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Theorem strss 13463
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 13462 . 2  |-  (  T. 
->  ( E `  T
)  =  ( E `
 S ) )
1110trud 1329 1  |-  ( E `
 T )  =  ( E `  S
)
Colors of variables: wff set class
Syntax hints:    T. wtru 1322    = wceq 1649    e. wcel 1721   _Vcvv 2920    C_ wss 3284   <.cop 3781   Fun wfun 5411   ` cfv 5417   ndxcnx 13425  Slot cslot 13427
This theorem is referenced by:  grpss  14785
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-slot 13432
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