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Theorem ressid 17290
Description: Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Hypothesis
Ref Expression
ressid.1 𝐵 = (Base‘𝑊)
Assertion
Ref Expression
ressid (𝑊𝑋 → (𝑊s 𝐵) = 𝑊)

Proof of Theorem ressid
StepHypRef Expression
1 ssid 4018 . 2 𝐵𝐵
2 ressid.1 . . 3 𝐵 = (Base‘𝑊)
32fvexi 6921 . 2 𝐵 ∈ V
4 eqid 2735 . . 3 (𝑊s 𝐵) = (𝑊s 𝐵)
54, 2ressid2 17278 . 2 ((𝐵𝐵𝑊𝑋𝐵 ∈ V) → (𝑊s 𝐵) = 𝑊)
61, 3, 5mp3an13 1451 1 (𝑊𝑋 → (𝑊s 𝐵) = 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-ress 17275
This theorem is referenced by:  ressval3d  17292  ressval3dOLD  17293  submgmid  18732  submid  18836  subgid  19159  gaid2  19334  subrngid  20566  subrgid  20590  sdrgid  20810  rlmval2  21217  rlmsca  21223  rlmsca2  21224  pjff  21750  dsmmfi  21776  frlmip  21816  evlrhm  22138  evlsscasrng  22139  evlsvarsrng  22141  evl1sca  22354  evl1var  22356  evls1scasrng  22359  evls1varsrng  22360  pf1ind  22375  evl1gsumadd  22378  evl1varpw  22381  ressply1evl  22390  cnstrcvs  25188  cncvs  25192  rlmbn  25409  ishl2  25418  rrxprds  25437  dchrptlem2  27324  evl1fpws  33570  resssra  33617  rgmoddimOLD  33638  qusdimsum  33656  fldextid  33687  riccrng1  42508  ricdrng1  42515  evlsevl  42558  evlvvval  42560  evlvvvallem  42561  mhphf4  42587  lnmfg  43071  lmhmfgsplit  43075  pwslnmlem2  43082  simpcntrab  46826
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