Theorem inxp 5780

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 2147, ax-12 2185. (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 5763	. 2 ⊢ Rel ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷))
2		relxp 5642	. 2 ⊢ Rel ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷))
3		an4 657	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)))
4		opelxp 5660	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
5		opelxp 5660	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))
6	4, 5	anbi12i 629	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
7		elin 3906	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶))
8		elin 3906	. . . . 5 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐷) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷))
9	7, 8	anbi12i 629	. . . 4 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)))
10	3, 6, 9	3bitr4i 303	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷)))
11		elin 3906	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷)))
12		opelxp 5660	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷)))
13	10, 11, 12	3bitr4i 303	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) ↔ ⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷)))
14	1, 2, 13	eqrelriiv 5739	1 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷))

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∩ cin 3889  ⟨cop 4574   × cxp 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-opab 5149  df-xp 5630  df-rel 5631

This theorem is referenced by:  xpindi  5782  xpindir  5783  dmxpin  5880  xpssres  5977  xpdisj1  6119  xpdisj2  6120  imainrect  6139  xpima  6140  cnvrescnv  6153  curry1  8047  curry2  8050  fpar  8059  marypha1lem  9339  fpwwe2lem12  10556  hashxplem  14386  sscres  17781  gsumxp  19942  pjfval  21696  pjpm  21698  txbas  23542  txcls  23579  txrest  23606  trust  24204  ressuss  24237  trcfilu  24268  metreslem  24337  ressxms  24500  ressms  24501  mbfmcst  34419  0rrv  34611  poimirlem26  37981