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Theorem dmxpinm 4502
Description: The domain of the intersection of two square cross products. Unlike dmin 4489, 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 4416 . . . 4 ((A × A) ∩ (B × B)) = ((AB) × (AB))
21dmeqi 4482 . . 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 4501 . 2 (x x (AB) → dom ((AB) × (AB)) = (AB))
53, 4eqtrd 2072 1 (x x (AB) → dom ((A × A) ∩ (B × B)) = (AB))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wex 1381   wcel 1393  cin 2913   × cxp 4289  dom cdm 4291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3869  ax-pow 3921  ax-pr 3938
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-pw 3356  df-sn 3376  df-pr 3377  df-op 3379  df-br 3759  df-opab 3813  df-xp 4297  df-rel 4298  df-dm 4301
This theorem is referenced by: (None)
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