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Theorem dmxpid 4808
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 4593 . . 3  |-  ( A  X.  A )  =  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  A ) }
21dmeqi 4788 . 2  |-  dom  ( A  X.  A )  =  dom  { <. y ,  x >.  |  (
y  e.  A  /\  x  e.  A ) }
3 elex2 2728 . . . 4  |-  ( y  e.  A  ->  E. x  x  e.  A )
43rgen 2510 . . 3  |-  A. y  e.  A  E. x  x  e.  A
5 dmopab3 4800 . . 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 2178 1  |-  dom  ( A  X.  A )  =  A
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1335   E.wex 1472    e. wcel 2128   A.wral 2435   {copab 4025    X. cxp 4585   dom cdm 4587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967  df-opab 4027  df-xp 4593  df-dm 4597
This theorem is referenced by:  dmxpin  4809  xpid11  4810  sqxpeq0  5010  xpider  6552  psmetdmdm  12766  xmetdmdm  12798
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