Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  opelinxp Structured version   Visualization version   GIF version

Theorem opelinxp 5625
 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)) ↔ ((𝐶𝐴𝐷𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑅))

Proof of Theorem opelinxp
StepHypRef Expression
1 brinxp2 5623 . 2 (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶𝐴𝐷𝐵) ∧ 𝐶𝑅𝐷))
2 df-br 5059 . 2 (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)))
3 df-br 5059 . . 3 (𝐶𝑅𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑅)
43anbi2i 624 . 2 (((𝐶𝐴𝐷𝐵) ∧ 𝐶𝑅𝐷) ↔ ((𝐶𝐴𝐷𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑅))
51, 2, 43bitr3i 303 1 (⟨𝐶, 𝐷⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝐶𝐴𝐷𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∧ wa 398   ∈ wcel 2110   ∩ cin 3934  ⟨cop 4566   class class class wbr 5058   × cxp 5547 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-xp 5555 This theorem is referenced by:  elinxp  5884  ssrnres  6029  iss2  35595
 Copyright terms: Public domain W3C validator