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Theorem ssdmres 5848
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 3860 . 2 (𝐴 ⊆ dom 𝐵 ↔ (𝐴 ∩ dom 𝐵) = 𝐴)
2 dmres 5847 . . 3 dom (𝐵𝐴) = (𝐴 ∩ dom 𝐵)
32eqeq1i 2743 . 2 (dom (𝐵𝐴) = 𝐴 ↔ (𝐴 ∩ dom 𝐵) = 𝐴)
41, 3bitr4i 281 1 (𝐴 ⊆ dom 𝐵 ↔ dom (𝐵𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1542  cin 3842  wss 3843  dom cdm 5525  cres 5527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-dm 5535  df-res 5537
This theorem is referenced by:  dmresi  5895  fnssresb  6458  fores  6602  foimacnv  6635  dffv2  6763  sbthlem4  8680  hashres  13891  hashimarn  13893  dvres3  24665  c1liplem1  24748  lhop1lem  24765  lhop  24768  usgrres  27250  vtxdginducedm1lem2  27482  wlkres  27612  trlreslem  27641  hhssabloi  29197  hhssnv  29199  hhshsslem1  29202  fresf1o  30540  fsupprnfi  30601  gsumhashmul  30893  cycpmconjvlem  30985  exidreslem  35658  divrngcl  35738  isdrngo2  35739  n0elqs2  36085  dvbdfbdioolem1  43011  fourierdlem48  43237  fourierdlem49  43238  fourierdlem71  43260  fourierdlem73  43262  fourierdlem94  43283  fourierdlem111  43300  fourierdlem112  43301  fourierdlem113  43302  fouriersw  43314  fouriercn  43315  dmvon  43686
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