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Theorem strss 13509
Description: Propagate component extraction to a structure  T from a subset structure  S. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 15-Jan-2014.)
Hypotheses
Ref Expression
strss.t  |-  T  e. 
_V
strss.f  |-  Fun  T
strss.s  |-  S  C_  T
strss.e  |-  E  = Slot  ( E `  ndx )
strss.n  |-  <. ( E `  ndx ) ,  C >.  e.  S
Assertion
Ref Expression
strss  |-  ( E `
 T )  =  ( E `  S
)

Proof of Theorem strss
StepHypRef Expression
1 strss.e . . 3  |-  E  = Slot  ( E `  ndx )
2 strss.t . . . 4  |-  T  e. 
_V
32a1i 11 . . 3  |-  (  T. 
->  T  e.  _V )
4 strss.f . . . 4  |-  Fun  T
54a1i 11 . . 3  |-  (  T. 
->  Fun  T )
6 strss.s . . . 4  |-  S  C_  T
76a1i 11 . . 3  |-  (  T. 
->  S  C_  T )
8 strss.n . . . 4  |-  <. ( E `  ndx ) ,  C >.  e.  S
98a1i 11 . . 3  |-  (  T. 
->  <. ( E `  ndx ) ,  C >.  e.  S )
101, 3, 5, 7, 9strssd 13508 . 2  |-  (  T. 
->  ( E `  T
)  =  ( E `
 S ) )
1110trud 1333 1  |-  ( E `
 T )  =  ( E `  S
)
Colors of variables: wff set class
Syntax hints:    T. wtru 1326    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   <.cop 3819   Fun wfun 5451   ` cfv 5457   ndxcnx 13471  Slot cslot 13473
This theorem is referenced by:  grpss  14831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-slot 13478
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