Theorem intxp 48810

Description: Intersection of Cartesian products is the Cartesian product of intersection of domains and ranges. See also inxp 5797 and iinxp 48809. (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 5025	. . . 4 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
2		intxp.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = (dom 𝑥 × ran 𝑥))
3	2	iineq2dv 4983	. . . 4 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝑥 = ∩ 𝑥 ∈ 𝐴 (dom 𝑥 × ran 𝑥))
4	1, 3	eqtrid 2777	. . 3 ⊢ (𝜑 → ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 (dom 𝑥 × ran 𝑥))
5		intxp.1	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
6		iinxp 48809	. . . 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 2765	. 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 5668	. 2 ⊢ (𝑋 × 𝑌) = (∩ 𝑥 ∈ 𝐴 dom 𝑥 × ∩ 𝑥 ∈ 𝐴 ran 𝑥)
12	8, 11	eqtr4di 2783	1 ⊢ (𝜑 → ∩ 𝐴 = (𝑋 × 𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∅c0 4298  ∩ cint 4912  ∩ ciin 4958   × cxp 5638  dom cdm 5640  ran crn 5641

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 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

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 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-int 4913  df-iin 4960  df-opab 5172  df-xp 5646  df-rel 5647

This theorem is referenced by: (None)