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Theorem inxp 5824
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.)
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 5821 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐷)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑥𝐶𝑦𝐷))}
2 an4 654 . . . . 5 (((𝑥𝐴𝑦𝐵) ∧ (𝑥𝐶𝑦𝐷)) ↔ ((𝑥𝐴𝑥𝐶) ∧ (𝑦𝐵𝑦𝐷)))
3 elin 3960 . . . . . 6 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
4 elin 3960 . . . . . 6 (𝑦 ∈ (𝐵𝐷) ↔ (𝑦𝐵𝑦𝐷))
53, 4anbi12i 627 . . . . 5 ((𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐵𝐷)) ↔ ((𝑥𝐴𝑥𝐶) ∧ (𝑦𝐵𝑦𝐷)))
62, 5bitr4i 277 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ (𝑥𝐶𝑦𝐷)) ↔ (𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐵𝐷)))
76opabbii 5208 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑥𝐶𝑦𝐷))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐵𝐷))}
81, 7eqtri 2759 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐷)}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐵𝐷))}
9 df-xp 5675 . . 3 (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
10 df-xp 5675 . . 3 (𝐶 × 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐷)}
119, 10ineq12i 4206 . 2 ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐷)})
12 df-xp 5675 . 2 ((𝐴𝐶) × (𝐵𝐷)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐵𝐷))}
138, 11, 123eqtr4i 2769 1 ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴𝐶) × (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  cin 3943  {copab 5203   × cxp 5667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-opab 5204  df-xp 5675  df-rel 5676
This theorem is referenced by:  xpindi  5825  xpindir  5826  dmxpin  5922  xpssres  6010  xpdisj1  6149  xpdisj2  6150  imainrect  6169  xpima  6170  cnvrescnv  6183  curry1  8072  curry2  8075  fpar  8084  marypha1lem  9410  fpwwe2lem12  10619  hashxplem  14375  sscres  17752  gsumxp  19803  pjfval  21194  pjpm  21196  txbas  23000  txcls  23037  txrest  23064  trust  23663  ressuss  23696  trcfilu  23728  metreslem  23797  ressxms  23963  ressms  23964  mbfmcst  33089  0rrv  33281  poimirlem26  36318
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