Theorem inxp 2496

Description: The intersection of two cross products. Exercise 9 of [TakeutiZaring] p. 25.

Assertion

Ref	Expression
inxp	⊢ ((A × B) ∩ (C × D)) = ((A ∩ C) × (B ∩ D))

Proof of Theorem inxp

Step	Hyp	Ref	Expression
1		relxp 2486	. . 3 ⊢ Rel (A × B)
2		relin 2491	. . 3 ⊢ (Rel (A × B) → Rel ((A × B) ∩ (C × D)))
3	1, 2	ax-mp 6	. 2 ⊢ Rel ((A × B) ∩ (C × D))
4		relxp 2486	. 2 ⊢ Rel ((A ∩ C) × (B ∩ D))
5		an4 388	. . . 4 ⊢ (((x ∈ A ∧ y ∈ B) ∧ (x ∈ C ∧ y ∈ D)) ↔ ((x ∈ A ∧ x ∈ C) ∧ (y ∈ B ∧ y ∈ D)))
6		visset 1350	. . . . . 6 ⊢ y ∈ V
7	6	opelxp 2452	. . . . 5 ⊢ (⟨x, y⟩ ∈ (A × B) ↔ (x ∈ A ∧ y ∈ B))
8	6	opelxp 2452	. . . . 5 ⊢ (⟨x, y⟩ ∈ (C × D) ↔ (x ∈ C ∧ y ∈ D))
9	7, 8	anbi12i 369	. . . 4 ⊢ ((⟨x, y⟩ ∈ (A × B) ∧ ⟨x, y⟩ ∈ (C × D)) ↔ ((x ∈ A ∧ y ∈ B) ∧ (x ∈ C ∧ y ∈ D)))
10		elin 1635	. . . . 5 ⊢ (x ∈ (A ∩ C) ↔ (x ∈ A ∧ x ∈ C))
11		elin 1635	. . . . 5 ⊢ (y ∈ (B ∩ D) ↔ (y ∈ B ∧ y ∈ D))
12	10, 11	anbi12i 369	. . . 4 ⊢ ((x ∈ (A ∩ C) ∧ y ∈ (B ∩ D)) ↔ ((x ∈ A ∧ x ∈ C) ∧ (y ∈ B ∧ y ∈ D)))
13	5, 9, 12	3bitr4 158	. . 3 ⊢ ((⟨x, y⟩ ∈ (A × B) ∧ ⟨x, y⟩ ∈ (C × D)) ↔ (x ∈ (A ∩ C) ∧ y ∈ (B ∩ D)))
14		elin 1635	. . 3 ⊢ (⟨x, y⟩ ∈ ((A × B) ∩ (C × D)) ↔ (⟨x, y⟩ ∈ (A × B) ∧ ⟨x, y⟩ ∈ (C × D)))
15	6	opelxp 2452	. . 3 ⊢ (⟨x, y⟩ ∈ ((A ∩ C) × (B ∩ D)) ↔ (x ∈ (A ∩ C) ∧ y ∈ (B ∩ D)))
16	13, 14, 15	3bitr4 158	. 2 ⊢ (⟨x, y⟩ ∈ ((A × B) ∩ (C × D)) ↔ ⟨x, y⟩ ∈ ((A ∩ C) × (B ∩ D)))
17	3, 4, 16	cleqreli 2484	1 ⊢ ((A × B) ∩ (C × D)) = ((A ∩ C) × (B ∩ D))

Syntax hints:   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ∩ cin 1486  ⟨cop 1810   × cxp 2408  Rel wrel 2415

This theorem is referenced by:  xpindi 2497  xpindir 2498  resabs1 2592  xpdisj1 2653  xpdisj2 2654

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-rel 2425

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