𝜑}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∩ {𝑥 ∣ 𝜑})
2		opex 4932	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
3	2	elintab 4487	. 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 1481   ∈ wcel 1990  {cab 2608  ⟨cop 4183  ∩ cint 4475   class class class wbr 4653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-int 4476  df-br 4654

This theorem is referenced by:  brtrclfv  13743