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Theorem ssdmres 5294
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 3490 . 2  |-  ( A 
C_  dom  B  <->  ( A  i^i  dom  B )  =  A )
2 dmres 5293 . . 3  |-  dom  ( B  |`  A )  =  ( A  i^i  dom  B )
32eqeq1i 2474 . 2  |-  ( dom  ( B  |`  A )  =  A  <->  ( A  i^i  dom  B )  =  A )
41, 3bitr4i 252 1  |-  ( A 
C_  dom  B  <->  dom  ( B  |`  A )  =  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379    i^i cin 3475    C_ wss 3476   dom cdm 4999    |` cres 5001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-dm 5009  df-res 5011
This theorem is referenced by:  dmresi  5328  fnssresb  5692  fores  5803  foimacnv  5832  dffv2  5939  sbthlem4  7630  hashimarn  12461  cnrest  19568  dvres3  22068  c1liplem1  22148  lhop1lem  22165  lhop  22168  usgrares1  24102  usgrafilem1  24103  resgrprn  24974  hhssabloi  25870  hhssnv  25872  hhshsslem1  25875  ghomfo  28522  exidreslem  29958  divrngcl  29979  isdrngo2  29980  dvbdfbdioolem1  31274  fourierdlem48  31471  fourierdlem49  31472  fourierdlem71  31494  fourierdlem73  31496  fourierdlem94  31517  fourierdlem111  31534  fourierdlem112  31535  fourierdlem113  31536  fouriersw  31548  fouriercn  31549
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