Theorem intxp 48793

Description: Intersection of Cartesian products is the Cartesian product of intersection of domains and ranges. See also inxp 5785 and iinxp 48792. (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 5018	. . . 4 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
2		intxp.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = (dom 𝑥 × ran 𝑥))
3	2	iineq2dv 4977	. . . 4 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝑥 = ∩ 𝑥 ∈ 𝐴 (dom 𝑥 × ran 𝑥))
4	1, 3	eqtrid 2776	. . 3 ⊢ (𝜑 → ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 (dom 𝑥 × ran 𝑥))
5		intxp.1	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
6		iinxp 48792	. . . 4 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (dom 𝑥 × ran 𝑥) = (∩ 𝑥 ∈ 𝐴 dom 𝑥 × ∩ 𝑥 ∈ 𝐴 ran 𝑥))
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 (dom 𝑥 × ran 𝑥) = (∩ 𝑥 ∈ 𝐴 dom 𝑥 × ∩ 𝑥 ∈ 𝐴 ran 𝑥))
8	4, 7	eqtrd 2764	. 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 5659	. 2 ⊢ (𝑋 × 𝑌) = (∩ 𝑥 ∈ 𝐴 dom 𝑥 × ∩ 𝑥 ∈ 𝐴 ran 𝑥)
12	8, 11	eqtr4di 2782	1 ⊢ (𝜑 → ∩ 𝐴 = (𝑋 × 𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∅c0 4292  ∩ cint 4906  ∩ ciin 4952   × cxp 5629  dom cdm 5631  ran crn 5632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-int 4907  df-iin 4954  df-opab 5165  df-xp 5637  df-rel 5638

This theorem is referenced by: (None)