ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmeq Unicode version

Theorem dmeq 4709
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 4708 . . 3  |-  ( A 
C_  B  ->  dom  A 
C_  dom  B )
2 dmss 4708 . . 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 3082 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 eqss 3082 . 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 1316    C_ wss 3041   dom cdm 4509
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-dm 4519
This theorem is referenced by:  dmeqi  4710  dmeqd  4711  xpid11  4732  sqxpeq0  4932  fneq1  5181  eqfnfv2  5487  offval  5957  ofrfval  5958  offval3  6000  smoeq  6155  tfrlemi14d  6198  tfr1onlemres  6214  tfrcllemres  6227  rdgivallem  6246  rdgon  6251  rdg0  6252  frec0g  6262  freccllem  6267  frecfcllem  6269  frecsuclem  6271  frecsuc  6272  ereq1  6404  fundmeng  6669  acfun  7031  ccfunen  7047  ennnfonelemj0  11841  ennnfonelemg  11843  ennnfonelemp1  11846  ennnfonelemom  11848  ennnfonelemnn0  11862  blfvalps  12481  reldvg  12744
  Copyright terms: Public domain W3C validator