Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ressid Structured version   Visualization version   GIF version

Theorem ressid 15762
 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 3587 . 2 𝐵𝐵
2 ressid.1 . . 3 𝐵 = (Base‘𝑊)
3 fvex 6113 . . 3 (Base‘𝑊) ∈ V
42, 3eqeltri 2684 . 2 𝐵 ∈ V
5 eqid 2610 . . 3 (𝑊s 𝐵) = (𝑊s 𝐵)
65, 2ressid2 15755 . 2 ((𝐵𝐵𝑊𝑋𝐵 ∈ V) → (𝑊s 𝐵) = 𝑊)
71, 4, 6mp3an13 1407 1 (𝑊𝑋 → (𝑊s 𝐵) = 𝑊)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾s cress 15696 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-ress 15702 This theorem is referenced by:  ressval3d  15764  submid  17174  subgid  17419  gaid2  17559  subrgid  18605  rlmval2  19015  rlmsca  19021  rlmsca2  19022  evlrhm  19346  evlsscasrng  19347  evlsvarsrng  19349  evl1sca  19519  evl1var  19521  evls1scasrng  19524  evls1varsrng  19525  pf1ind  19540  evl1gsumadd  19543  evl1varpw  19546  pjff  19875  dsmmfi  19901  frlmip  19936  cnstrcvs  22749  cncvs  22753  rlmbn  22965  ishl2  22974  rrxprds  22985  dchrptlem2  24790  lnmfg  36670  lmhmfgsplit  36674  pwslnmlem2  36681  submgmid  41583
 Copyright terms: Public domain W3C validator