Theorem opelinxp 5384

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 5381	. 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷))
2		df-br 4842	. 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)))
3		df-br 4842	. . 3 ⊢ (𝐶𝑅𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑅)
4	3	anbi2i 617	. 2 ⊢ (((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑅))
5	1, 2, 4	3bitr3i 293	1 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑅))

Syntax hints:   ↔ wb 198   ∧ wa 385   ∈ wcel 2157   ∩ cin 3766  ⟨cop 4372   class class class wbr 4841   × cxp 5308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-xp 5316

This theorem is referenced by:  elinxp  5642  ssrnres  5787  iss2  34597