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Theorem ssdmres 5169
Description: A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
Assertion
Ref Expression
ssdmres  |-  ( A 
C_  dom  B  <->  dom  ( B  |`  A )  =  A )

Proof of Theorem ssdmres
StepHypRef Expression
1 df-ss 3335 . 2  |-  ( A 
C_  dom  B  <->  ( A  i^i  dom  B )  =  A )
2 dmres 5168 . . 3  |-  dom  ( B  |`  A )  =  ( A  i^i  dom  B )
32eqeq1i 2444 . 2  |-  ( dom  ( B  |`  A )  =  A  <->  ( A  i^i  dom  B )  =  A )
41, 3bitr4i 245 1  |-  ( A 
C_  dom  B  <->  dom  ( B  |`  A )  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    i^i cin 3320    C_ wss 3321   dom cdm 4879    |` cres 4881
This theorem is referenced by:  dmresi  5197  fnssresb  5558  fores  5663  foimacnv  5693  dffv2  5797  sbthlem4  7221  cnrest  17350  dvres3  19801  c1liplem1  19881  lhop1lem  19898  lhop  19901  usgrares1  21425  usgrafilem1  21426  resgrprn  21869  hhssabloi  22763  hhssnv  22765  hhshsslem1  22768  ghomfo  25103  exidreslem  26553  divrngcl  26574  isdrngo2  26575  hashimarn  28164
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-xp 4885  df-dm 4889  df-res 4891
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