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Theorem ressid 17130
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 3967 . 2 𝐵𝐵
2 ressid.1 . . 3 𝐵 = (Base‘𝑊)
32fvexi 6857 . 2 𝐵 ∈ V
4 eqid 2733 . . 3 (𝑊s 𝐵) = (𝑊s 𝐵)
54, 2ressid2 17121 . 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 3444  wss 3911  cfv 6497  (class class class)co 7358  Basecbs 17088  s cress 17117
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 5257  ax-nul 5264  ax-pr 5385
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-ress 17118
This theorem is referenced by:  ressval3d  17132  ressval3dOLD  17133  submid  18626  subgid  18935  gaid2  19088  subrgid  20238  sdrgid  20276  rlmval2  20679  rlmsca  20685  rlmsca2  20686  pjff  21134  dsmmfi  21160  frlmip  21200  evlrhm  21522  evlsscasrng  21523  evlsvarsrng  21525  evl1sca  21716  evl1var  21718  evls1scasrng  21721  evls1varsrng  21722  pf1ind  21737  evl1gsumadd  21740  evl1varpw  21743  cnstrcvs  24520  cncvs  24524  rlmbn  24741  ishl2  24750  rrxprds  24769  dchrptlem2  26629  ressply1evl  32321  rgmoddim  32362  qusdimsum  32380  fldextid  32405  riccrng1  40748  ricdrng1  40762  evlsevl  40795  mhphf4  40817  lnmfg  41452  lmhmfgsplit  41456  pwslnmlem2  41463  simpcntrab  45197  submgmid  46173
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