Theorem dmxpinm 4670

Description: The domain of the intersection of two square cross products. Unlike dmin 4657, 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 4583	. . . 4 ⊢ ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵))
2	1	dmeqi 4650	. . 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 4669	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) → dom ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵)) = (𝐴 ∩ 𝐵))
5	3, 4	eqtrd 2121	1 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) → dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   = wceq 1290  ∃wex 1427   ∈ wcel 1439   ∩ cin 2999   × cxp 4450  dom cdm 4452

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4458  df-rel 4459  df-dm 4462

This theorem is referenced by: (None)