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Theorem dmxpin 4842
Description: The domain of the intersection of two square Cartesian products. Unlike dmin 4828, equality holds. (Contributed by NM, 29-Jan-2008.)
Assertion
Ref Expression
dmxpin dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴𝐵)

Proof of Theorem dmxpin
StepHypRef Expression
1 inxp 4754 . . 3 ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = ((𝐴𝐵) × (𝐴𝐵))
21dmeqi 4821 . 2 dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = dom ((𝐴𝐵) × (𝐴𝐵))
3 dmxpid 4841 . 2 dom ((𝐴𝐵) × (𝐴𝐵)) = (𝐴𝐵)
42, 3eqtri 2196 1 dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1353  cin 3126   × cxp 4618  dom cdm 4620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-dm 4630
This theorem is referenced by: (None)
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