Theorem intxp 48871

Description: Intersection of Cartesian products is the Cartesian product of intersection of domains and ranges. See also inxp 5770 and iinxp 48870. (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 5006	. . . 4 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
2		intxp.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = (dom 𝑥 × ran 𝑥))
3	2	iineq2dv 4965	. . . 4 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝑥 = ∩ 𝑥 ∈ 𝐴 (dom 𝑥 × ran 𝑥))
4	1, 3	eqtrid 2778	. . 3 ⊢ (𝜑 → ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 (dom 𝑥 × ran 𝑥))
5		intxp.1	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
6		iinxp 48870	. . . 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 2766	. 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 5642	. 2 ⊢ (𝑋 × 𝑌) = (∩ 𝑥 ∈ 𝐴 dom 𝑥 × ∩ 𝑥 ∈ 𝐴 ran 𝑥)
12	8, 11	eqtr4di 2784	1 ⊢ (𝜑 → ∩ 𝐴 = (𝑋 × 𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∅c0 4280  ∩ cint 4895  ∩ ciin 4940   × cxp 5612  dom cdm 5614  ran crn 5615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-int 4896  df-iin 4942  df-opab 5152  df-xp 5620  df-rel 5621

This theorem is referenced by: (None)