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Theorem ressid 14539
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 3516 . 2  |-  B  C_  B
2 ressid.1 . . 3  |-  B  =  ( Base `  W
)
3 fvex 5867 . . 3  |-  ( Base `  W )  e.  _V
42, 3eqeltri 2544 . 2  |-  B  e. 
_V
5 eqid 2460 . . 3  |-  ( Ws  B )  =  ( Ws  B )
65, 2ressid2 14532 . 2  |-  ( ( B  C_  B  /\  W  e.  X  /\  B  e.  _V )  ->  ( Ws  B )  =  W )
71, 4, 6mp3an13 1310 1  |-  ( W  e.  X  ->  ( Ws  B )  =  W )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   _Vcvv 3106    C_ wss 3469   ` cfv 5579  (class class class)co 6275   Basecbs 14479   ↾s cress 14480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-ress 14486
This theorem is referenced by:  submid  15785  subgid  15991  gaid2  16129  subrgid  17207  rlmval2  17616  rlmsca  17622  rlmsca2  17623  evlrhm  17958  evlsscasrng  17959  evlsvarsrng  17961  evl1sca  18134  evl1var  18136  evls1scasrng  18139  evls1varsrng  18140  pf1ind  18155  evl1gsumadd  18158  evl1varpw  18161  pjff  18503  dsmmfi  18529  frlmip  18569  rlmbn  21529  ishl2  21538  rrxprds  21549  dchrptlem2  23261  lnmfg  30621  lmhmfgsplit  30625  pwslnmlem2  30632
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