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Theorem dmxpinm 4499
 Description: The domain of the intersection of two square cross products. Unlike dmin 4486, 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 4413 . . . 4 ((A × A) ∩ (B × B)) = ((AB) × (AB))
21dmeqi 4479 . . 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 4498 . 2 (x x (AB) → dom ((AB) × (AB)) = (AB))
53, 4eqtrd 2069 1 (x x (AB) → dom ((A × A) ∩ (B × B)) = (AB))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242  ∃wex 1378   ∈ wcel 1390   ∩ cin 2910   × cxp 4286  dom cdm 4288 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-dm 4298 This theorem is referenced by: (None)
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