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Theorem ssdmres 5914
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 3904 . 2 (𝐴 ⊆ dom 𝐵 ↔ (𝐴 ∩ dom 𝐵) = 𝐴)
2 dmres 5913 . . 3 dom (𝐵𝐴) = (𝐴 ∩ dom 𝐵)
32eqeq1i 2743 . 2 (dom (𝐵𝐴) = 𝐴 ↔ (𝐴 ∩ dom 𝐵) = 𝐴)
41, 3bitr4i 277 1 (𝐴 ⊆ dom 𝐵 ↔ dom (𝐵𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1539  cin 3886  wss 3887  dom cdm 5589  cres 5591
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-dm 5599  df-res 5601
This theorem is referenced by:  dmresi  5961  fnssresb  6554  fores  6698  foimacnv  6733  dffv2  6863  sbthlem4  8873  hashres  14153  hashimarn  14155  dvres3  25077  c1liplem1  25160  lhop1lem  25177  lhop  25180  usgrres  27675  vtxdginducedm1lem2  27907  wlkres  28038  trlreslem  28067  hhssabloi  29624  hhssnv  29626  hhshsslem1  29629  fresf1o  30966  fsupprnfi  31026  gsumhashmul  31316  cycpmconjvlem  31408  exidreslem  36035  divrngcl  36115  isdrngo2  36116  n0elqs2  36462  dvbdfbdioolem1  43469  fourierdlem48  43695  fourierdlem49  43696  fourierdlem71  43718  fourierdlem73  43720  fourierdlem94  43741  fourierdlem111  43758  fourierdlem112  43759  fourierdlem113  43760  fouriersw  43772  fouriercn  43773  dmvon  44144
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