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Theorem ssdmres 5340
 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 3554 . 2 (𝐴 ⊆ dom 𝐵 ↔ (𝐴 ∩ dom 𝐵) = 𝐴)
2 dmres 5339 . . 3 dom (𝐵𝐴) = (𝐴 ∩ dom 𝐵)
32eqeq1i 2615 . 2 (dom (𝐵𝐴) = 𝐴 ↔ (𝐴 ∩ dom 𝐵) = 𝐴)
41, 3bitr4i 266 1 (𝐴 ⊆ dom 𝐵 ↔ dom (𝐵𝐴) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∩ cin 3539   ⊆ wss 3540  dom cdm 5038   ↾ cres 5040 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048  df-res 5050 This theorem is referenced by:  dmresi  5376  fnssresb  5917  fores  6037  foimacnv  6067  dffv2  6181  sbthlem4  7958  hashimarn  13085  dvres3  23483  c1liplem1  23563  lhop1lem  23580  lhop  23583  usgrares1  25939  usgrafilem1  25940  hhssabloi  27503  hhssnv  27505  hhshsslem1  27508  fresf1o  28815  exidreslem  32846  divrngcl  32926  isdrngo2  32927  dvbdfbdioolem1  38818  fourierdlem48  39047  fourierdlem49  39048  fourierdlem71  39070  fourierdlem73  39072  fourierdlem94  39093  fourierdlem111  39110  fourierdlem112  39111  fourierdlem113  39112  fouriersw  39124  fouriercn  39125  dmvon  39496  trlreslem  40907
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