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Theorem opelinxp 34453
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 brinxp2ALTV 34377 . 2 (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶𝐴𝐷𝐵) ∧ 𝐶𝑅𝐷))
2 df-br 4787 . 2 (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)))
3 df-br 4787 . . 3 (𝐶𝑅𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑅)
43anbi2i 609 . 2 (((𝐶𝐴𝐷𝐵) ∧ 𝐶𝑅𝐷) ↔ ((𝐶𝐴𝐷𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑅))
51, 2, 43bitr3i 290 1 (⟨𝐶, 𝐷⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝐶𝐴𝐷𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  wcel 2145  cin 3722  cop 4322   class class class wbr 4786   × cxp 5247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255
This theorem is referenced by:  iss2  34454
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