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Theorem dmxpid 4914
Description: The domain of a square cross product. (Contributed by NM, 28-Jul-1995.)
Assertion
Ref Expression
dmxpid  |-  dom  ( A  X.  A )  =  A

Proof of Theorem dmxpid
StepHypRef Expression
1 dm0 4908 . . 3  |-  dom  (/)  =  (/)
2 xpeq1 4719 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  A )  =  ( (/)  X.  A
) )
3 xp0r 4784 . . . . 5  |-  ( (/)  X.  A )  =  (/)
42, 3syl6eq 2344 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  A )  =  (/) )
54dmeqd 4897 . . 3  |-  ( A  =  (/)  ->  dom  ( A  X.  A )  =  dom  (/) )
6 id 19 . . 3  |-  ( A  =  (/)  ->  A  =  (/) )
71, 5, 63eqtr4a 2354 . 2  |-  ( A  =  (/)  ->  dom  ( A  X.  A )  =  A )
8 dmxp 4913 . 2  |-  ( A  =/=  (/)  ->  dom  ( A  X.  A )  =  A )
97, 8pm2.61ine 2535 1  |-  dom  ( A  X.  A )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1632   (/)c0 3468    X. cxp 4703   dom cdm 4705
This theorem is referenced by:  dmxpin  4915  xpid11  4916  sofld  5137  xpider  6746  hartogslem1  7273  unxpwdom2  7318  infxpenlem  7657  fpwwe2lem13  8280  fpwwe2  8281  canth4  8285  dmrecnq  8608  homfeqbas  13615  sscfn1  13710  sscfn2  13711  ssclem  13712  isssc  13713  rescval2  13721  issubc2  13729  cofuval  13772  resfval2  13783  resf1st  13784  psssdm2  14340  tsrss  14348  xmetdmdm  17916  setsmstopn  18040  tmsval  18043  tngtopn  18182  caufval  18717  grporndm  20893  isabloda  20982  ismndo2  21028  vcoprne  21151  dfhnorm2  21717  hhshsslem1  21860  nZdef  25283  islatalg  25286  rngodmeqrn  25522  zintdom  25541  svs2  25590  reldded  25844  reldcat  25865  filnetlem4  26433  ssbnd  26615  bnd2lem  26618  ismtyval  26627  exidreslem  26670  divrngcl  26691  isdrngo2  26692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-dm 4715
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