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Theorem dmeq 4786
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 4785 . . 3  |-  ( A 
C_  B  ->  dom  A 
C_  dom  B )
2 dmss 4785 . . 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 3143 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 eqss 3143 . 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 1335    C_ wss 3102   dom cdm 4586
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-dm 4596
This theorem is referenced by:  dmeqi  4787  dmeqd  4788  xpid11  4809  sqxpeq0  5009  fneq1  5258  eqfnfv2  5566  offval  6039  ofrfval  6040  offval3  6082  smoeq  6237  tfrlemi14d  6280  tfr1onlemres  6296  tfrcllemres  6309  rdgivallem  6328  rdgon  6333  rdg0  6334  frec0g  6344  freccllem  6349  frecfcllem  6351  frecsuclem  6353  frecsuc  6354  ereq1  6487  fundmeng  6752  acfun  7142  ccfunen  7184  ennnfonelemj0  12141  ennnfonelemg  12143  ennnfonelemp1  12146  ennnfonelemom  12148  ennnfonelemnn0  12162  blfvalps  12796  reldvg  13059
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