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Theorem dmxpid 4832
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 4617 . . 3  |-  ( A  X.  A )  =  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  A ) }
21dmeqi 4812 . 2  |-  dom  ( A  X.  A )  =  dom  { <. y ,  x >.  |  (
y  e.  A  /\  x  e.  A ) }
3 elex2 2746 . . . 4  |-  ( y  e.  A  ->  E. x  x  e.  A )
43rgen 2523 . . 3  |-  A. y  e.  A  E. x  x  e.  A
5 dmopab3 4824 . . 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 2191 1  |-  dom  ( A  X.  A )  =  A
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1348   E.wex 1485    e. wcel 2141   A.wral 2448   {copab 4049    X. cxp 4609   dom cdm 4611
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-dm 4621
This theorem is referenced by:  dmxpin  4833  xpid11  4834  sqxpeq0  5034  xpider  6584  psmetdmdm  13118  xmetdmdm  13150
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