Theorem opelinxp 5698

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 5696	. 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷))
2		df-br 5073	. 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)))
3		df-br 5073	. . 3 ⊢ (𝐶𝑅𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑅)
4	3	anbi2i 629	. 2 ⊢ (((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑅))
5	1, 2, 4	3bitr3i 302	1 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑅))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∈ wcel 2119   ∩ cin 3882  ⟨cop 4561   class class class wbr 5072   × cxp 5616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624

This theorem is referenced by:  elinxp  5971  ssrnres  6129  iss2  38711