Theorem bj-inxpALT 38169

Description: Alternate proof of inxp 5817 not depending on df-rel 5663. The main proof inxp 5817 is shorter because it benefits from the efficient machinery developed for relations: one shows that a class is a relation using theorems like relxp 5674, relinxp 5799, relin1 5797, relres 6010, and then uses the generic properties of relations, in particular ordered-pair extensionality eqrelriiv 5773. This illustrates the fact that the definitions of Cartesian product df-xp 5662, restriction df-res 5668, and relation df-rel 5663 are closely intertwined. Note however that one could use a variant of eqrelriiv 5773 not using the Rel predicate, as in bj-ssrel 38168. (Contributed by BJ, 24-Oct-2026.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-inxpALT	⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷))

Proof of Theorem bj-inxpALT

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2852	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) ↔ ⟨𝑥, 𝑦⟩ ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷))))
2		elin 4096	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷)))
3	1, 2	bitrdi 290	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷))))
4		an4 669	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
5		elin 4096	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶))
6		elin 4096	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐷) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷))
7	5, 6	anbi12i 640	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)))
8		opelxp 5692	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
9		opelxp 5692	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))
10	8, 9	anbi12i 640	. . . . . 6 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
11	4, 7, 10	3bitr4ri 307	. . . . 5 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 × 𝐷)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷)))
12	3, 11	bitrdi 290	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷))))
13	12	copsex2gb 5791	. . 3 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷))) ↔ (𝑧 ∈ (V × V) ∧ 𝑧 ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷))))
14		elxp 5679	. . 3 ⊢ (𝑧 ∈ ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷)) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷))))
15		xpss 5672	. . . . 5 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
16		elinel1 4101	. . . . 5 ⊢ (𝑧 ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) → 𝑧 ∈ (𝐴 × 𝐵))
17	15, 16	sselid 3911	. . . 4 ⊢ (𝑧 ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) → 𝑧 ∈ (V × V))
18	17	pm4.71ri 570	. . 3 ⊢ (𝑧 ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) ↔ (𝑧 ∈ (V × V) ∧ 𝑧 ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷))))
19	13, 14, 18	3bitr4ri 307	. 2 ⊢ (𝑧 ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) ↔ 𝑧 ∈ ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷)))
20	19	eqriv 2760	1 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷))

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454   ∩ cin 4092  ⟨cop 4838   × cxp 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5172  df-xp 5662

This theorem is used by: (None)