Theorem intxp 49667

Description: Intersection of Cartesian products is the Cartesian product of intersection of domains and ranges. See also inxp 5820 and iinxp 49666. (Contributed by Zhi Wang, 30-Oct-2025.)

Hypotheses

Ref	Expression
intxp.1	⊢ (𝜑 → 𝐴 ≠ ∅)
intxp.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = (dom 𝑥 × ran 𝑥))
intxp.3	⊢ 𝑋 = ∩ 𝑥 ∈ 𝐴 dom 𝑥
intxp.4	⊢ 𝑌 = ∩ 𝑥 ∈ 𝐴 ran 𝑥

Assertion

Ref	Expression
intxp	⊢ (𝜑 → ∩ 𝐴 = (𝑋 × 𝑌))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Allowed substitution hints:   𝑋(𝑥)   𝑌(𝑥)

Proof of Theorem intxp

Step	Hyp	Ref	Expression
1		intiin 5026	. . . 4 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
2		intxp.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = (dom 𝑥 × ran 𝑥))
3	2	iineq2dv 4984	. . . 4 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝑥 = ∩ 𝑥 ∈ 𝐴 (dom 𝑥 × ran 𝑥))
4	1, 3	eqtrid 2812	. . 3 ⊢ (𝜑 → ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 (dom 𝑥 × ran 𝑥))
5		intxp.1	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
6		iinxp 49666	. . . 4 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (dom 𝑥 × ran 𝑥) = (∩ 𝑥 ∈ 𝐴 dom 𝑥 × ∩ 𝑥 ∈ 𝐴 ran 𝑥))
7	5, 6	syl 18	. . 3 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 (dom 𝑥 × ran 𝑥) = (∩ 𝑥 ∈ 𝐴 dom 𝑥 × ∩ 𝑥 ∈ 𝐴 ran 𝑥))
8	4, 7	eqtrd 2800	. 2 ⊢ (𝜑 → ∩ 𝐴 = (∩ 𝑥 ∈ 𝐴 dom 𝑥 × ∩ 𝑥 ∈ 𝐴 ran 𝑥))
9		intxp.3	. . 3 ⊢ 𝑋 = ∩ 𝑥 ∈ 𝐴 dom 𝑥
10		intxp.4	. . 3 ⊢ 𝑌 = ∩ 𝑥 ∈ 𝐴 ran 𝑥
11	9, 10	xpeq12i 5691	. 2 ⊢ (𝑋 × 𝑌) = (∩ 𝑥 ∈ 𝐴 dom 𝑥 × ∩ 𝑥 ∈ 𝐴 ran 𝑥)
12	8, 11	eqtr4di 2818	1 ⊢ (𝜑 → ∩ 𝐴 = (𝑋 × 𝑌))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∅c0 4286  ∩ cint 4914  ∩ ciin 4959   × cxp 5661  dom cdm 5663  ran crn 5664

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

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 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-int 4915  df-iin 4961  df-opab 5176  df-xp 5669  df-rel 5670

This theorem is used by: (None)