Theorem brintclab 15040

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 5144	. 2 ⊢ (𝐴∩ {𝑥 ∣ 𝜑}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∩ {𝑥 ∣ 𝜑})
2		opex 5469	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
3	2	elintab 4958	. 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 1538   ∈ wcel 2108  {cab 2714  ⟨cop 4632  ∩ cint 4946   class class class wbr 5143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-int 4947  df-br 5144

This theorem is referenced by:  brtrclfv  15041