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Theorem inxp 5842
Description: Intersection of two Cartesian products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-10 2141, ax-12 2177. (Revised by SN, 5-May-2025.)
Assertion
Ref Expression
inxp ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴𝐶) × (𝐵𝐷))

Proof of Theorem inxp
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relinxp 5824 . 2 Rel ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷))
2 relxp 5703 . 2 Rel ((𝐴𝐶) × (𝐵𝐷))
3 an4 656 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ (𝑥𝐶𝑦𝐷)) ↔ ((𝑥𝐴𝑥𝐶) ∧ (𝑦𝐵𝑦𝐷)))
4 opelxp 5721 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
5 opelxp 5721 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ↔ (𝑥𝐶𝑦𝐷))
64, 5anbi12i 628 . . . 4 ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷)) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑥𝐶𝑦𝐷)))
7 elin 3967 . . . . 5 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
8 elin 3967 . . . . 5 (𝑦 ∈ (𝐵𝐷) ↔ (𝑦𝐵𝑦𝐷))
97, 8anbi12i 628 . . . 4 ((𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐵𝐷)) ↔ ((𝑥𝐴𝑥𝐶) ∧ (𝑦𝐵𝑦𝐷)))
103, 6, 93bitr4i 303 . . 3 ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷)) ↔ (𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐵𝐷)))
11 elin 3967 . . 3 (⟨𝑥, 𝑦⟩ ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷)))
12 opelxp 5721 . . 3 (⟨𝑥, 𝑦⟩ ∈ ((𝐴𝐶) × (𝐵𝐷)) ↔ (𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐵𝐷)))
1310, 11, 123bitr4i 303 . 2 (⟨𝑥, 𝑦⟩ ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) ↔ ⟨𝑥, 𝑦⟩ ∈ ((𝐴𝐶) × (𝐵𝐷)))
141, 2, 13eqrelriiv 5800 1 ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴𝐶) × (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2108  cin 3950  cop 4632   × cxp 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-opab 5206  df-xp 5691  df-rel 5692
This theorem is referenced by:  xpindi  5844  xpindir  5845  dmxpin  5942  xpssres  6036  xpdisj1  6181  xpdisj2  6182  imainrect  6201  xpima  6202  cnvrescnv  6215  curry1  8129  curry2  8132  fpar  8141  marypha1lem  9473  fpwwe2lem12  10682  hashxplem  14472  sscres  17867  gsumxp  19994  pjfval  21726  pjpm  21728  txbas  23575  txcls  23612  txrest  23639  trust  24238  ressuss  24271  trcfilu  24303  metreslem  24372  ressxms  24538  ressms  24539  mbfmcst  34261  0rrv  34453  poimirlem26  37653
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