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Theorem dmeq 4597
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 4596 . . 3  |-  ( A 
C_  B  ->  dom  A 
C_  dom  B )
2 dmss 4596 . . 3  |-  ( B 
C_  A  ->  dom  B 
C_  dom  A )
31, 2anim12i 331 . 2  |-  ( ( A  C_  B  /\  B  C_  A )  -> 
( dom  A  C_  dom  B  /\  dom  B  C_  dom  A ) )
4 eqss 3027 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 eqss 3027 . 2  |-  ( dom 
A  =  dom  B  <->  ( dom  A  C_  dom  B  /\  dom  B  C_  dom  A ) )
63, 4, 53imtr4i 199 1  |-  ( A  =  B  ->  dom  A  =  dom  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1287    C_ wss 2986   dom cdm 4404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-un 2990  df-in 2992  df-ss 2999  df-sn 3431  df-pr 3432  df-op 3434  df-br 3815  df-dm 4414
This theorem is referenced by:  dmeqi  4598  dmeqd  4599  xpid11m  4619  fneq1  5058  eqfnfv2  5346  offval  5801  ofrfval  5802  offval3  5843  smoeq  5990  tfrlemi14d  6033  tfr1onlemres  6049  tfrcllemres  6062  rdgivallem  6081  rdgon  6086  rdg0  6087  frec0g  6097  freccllem  6102  frecfcllem  6104  frecsuclem  6106  frecsuc  6107  ereq1  6232  fundmeng  6457
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