Theorem inxp 5809

Description: Intersection of two Cartesian products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-10 2178, ax-12 2215. (Revised by SN, 5-May-2025.)

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		relinxp 5792	. 2 ⊢ Rel ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷))
2		relxp 5670	. 2 ⊢ Rel ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷))
3		an4 668	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)))
4		opelxp 5688	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
5		opelxp 5688	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))
6	4, 5	anbi12i 639	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
7		elin 3923	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶))
8		elin 3923	. . . . 5 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐷) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷))
9	7, 8	anbi12i 639	. . . 4 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)))
10	3, 6, 9	3bitr4i 306	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷)))
11		elin 3923	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷)))
12		opelxp 5688	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷)))
13	10, 11, 12	3bitr4i 306	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) ↔ ⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷)))
14	1, 2, 13	eqrelriiv 5767	1 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷))

Syntax hints:   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ∩ cin 3906  ⟨cop 4591   × cxp 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-opab 5168  df-xp 5658  df-rel 5659

This theorem is referenced by:  xpindi  5810  xpindir  5811  dmxpin  5912  xpssres  6008  xpdisj1  6150  xpdisj2  6151  imainrect  6171  xpima  6172  cnvrescnv  6186  curry1  8087  curry2  8090  fpar  8099  marypha1lem  9381  fpwwe2lem12  10615  hashxplem  14460  sscres  17870  gsumxp  20037  pjfval  21816  pjpm  21818  txbas  23685  txcls  23722  txrest  23749  trust  24347  ressuss  24380  trcfilu  24411  metreslem  24480  ressxms  24643  ressms  24644  mbfmcst  34566  0rrv  34758  poimirlem26  38157