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Theorem dmxpid 5074
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 5068 . . 3  |-  dom  (/)  =  (/)
2 xpeq1 4868 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  A )  =  ( (/)  X.  A
) )
3 0xp 4935 . . . . 5  |-  ( (/)  X.  A )  =  (/)
42, 3syl6eq 2486 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  A )  =  (/) )
54dmeqd 5057 . . 3  |-  ( A  =  (/)  ->  dom  ( A  X.  A )  =  dom  (/) )
6 id 23 . . 3  |-  ( A  =  (/)  ->  A  =  (/) )
71, 5, 63eqtr4a 2496 . 2  |-  ( A  =  (/)  ->  dom  ( A  X.  A )  =  A )
8 dmxp 5073 . 2  |-  ( A  =/=  (/)  ->  dom  ( A  X.  A )  =  A )
97, 8pm2.61ine 2744 1  |-  dom  ( A  X.  A )  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   (/)c0 3767    X. cxp 4852   dom cdm 4854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-dm 4864
This theorem is referenced by:  dmxpin  5075  xpid11  5076  sofld  5304  xpider  7442  hartogslem1  8057  unxpwdom2  8103  infxpenlem  8443  fpwwe2lem13  9066  fpwwe2  9067  canth4  9071  dmrecnq  9392  homfeqbas  15552  sscfn1  15673  sscfn2  15674  ssclem  15675  isssc  15676  rescval2  15684  issubc2  15692  cofuval  15738  resfval2  15749  resf1st  15750  psssdm2  16412  tsrss  16420  decpmatval  19720  pmatcollpw3lem  19738  ustssco  21160  ustbas2  21171  psmetdmdm  21252  xmetdmdm  21281  setsmstopn  21424  tmsval  21427  tngtopn  21589  caufval  22138  grporndm  25783  isabloda  25872  ismndo2  25918  vcoprne  26043  dfhnorm2  26610  hhshsslem1  26753  metideq  28535  filnetlem4  30822  poimirlem3  31646  ssbnd  31823  bnd2lem  31826  ismtyval  31835  exidreslem  31878  divrngcl  31899  isdrngo2  31900  fnxpdmdm  38525
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