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Theorem inxp 5844
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 2138, ax-12 2174. (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 5826 . 2 Rel ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷))
2 relxp 5706 . 2 Rel ((𝐴𝐶) × (𝐵𝐷))
3 an4 656 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ (𝑥𝐶𝑦𝐷)) ↔ ((𝑥𝐴𝑥𝐶) ∧ (𝑦𝐵𝑦𝐷)))
4 opelxp 5724 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
5 opelxp 5724 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ↔ (𝑥𝐶𝑦𝐷))
64, 5anbi12i 628 . . . 4 ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷)) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑥𝐶𝑦𝐷)))
7 elin 3978 . . . . 5 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
8 elin 3978 . . . . 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 3978 . . 3 (⟨𝑥, 𝑦⟩ ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷)))
12 opelxp 5724 . . 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 5802 1 ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴𝐶) × (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1536  wcel 2105  cin 3961  cop 4636   × cxp 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-opab 5210  df-xp 5694  df-rel 5695
This theorem is referenced by:  xpindi  5846  xpindir  5847  dmxpin  5944  xpssres  6037  xpdisj1  6182  xpdisj2  6183  imainrect  6202  xpima  6203  cnvrescnv  6216  curry1  8127  curry2  8130  fpar  8139  marypha1lem  9470  fpwwe2lem12  10679  hashxplem  14468  sscres  17870  gsumxp  20008  pjfval  21743  pjpm  21745  txbas  23590  txcls  23627  txrest  23654  trust  24253  ressuss  24286  trcfilu  24318  metreslem  24387  ressxms  24553  ressms  24554  mbfmcst  34240  0rrv  34432  poimirlem26  37632
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