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Theorem dmeq 4562
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 4561 . . 3  |-  ( A 
C_  B  ->  dom  A 
C_  dom  B )
2 dmss 4561 . . 3  |-  ( B 
C_  A  ->  dom  B 
C_  dom  A )
31, 2anim12i 325 . 2  |-  ( ( A  C_  B  /\  B  C_  A )  -> 
( dom  A  C_  dom  B  /\  dom  B  C_  dom  A ) )
4 eqss 2987 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 eqss 2987 . 2  |-  ( dom 
A  =  dom  B  <->  ( dom  A  C_  dom  B  /\  dom  B  C_  dom  A ) )
63, 4, 53imtr4i 194 1  |-  ( A  =  B  ->  dom  A  =  dom  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259    C_ wss 2944   dom cdm 4372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-dm 4382
This theorem is referenced by:  dmeqi  4563  dmeqd  4564  xpid11m  4584  fneq1  5014  eqfnfv2  5293  offval  5746  ofrfval  5747  offval3  5788  smoeq  5935  tfrlemi14d  5977  rdgivallem  5998  rdg0  6004  frec0g  6013  frecsuclem3  6020  frecsuc  6021  ereq1  6143  fundmeng  6317
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