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Theorem dmxpid 5222
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 5216 . . 3  |-  dom  (/)  =  (/)
2 xpeq1 5013 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  A )  =  ( (/)  X.  A
) )
3 0xp 5080 . . . . 5  |-  ( (/)  X.  A )  =  (/)
42, 3syl6eq 2524 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  A )  =  (/) )
54dmeqd 5205 . . 3  |-  ( A  =  (/)  ->  dom  ( A  X.  A )  =  dom  (/) )
6 id 22 . . 3  |-  ( A  =  (/)  ->  A  =  (/) )
71, 5, 63eqtr4a 2534 . 2  |-  ( A  =  (/)  ->  dom  ( A  X.  A )  =  A )
8 dmxp 5221 . 2  |-  ( A  =/=  (/)  ->  dom  ( A  X.  A )  =  A )
97, 8pm2.61ine 2780 1  |-  dom  ( A  X.  A )  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   (/)c0 3785    X. cxp 4997   dom cdm 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-dm 5009
This theorem is referenced by:  dmxpin  5223  xpid11  5224  sofld  5455  xpider  7382  hartogslem1  7967  unxpwdom2  8014  infxpenlem  8391  fpwwe2lem13  9020  fpwwe2  9021  canth4  9025  dmrecnq  9346  homfeqbas  14952  sscfn1  15047  sscfn2  15048  ssclem  15049  isssc  15050  rescval2  15058  issubc2  15066  cofuval  15109  resfval2  15120  resf1st  15121  psssdm2  15702  tsrss  15710  decpmatval  19061  pmatcollpw3lem  19079  ustssco  20480  ustbas2  20491  psmetdmdm  20572  xmetdmdm  20601  setsmstopn  20744  tmsval  20747  tngtopn  20927  caufval  21477  grporndm  24916  isabloda  25005  ismndo2  25051  vcoprne  25176  dfhnorm2  25743  hhshsslem1  25887  metideq  27536  filnetlem4  29830  ssbnd  29915  bnd2lem  29918  ismtyval  29927  exidreslem  29970  divrngcl  29991  isdrngo2  29992  fnxpdmdm  31959
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