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Theorem dmeq 4739
Description: Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
Assertion
Ref Expression
dmeq  |-  ( A  =  B  ->  dom  A  =  dom  B )

Proof of Theorem dmeq
StepHypRef Expression
1 dmss 4738 . . 3  |-  ( A 
C_  B  ->  dom  A 
C_  dom  B )
2 dmss 4738 . . 3  |-  ( B 
C_  A  ->  dom  B 
C_  dom  A )
31, 2anim12i 336 . 2  |-  ( ( A  C_  B  /\  B  C_  A )  -> 
( dom  A  C_  dom  B  /\  dom  B  C_  dom  A ) )
4 eqss 3112 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 eqss 3112 . 2  |-  ( dom 
A  =  dom  B  <->  ( dom  A  C_  dom  B  /\  dom  B  C_  dom  A ) )
63, 4, 53imtr4i 200 1  |-  ( A  =  B  ->  dom  A  =  dom  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    C_ wss 3071   dom cdm 4539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-dm 4549
This theorem is referenced by:  dmeqi  4740  dmeqd  4741  xpid11  4762  sqxpeq0  4962  fneq1  5211  eqfnfv2  5519  offval  5989  ofrfval  5990  offval3  6032  smoeq  6187  tfrlemi14d  6230  tfr1onlemres  6246  tfrcllemres  6259  rdgivallem  6278  rdgon  6283  rdg0  6284  frec0g  6294  freccllem  6299  frecfcllem  6301  frecsuclem  6303  frecsuc  6304  ereq1  6436  fundmeng  6701  acfun  7063  ccfunen  7079  ennnfonelemj0  11914  ennnfonelemg  11916  ennnfonelemp1  11919  ennnfonelemom  11921  ennnfonelemnn0  11935  blfvalps  12554  reldvg  12817
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