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Theorem dmxpid 4718
Description: The domain of a square Cartesian product. (Contributed by NM, 28-Jul-1995.) (Revised by Jim Kingdon, 11-Apr-2023.)
Assertion
Ref Expression
dmxpid  |-  dom  ( A  X.  A )  =  A

Proof of Theorem dmxpid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xp 4503 . . 3  |-  ( A  X.  A )  =  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  A ) }
21dmeqi 4698 . 2  |-  dom  ( A  X.  A )  =  dom  { <. y ,  x >.  |  (
y  e.  A  /\  x  e.  A ) }
3 elex2 2671 . . . 4  |-  ( y  e.  A  ->  E. x  x  e.  A )
43rgen 2457 . . 3  |-  A. y  e.  A  E. x  x  e.  A
5 dmopab3 4710 . . 3  |-  ( A. y  e.  A  E. x  x  e.  A  <->  dom 
{ <. y ,  x >.  |  ( y  e.  A  /\  x  e.  A ) }  =  A )
64, 5mpbi 144 . 2  |-  dom  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  A
) }  =  A
72, 6eqtri 2133 1  |-  dom  ( A  X.  A )  =  A
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1312   E.wex 1449    e. wcel 1461   A.wral 2388   {copab 3946    X. cxp 4495   dom cdm 4497
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 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-xp 4503  df-dm 4507
This theorem is referenced by:  dmxpin  4719  xpid11  4720  sqxpeq0  4918  xpider  6452  psmetdmdm  12307  xmetdmdm  12339
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