Theorem brintclab 14936

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 5101	. 2 ⊢ (𝐴∩ {𝑥 ∣ 𝜑}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∩ {𝑥 ∣ 𝜑})
2		opex 5419	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
3	2	elintab 4916	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑥))
4	1, 3	bitri 275	1 ⊢ (𝐴∩ {𝑥 ∣ 𝜑}𝐵 ↔ ∀𝑥(𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540   ∈ wcel 2114  {cab 2715  ⟨cop 4588  ∩ cint 4904   class class class wbr 5100

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3402  df-v 3444  df-un 3908  df-in 3910  df-ss 3920  df-sn 4583  df-pr 4585  df-op 4589  df-int 4905  df-br 5101

This theorem is referenced by:  brtrclfv  14937