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Theorem ssdmres 6005
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 3966 . 2 (𝐴 ⊆ dom 𝐵 ↔ (𝐴 ∩ dom 𝐵) = 𝐴)
2 dmres 6004 . . 3 dom (𝐵𝐴) = (𝐴 ∩ dom 𝐵)
32eqeq1i 2738 . 2 (dom (𝐵𝐴) = 𝐴 ↔ (𝐴 ∩ dom 𝐵) = 𝐴)
41, 3bitr4i 278 1 (𝐴 ⊆ dom 𝐵 ↔ dom (𝐵𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  cin 3948  wss 3949  dom cdm 5677  cres 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-dm 5687  df-res 5689
This theorem is referenced by:  dmresi  6052  fnssresb  6673  fores  6816  foimacnv  6851  dffv2  6987  sbthlem4  9086  hashres  14398  hashimarn  14400  dvres3  25430  c1liplem1  25513  lhop1lem  25530  lhop  25533  usgrres  28565  vtxdginducedm1lem2  28797  wlkres  28927  trlreslem  28956  hhssabloi  30515  hhssnv  30517  hhshsslem1  30520  fresf1o  31855  fsupprnfi  31914  gsumhashmul  32208  cycpmconjvlem  32300  exidreslem  36745  divrngcl  36825  isdrngo2  36826  n0elqs2  37196  dvbdfbdioolem1  44644  fourierdlem48  44870  fourierdlem49  44871  fourierdlem71  44893  fourierdlem73  44895  fourierdlem94  44916  fourierdlem111  44933  fourierdlem112  44934  fourierdlem113  44935  fouriersw  44947  fouriercn  44948  dmvon  45322
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