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Theorem dmxpid 5212
Description: The domain of a square Cartesian product. (Contributed by NM, 28-Jul-1995.)
Assertion
Ref Expression
dmxpid  |-  dom  ( A  X.  A )  =  A

Proof of Theorem dmxpid
StepHypRef Expression
1 dm0 5206 . . 3  |-  dom  (/)  =  (/)
2 xpeq1 5003 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  A )  =  ( (/)  X.  A
) )
3 0xp 5070 . . . . 5  |-  ( (/)  X.  A )  =  (/)
42, 3syl6eq 2500 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  A )  =  (/) )
54dmeqd 5195 . . 3  |-  ( A  =  (/)  ->  dom  ( A  X.  A )  =  dom  (/) )
6 id 22 . . 3  |-  ( A  =  (/)  ->  A  =  (/) )
71, 5, 63eqtr4a 2510 . 2  |-  ( A  =  (/)  ->  dom  ( A  X.  A )  =  A )
8 dmxp 5211 . 2  |-  ( A  =/=  (/)  ->  dom  ( A  X.  A )  =  A )
97, 8pm2.61ine 2756 1  |-  dom  ( A  X.  A )  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383   (/)c0 3770    X. cxp 4987   dom cdm 4989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-dm 4999
This theorem is referenced by:  dmxpin  5213  xpid11  5214  sofld  5445  xpider  7384  hartogslem1  7970  unxpwdom2  8017  infxpenlem  8394  fpwwe2lem13  9023  fpwwe2  9024  canth4  9028  dmrecnq  9349  homfeqbas  15073  sscfn1  15168  sscfn2  15169  ssclem  15170  isssc  15171  rescval2  15179  issubc2  15187  cofuval  15230  resfval2  15241  resf1st  15242  psssdm2  15824  tsrss  15832  decpmatval  19244  pmatcollpw3lem  19262  ustssco  20695  ustbas2  20706  psmetdmdm  20787  xmetdmdm  20816  setsmstopn  20959  tmsval  20962  tngtopn  21142  caufval  21692  grporndm  25190  isabloda  25279  ismndo2  25325  vcoprne  25450  dfhnorm2  26017  hhshsslem1  26161  metideq  27850  filnetlem4  30175  ssbnd  30260  bnd2lem  30263  ismtyval  30272  exidreslem  30315  divrngcl  30336  isdrngo2  30337  fnxpdmdm  32410
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