Theorem inxp 4912

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)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷))

Proof of Theorem inxp

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inopab 4910	. . 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 592	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)))
3		elin 3412	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶))
4		elin 3412	. . . . . 6 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐷) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷))
5	3, 4	anbi12i 464	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)))
6	2, 5	bitr4i 187	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷)))
7	6	opabbii 4196	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷))}
8	1, 7	eqtri 2259	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷))}
9		df-xp 4778	. . 3 ⊢ (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}
10		df-xp 4778	. . 3 ⊢ (𝐶 × 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}
11	9, 10	ineq12i 3430	. 2 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)})
12		df-xp 4778	. 2 ⊢ ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷))}
13	8, 11, 12	3eqtr4i 2269	1 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷))

Syntax hints:   ∧ wa 104   = wceq 1402   ∈ wcel 2209   ∩ cin 3219  {copab 4189   × cxp 4770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-opab 4191  df-xp 4778  df-rel 4779

This theorem is referenced by:  xpindi  4913  xpindir  4914  dmxpin  5002  xpssres  5096  xpdisj1  5210  xpdisj2  5211  imainrect  5231  xpima1  5232  xpima2m  5233  hashxp  11250  txbas  15342  txrest  15360  metreslem  15464