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Theorem ressid 14353
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  |-  ( W  e.  X  ->  ( Ws  B )  =  W )

Proof of Theorem ressid
StepHypRef Expression
1 ssid 3484 . 2  |-  B  C_  B
2 ressid.1 . . 3  |-  B  =  ( Base `  W
)
3 fvex 5810 . . 3  |-  ( Base `  W )  e.  _V
42, 3eqeltri 2538 . 2  |-  B  e. 
_V
5 eqid 2454 . . 3  |-  ( Ws  B )  =  ( Ws  B )
65, 2ressid2 14346 . 2  |-  ( ( B  C_  B  /\  W  e.  X  /\  B  e.  _V )  ->  ( Ws  B )  =  W )
71, 4, 6mp3an13 1306 1  |-  ( W  e.  X  ->  ( Ws  B )  =  W )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3078    C_ wss 3437   ` cfv 5527  (class class class)co 6201   Basecbs 14293   ↾s cress 14294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-ress 14300
This theorem is referenced by:  submid  15599  subgid  15803  gaid2  15941  subrgid  16991  rlmval2  17399  rlmsca  17405  rlmsca2  17406  evlrhm  17736  evlsscasrng  17737  evlsvarsrng  17739  evl1sca  17894  evl1var  17896  evls1scasrng  17899  evls1varsrng  17900  pf1ind  17915  evl1gsumadd  17918  evl1varpw  17921  pjff  18263  dsmmfi  18289  frlmip  18329  rlmbn  21006  ishl2  21015  rrxprds  21026  dchrptlem2  22738  lnmfg  29584  lmhmfgsplit  29588  pwslnmlem2  29595
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