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Theorem inxp 4413
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 4411 . . 3 ({⟨x, y⟩ ∣ (x A y B)} ∩ {⟨x, y⟩ ∣ (x 𝐶 y 𝐷)}) = {⟨x, y⟩ ∣ ((x A y B) (x 𝐶 y 𝐷))}
2 an4 520 . . . . 5 (((x A y B) (x 𝐶 y 𝐷)) ↔ ((x A x 𝐶) (y B y 𝐷)))
3 elin 3120 . . . . . 6 (x (A𝐶) ↔ (x A x 𝐶))
4 elin 3120 . . . . . 6 (y (B𝐷) ↔ (y B y 𝐷))
53, 4anbi12i 433 . . . . 5 ((x (A𝐶) y (B𝐷)) ↔ ((x A x 𝐶) (y B y 𝐷)))
62, 5bitr4i 176 . . . 4 (((x A y B) (x 𝐶 y 𝐷)) ↔ (x (A𝐶) y (B𝐷)))
76opabbii 3815 . . 3 {⟨x, y⟩ ∣ ((x A y B) (x 𝐶 y 𝐷))} = {⟨x, y⟩ ∣ (x (A𝐶) y (B𝐷))}
81, 7eqtri 2057 . 2 ({⟨x, y⟩ ∣ (x A y B)} ∩ {⟨x, y⟩ ∣ (x 𝐶 y 𝐷)}) = {⟨x, y⟩ ∣ (x (A𝐶) y (B𝐷))}
9 df-xp 4294 . . 3 (A × B) = {⟨x, y⟩ ∣ (x A y B)}
10 df-xp 4294 . . 3 (𝐶 × 𝐷) = {⟨x, y⟩ ∣ (x 𝐶 y 𝐷)}
119, 10ineq12i 3130 . 2 ((A × B) ∩ (𝐶 × 𝐷)) = ({⟨x, y⟩ ∣ (x A y B)} ∩ {⟨x, y⟩ ∣ (x 𝐶 y 𝐷)})
12 df-xp 4294 . 2 ((A𝐶) × (B𝐷)) = {⟨x, y⟩ ∣ (x (A𝐶) y (B𝐷))}
138, 11, 123eqtr4i 2067 1 ((A × B) ∩ (𝐶 × 𝐷)) = ((A𝐶) × (B𝐷))
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242   wcel 1390  cin 2910  {copab 3808   × cxp 4286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-xp 4294  df-rel 4295
This theorem is referenced by:  xpindi  4414  xpindir  4415  dmxpinm  4499  xpssres  4588  xpdisj1  4690  xpdisj2  4691  imainrect  4709  xpima1  4710  xpima2m  4711
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