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Theorem inxp 4497
Description: The intersection of two cross 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 4495 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐷)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑥𝐶𝑦𝐷))}
2 an4 528 . . . . 5 (((𝑥𝐴𝑦𝐵) ∧ (𝑥𝐶𝑦𝐷)) ↔ ((𝑥𝐴𝑥𝐶) ∧ (𝑦𝐵𝑦𝐷)))
3 elin 3153 . . . . . 6 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
4 elin 3153 . . . . . 6 (𝑦 ∈ (𝐵𝐷) ↔ (𝑦𝐵𝑦𝐷))
53, 4anbi12i 441 . . . . 5 ((𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐵𝐷)) ↔ ((𝑥𝐴𝑥𝐶) ∧ (𝑦𝐵𝑦𝐷)))
62, 5bitr4i 180 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ (𝑥𝐶𝑦𝐷)) ↔ (𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐵𝐷)))
76opabbii 3851 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑥𝐶𝑦𝐷))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐵𝐷))}
81, 7eqtri 2076 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐷)}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐵𝐷))}
9 df-xp 4378 . . 3 (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
10 df-xp 4378 . . 3 (𝐶 × 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐷)}
119, 10ineq12i 3163 . 2 ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐷)})
12 df-xp 4378 . 2 ((𝐴𝐶) × (𝐵𝐷)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐶) ∧ 𝑦 ∈ (𝐵𝐷))}
138, 11, 123eqtr4i 2086 1 ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴𝐶) × (𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wcel 1409  cin 2943  {copab 3844   × cxp 4370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-opab 3846  df-xp 4378  df-rel 4379
This theorem is referenced by:  xpindi  4498  xpindir  4499  dmxpinm  4583  xpssres  4672  xpdisj1  4774  xpdisj2  4775  imainrect  4793  xpima1  4794  xpima2m  4795
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