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Theorem inxp 5410
Description: The 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.)
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 inopab 5408 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐷)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑥𝐶𝑦𝐷))}
2 an4 900 . . . . 5 (((𝑥𝐴𝑦𝐵) ∧ (𝑥𝐶𝑦𝐷)) ↔ ((𝑥𝐴𝑥𝐶) ∧ (𝑦𝐵𝑦𝐷)))
3 elin 3939 . . . . . 6 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
4 elin 3939 . . . . . 6 (𝑦 ∈ (𝐵𝐷) ↔ (𝑦𝐵𝑦𝐷))
53, 4anbi12i 735 . . . . 5 ((𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐵𝐷)) ↔ ((𝑥𝐴𝑥𝐶) ∧ (𝑦𝐵𝑦𝐷)))
62, 5bitr4i 267 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ (𝑥𝐶𝑦𝐷)) ↔ (𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐵𝐷)))
76opabbii 4869 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑥𝐶𝑦𝐷))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐵𝐷))}
81, 7eqtri 2782 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐷)}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐵𝐷))}
9 df-xp 5272 . . 3 (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
10 df-xp 5272 . . 3 (𝐶 × 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐷)}
119, 10ineq12i 3955 . 2 ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐷)})
12 df-xp 5272 . 2 ((𝐴𝐶) × (𝐵𝐷)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐵𝐷))}
138, 11, 123eqtr4i 2792 1 ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴𝐶) × (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wcel 2139  cin 3714  {copab 4864   × cxp 5264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-xp 5272  df-rel 5273
This theorem is referenced by:  xpindi  5411  xpindir  5412  dmxpin  5501  xpssres  5592  xpdisj1  5713  xpdisj2  5714  imainrect  5733  xpima  5734  curry1  7437  curry2  7440  fpar  7449  marypha1lem  8504  fpwwe2lem13  9656  hashxplem  13412  sscres  16684  gsumxp  18575  pjfval  20252  pjpm  20254  txbas  21572  txcls  21609  txrest  21636  trust  22234  ressuss  22268  trcfilu  22299  metreslem  22368  ressxms  22531  ressms  22532  mbfmcst  30630  0rrv  30822  poimirlem26  33748
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