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Theorem strss 13274
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 10 . . 3  |-  (  T. 
->  T  e.  _V )
4 strss.f . . . 4  |-  Fun  T
54a1i 10 . . 3  |-  (  T. 
->  Fun  T )
6 strss.s . . . 4  |-  S  C_  T
76a1i 10 . . 3  |-  (  T. 
->  S  C_  T )
8 strss.n . . . 4  |-  <. ( E `  ndx ) ,  C >.  e.  S
98a1i 10 . . 3  |-  (  T. 
->  <. ( E `  ndx ) ,  C >.  e.  S )
101, 3, 5, 7, 9strssd 13273 . 2  |-  (  T. 
->  ( E `  T
)  =  ( E `
 S ) )
1110trud 1323 1  |-  ( E `
 T )  =  ( E `  S
)
Colors of variables: wff set class
Syntax hints:    T. wtru 1316    = wceq 1642    e. wcel 1710   _Vcvv 2864    C_ wss 3228   <.cop 3719   Fun wfun 5328   ` cfv 5334   ndxcnx 13236  Slot cslot 13238
This theorem is referenced by:  grpss  14596
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-iota 5298  df-fun 5336  df-fv 5342  df-slot 13243
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