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Theorem ressid 17189
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 4005 . 2 𝐵𝐵
2 ressid.1 . . 3 𝐵 = (Base‘𝑊)
32fvexi 6906 . 2 𝐵 ∈ V
4 eqid 2733 . . 3 (𝑊s 𝐵) = (𝑊s 𝐵)
54, 2ressid2 17177 . 2 ((𝐵𝐵𝑊𝑋𝐵 ∈ V) → (𝑊s 𝐵) = 𝑊)
61, 3, 5mp3an13 1453 1 (𝑊𝑋 → (𝑊s 𝐵) = 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3475  wss 3949  cfv 6544  (class class class)co 7409  Basecbs 17144  s cress 17173
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-ress 17174
This theorem is referenced by:  ressval3d  17191  ressval3dOLD  17192  submid  18691  subgid  19008  gaid2  19167  subrgid  20321  sdrgid  20408  rlmval2  20816  rlmsca  20822  rlmsca2  20823  pjff  21267  dsmmfi  21293  frlmip  21333  evlrhm  21659  evlsscasrng  21660  evlsvarsrng  21662  evl1sca  21853  evl1var  21855  evls1scasrng  21858  evls1varsrng  21859  pf1ind  21874  evl1gsumadd  21877  evl1varpw  21880  cnstrcvs  24657  cncvs  24661  rlmbn  24878  ishl2  24887  rrxprds  24906  dchrptlem2  26768  ressply1evl  32678  rgmoddimOLD  32726  qusdimsum  32744  fldextid  32769  riccrng1  41144  ricdrng1  41150  evlsevl  41191  evlvvval  41193  evlvvvallem  41194  mhphf4  41220  lnmfg  41872  lmhmfgsplit  41876  pwslnmlem2  41883  simpcntrab  45634  submgmid  46611  subrngid  46776
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