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Theorem ssdmres 5578
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 3729 . 2 (𝐴 ⊆ dom 𝐵 ↔ (𝐴 ∩ dom 𝐵) = 𝐴)
2 dmres 5577 . . 3 dom (𝐵𝐴) = (𝐴 ∩ dom 𝐵)
32eqeq1i 2765 . 2 (dom (𝐵𝐴) = 𝐴 ↔ (𝐴 ∩ dom 𝐵) = 𝐴)
41, 3bitr4i 267 1 (𝐴 ⊆ dom 𝐵 ↔ dom (𝐵𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1632  cin 3714  wss 3715  dom cdm 5266  cres 5268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-dm 5276  df-res 5278
This theorem is referenced by:  dmresi  5615  fnssresb  6164  fores  6285  foimacnv  6315  dffv2  6433  sbthlem4  8238  hashres  13417  hashimarn  13419  dvres3  23876  c1liplem1  23958  lhop1lem  23975  lhop  23978  usgrres  26399  vtxdginducedm1lem2  26646  trlreslem  26806  hhssabloi  28428  hhssnv  28430  hhshsslem1  28433  fresf1o  29742  exidreslem  33989  divrngcl  34069  isdrngo2  34070  n0elqs2  34423  dvbdfbdioolem1  40646  fourierdlem48  40874  fourierdlem49  40875  fourierdlem71  40897  fourierdlem73  40899  fourierdlem94  40920  fourierdlem111  40937  fourierdlem112  40938  fourierdlem113  40939  fouriersw  40951  fouriercn  40952  dmvon  41326
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