MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssdmres Structured version   Visualization version   GIF version

Theorem ssdmres 5869
Description: A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
Assertion
Ref Expression
ssdmres (𝐴 ⊆ dom 𝐵 ↔ dom (𝐵𝐴) = 𝐴)

Proof of Theorem ssdmres
StepHypRef Expression
1 df-ss 3949 . 2 (𝐴 ⊆ dom 𝐵 ↔ (𝐴 ∩ dom 𝐵) = 𝐴)
2 dmres 5868 . . 3 dom (𝐵𝐴) = (𝐴 ∩ dom 𝐵)
32eqeq1i 2823 . 2 (dom (𝐵𝐴) = 𝐴 ↔ (𝐴 ∩ dom 𝐵) = 𝐴)
41, 3bitr4i 279 1 (𝐴 ⊆ dom 𝐵 ↔ dom (𝐵𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  cin 3932  wss 3933  dom cdm 5548  cres 5550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-dm 5558  df-res 5560
This theorem is referenced by:  dmresi  5914  fnssresb  6462  fores  6593  foimacnv  6625  dffv2  6749  sbthlem4  8618  hashres  13787  hashimarn  13789  dvres3  24438  c1liplem1  24520  lhop1lem  24537  lhop  24540  usgrres  27017  vtxdginducedm1lem2  27249  wlkres  27379  trlreslem  27408  hhssabloi  28966  hhssnv  28968  hhshsslem1  28971  fresf1o  30304  cycpmconjvlem  30710  exidreslem  35036  divrngcl  35116  isdrngo2  35117  n0elqs2  35465  dvbdfbdioolem1  42089  fourierdlem48  42316  fourierdlem49  42317  fourierdlem71  42339  fourierdlem73  42341  fourierdlem94  42362  fourierdlem111  42379  fourierdlem112  42380  fourierdlem113  42381  fouriersw  42393  fouriercn  42394  dmvon  42765
  Copyright terms: Public domain W3C validator