Theorem opelinxp 5666

Description: Ordered pair element in an intersection with Cartesian product. (Contributed by Peter Mazsa, 21-Jul-2019.)

Assertion

Ref	Expression
opelinxp	⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑅))

Proof of Theorem opelinxp

Step	Hyp	Ref	Expression
1		brinxp2 5664	. 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷))
2		df-br 5075	. 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)))
3		df-br 5075	. . 3 ⊢ (𝐶𝑅𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑅)
4	3	anbi2i 623	. 2 ⊢ (((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑅))
5	1, 2, 4	3bitr3i 301	1 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑅))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∈ wcel 2106   ∩ cin 3886  ⟨cop 4567   class class class wbr 5074   × cxp 5587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595

This theorem is referenced by:  elinxp  5929  ssrnres  6081  iss2  36479