Theorem inxp 5808

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 2177, ax-12 2212. (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 5791	. 2 ⊢ Rel ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷))
2		relxp 5668	. 2 ⊢ Rel ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷))
3		an4 669	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)))
4		opelxp 5686	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
5		opelxp 5686	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))
6	4, 5	anbi12i 640	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
7		elin 4093	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶))
8		elin 4093	. . . . 5 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐷) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷))
9	7, 8	anbi12i 640	. . . 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 4093	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷)))
12		opelxp 5686	. . 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 5765	1 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷))

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ∩ cin 4089  ⟨cop 4834   × cxp 5648

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-opab 5167  df-xp 5656  df-rel 5657

This theorem is used by:  xpindi  5809  xpindir  5810  dmxpin  5912  xpssres  6006  xpdisj1  6151  xpdisj2  6152  imainrect  6172  xpima  6173  cnvrescnv  6187  curry1  8117  curry2  8120  fpar  8129  marypha1lem  9428  fpwwe2lem12  10730  hashxplem  14581  sscres  18003  gsumxp  20195  pjfval  22017  pjpm  22019  txbas  23891  txcls  23928  txrest  23955  trust  24553  ressuss  24586  trcfilu  24617  metreslem  24686  ressxms  24849  ressms  24850  mbfmcst  34937  0rrv  35129  poimirlem26  38657