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Theorem ssdmres 4993
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 3179 . 2  |-  ( A 
C_  dom  B  <->  ( A  i^i  dom  B )  =  A )
2 dmres 4992 . . 3  |-  dom  ( B  |`  A )  =  ( A  i^i  dom  B )
32eqeq1i 2303 . 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 1632    i^i cin 3164    C_ wss 3165   dom cdm 4705    |` cres 4707
This theorem is referenced by:  dmresi  5021  fnssresb  5372  fores  5476  foimacnv  5506  dffv2  5608  sbthlem4  6990  cnrest  17029  dvres3  19279  c1liplem1  19359  lhop1lem  19376  lhop  19379  resgrprn  20963  hhssabloi  21855  hhssnv  21857  hhshsslem1  21860  ghomfo  24013  svs2  25590  exidreslem  26670  divrngcl  26691  isdrngo2  26692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-dm 4715  df-res 4717
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