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

Theorem ressid 16547
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 3986 . 2 𝐵𝐵
2 ressid.1 . . 3 𝐵 = (Base‘𝑊)
32fvexi 6677 . 2 𝐵 ∈ V
4 eqid 2818 . . 3 (𝑊s 𝐵) = (𝑊s 𝐵)
54, 2ressid2 16540 . 2 ((𝐵𝐵𝑊𝑋𝐵 ∈ V) → (𝑊s 𝐵) = 𝑊)
61, 3, 5mp3an13 1443 1 (𝑊𝑋 → (𝑊s 𝐵) = 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  wss 3933  cfv 6348  (class class class)co 7145  Basecbs 16471  s cress 16472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-ress 16479
This theorem is referenced by:  ressval3d  16549  submid  17963  subgid  18219  gaid2  18371  subrgid  19466  sdrgid  19504  rlmval2  19895  rlmsca  19901  rlmsca2  19902  evlrhm  20237  evlsscasrng  20238  evlsvarsrng  20240  evl1sca  20425  evl1var  20427  evls1scasrng  20430  evls1varsrng  20431  pf1ind  20446  evl1gsumadd  20449  evl1varpw  20452  pjff  20784  dsmmfi  20810  frlmip  20850  cnstrcvs  23672  cncvs  23676  rlmbn  23891  ishl2  23900  rrxprds  23919  dchrptlem2  25768  rgmoddim  30907  qusdimsum  30923  fldextid  30948  lnmfg  39560  lmhmfgsplit  39564  pwslnmlem2  39571  simpcntrab  43004  submgmid  43937
  Copyright terms: Public domain W3C validator