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Theorem dmxpinm 4583
Description: The domain of the intersection of two square cross products. Unlike dmin 4570, equality holds. (Contributed by NM, 29-Jan-2008.)
Assertion
Ref Expression
dmxpinm (∃𝑥 𝑥 ∈ (𝐴𝐵) → dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dmxpinm
StepHypRef Expression
1 inxp 4497 . . . 4 ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = ((𝐴𝐵) × (𝐴𝐵))
21dmeqi 4563 . . 3 dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = dom ((𝐴𝐵) × (𝐴𝐵))
32a1i 9 . 2 (∃𝑥 𝑥 ∈ (𝐴𝐵) → dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = dom ((𝐴𝐵) × (𝐴𝐵)))
4 dmxpm 4582 . 2 (∃𝑥 𝑥 ∈ (𝐴𝐵) → dom ((𝐴𝐵) × (𝐴𝐵)) = (𝐴𝐵))
53, 4eqtrd 2088 1 (∃𝑥 𝑥 ∈ (𝐴𝐵) → dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wex 1397  wcel 1409  cin 2943   × cxp 4370  dom cdm 4372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-dm 4382
This theorem is referenced by: (None)
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