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Theorem dmxpin 4721
 Description: The domain of the intersection of two square Cartesian products. Unlike dmin 4707, 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 4633 . . 3 ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = ((𝐴𝐵) × (𝐴𝐵))
21dmeqi 4700 . 2 dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = dom ((𝐴𝐵) × (𝐴𝐵))
3 dmxpid 4720 . 2 dom ((𝐴𝐵) × (𝐴𝐵)) = (𝐴𝐵)
42, 3eqtri 2135 1 dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   = wceq 1314   ∩ cin 3036   × cxp 4497  dom cdm 4499 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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-xp 4505  df-rel 4506  df-dm 4509 This theorem is referenced by: (None)
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