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Theorem ssdmres 5146
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 3456 . 2  |-  ( A 
C_  dom  B  <->  ( A  i^i  dom  B )  =  A )
2 dmres 5145 . . 3  |-  dom  ( B  |`  A )  =  ( A  i^i  dom  B )
32eqeq1i 2436 . 2  |-  ( dom  ( B  |`  A )  =  A  <->  ( A  i^i  dom  B )  =  A )
41, 3bitr4i 255 1  |-  ( A 
C_  dom  B  <->  dom  ( B  |`  A )  =  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    = wceq 1437    i^i cin 3441    C_ wss 3442   dom cdm 4854    |` cres 4856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-dm 4864  df-res 4866
This theorem is referenced by:  dmresi  5180  fnssresb  5706  fores  5819  foimacnv  5848  dffv2  5954  sbthlem4  7691  hashimarn  12605  dvres3  22745  c1liplem1  22825  lhop1lem  22842  lhop  22845  usgrares1  24983  usgrafilem1  24984  resgrprn  25853  hhssabloi  26748  hhssnv  26750  hhshsslem1  26753  fresf1o  28071  ghomfo  30097  exidreslem  31878  divrngcl  31899  isdrngo2  31900  dvbdfbdioolem1  37371  fourierdlem48  37585  fourierdlem49  37586  fourierdlem71  37608  fourierdlem73  37610  fourierdlem94  37631  fourierdlem111  37648  fourierdlem112  37649  fourierdlem113  37650  fouriersw  37662  fouriercn  37663
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