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Theorem dmeqd 4651
Description: Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
Hypothesis
Ref Expression
dmeqd.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
dmeqd  |-  ( ph  ->  dom  A  =  dom  B )

Proof of Theorem dmeqd
StepHypRef Expression
1 dmeqd.1 . 2  |-  ( ph  ->  A  =  B )
2 dmeq 4649 . 2  |-  ( A  =  B  ->  dom  A  =  dom  B )
31, 2syl 14 1  |-  ( ph  ->  dom  A  =  dom  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1290   dom cdm 4452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-dm 4462
This theorem is referenced by:  rneq  4675  dmsnsnsng  4921  elxp4  4931  fndmin  5420  1stvalg  5927  fo1st  5942  f1stres  5944  errn  6328  xpassen  6600  xpdom2  6601  frecuzrdgtclt  9889  shftdm  10317  isstruct2im  11565  isstruct2r  11566  setsvalg  11585
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