Theorem dmxpinm 4574

Description: The domain of the intersection of two square cross products. Unlike dmin 4561, equality holds. (Contributed by NM, 29-Jan-2008.)

Assertion

Ref	Expression
dmxpinm	⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) → dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴 ∩ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dmxpinm

Step	Hyp	Ref	Expression
1		inxp 4488	. . . 4 ⊢ ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵))
2	1	dmeqi 4554	. . 3 ⊢ dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = dom ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵))
3	2	a1i 9	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) → dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = dom ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵)))
4		dmxpm 4573	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) → dom ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵)) = (𝐴 ∩ 𝐵))
5	3, 4	eqtrd 2113	1 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) → dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   = wceq 1284  ∃wex 1421   ∈ wcel 1433   ∩ cin 2972   × cxp 4361  dom cdm 4363

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-dm 4373

This theorem is referenced by: (None)