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Theorem ssdmres 4977
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 3166 . 2  |-  ( A 
C_  dom  B  <->  ( A  i^i  dom  B )  =  A )
2 dmres 4976 . . 3  |-  dom  ( B  |`  A )  =  ( A  i^i  dom  B )
32eqeq1i 2290 . 2  |-  ( dom  ( B  |`  A )  =  A  <->  ( A  i^i  dom  B )  =  A )
41, 3bitr4i 243 1  |-  ( A 
C_  dom  B  <->  dom  ( B  |`  A )  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    i^i cin 3151    C_ wss 3152   dom cdm 4689    |` cres 4691
This theorem is referenced by:  dmresi  5005  fnssresb  5356  fores  5460  foimacnv  5490  dffv2  5592  sbthlem4  6974  cnrest  17013  dvres3  19263  c1liplem1  19343  lhop1lem  19360  lhop  19363  resgrprn  20947  hhssabloi  21839  hhssnv  21841  hhshsslem1  21844  ghomfo  23998  svs2  25487  exidreslem  26567  divrngcl  26588  isdrngo2  26589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-dm 4699  df-res 4701
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