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Theorem inxpk 4278

Description: The intersection of two Kuratowski cross products. (Contributed by SF, 13-Jan-2015.)

**Assertion**
Ref	Expression
inxpk	⊢ ((A ×_k B) ∩ (C ×_k D)) = ((A ∩ C) ×_k (B ∩ D))

Proof of Theorem inxpk Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 3476	. . 3 ⊢ ((A ×_k B) ∩ (C ×_k D)) ⊆ (A ×_k B)
2		xpkssvvk 4211	. . 3 ⊢ (A ×_k B) ⊆ (V ×_k V)
3	1, 2	sstri 3282	. 2 ⊢ ((A ×_k B) ∩ (C ×_k D)) ⊆ (V ×_k V)
4		xpkssvvk 4211	. 2 ⊢ ((A ∩ C) ×_k (B ∩ D)) ⊆ (V ×_k V)
5		an4 797	. . 3 ⊢ (((x ∈ A ∧ y ∈ B) ∧ (x ∈ C ∧ y ∈ D)) ↔ ((x ∈ A ∧ x ∈ C) ∧ (y ∈ B ∧ y ∈ D)))
6		elin 3220	. . . 4 ⊢ (⟪x, y⟫ ∈ ((A ×_k B) ∩ (C ×_k D)) ↔ (⟪x, y⟫ ∈ (A ×_k B) ∧ ⟪x, y⟫ ∈ (C ×_k D)))
7		vex 2863	. . . . . 6 ⊢ x ∈ V
8		vex 2863	. . . . . 6 ⊢ y ∈ V
9	7, 8	opkelxpk 4249	. . . . 5 ⊢ (⟪x, y⟫ ∈ (A ×_k B) ↔ (x ∈ A ∧ y ∈ B))
10	7, 8	opkelxpk 4249	. . . . 5 ⊢ (⟪x, y⟫ ∈ (C ×_k D) ↔ (x ∈ C ∧ y ∈ D))
11	9, 10	anbi12i 678	. . . 4 ⊢ ((⟪x, y⟫ ∈ (A ×_k B) ∧ ⟪x, y⟫ ∈ (C ×_k D)) ↔ ((x ∈ A ∧ y ∈ B) ∧ (x ∈ C ∧ y ∈ D)))
12	6, 11	bitri 240	. . 3 ⊢ (⟪x, y⟫ ∈ ((A ×_k B) ∩ (C ×_k D)) ↔ ((x ∈ A ∧ y ∈ B) ∧ (x ∈ C ∧ y ∈ D)))
13	7, 8	opkelxpk 4249	. . . 4 ⊢ (⟪x, y⟫ ∈ ((A ∩ C) ×_k (B ∩ D)) ↔ (x ∈ (A ∩ C) ∧ y ∈ (B ∩ D)))
14		elin 3220	. . . . 5 ⊢ (x ∈ (A ∩ C) ↔ (x ∈ A ∧ x ∈ C))
15		elin 3220	. . . . 5 ⊢ (y ∈ (B ∩ D) ↔ (y ∈ B ∧ y ∈ D))
16	14, 15	anbi12i 678	. . . 4 ⊢ ((x ∈ (A ∩ C) ∧ y ∈ (B ∩ D)) ↔ ((x ∈ A ∧ x ∈ C) ∧ (y ∈ B ∧ y ∈ D)))
17	13, 16	bitri 240	. . 3 ⊢ (⟪x, y⟫ ∈ ((A ∩ C) ×_k (B ∩ D)) ↔ ((x ∈ A ∧ x ∈ C) ∧ (y ∈ B ∧ y ∈ D)))
18	5, 12, 17	3bitr4i 268	. 2 ⊢ (⟪x, y⟫ ∈ ((A ×_k B) ∩ (C ×_k D)) ↔ ⟪x, y⟫ ∈ ((A ∩ C) ×_k (B ∩ D)))
19	3, 4, 18	eqrelkriiv 4214	1 ⊢ ((A ×_k B) ∩ (C ×_k D)) = ((A ∩ C) ×_k (B ∩ D))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 358 = wceq 1642 ∈ wcel 1710 Vcvv 2860 ∩ cin 3209 ⟪copk 4058 ×_k cxpk 4175

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4079 ax-sn 4088

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2479 df-ne 2519 df-ral 2620 df-v 2862 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-ss 3260 df-nul 3552 df-sn 3742 df-pr 3743 df-opk 4059 df-xpk 4186

This theorem is referenced by: xpkexg 4289

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