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Theorem strslss 12523
Description: Propagate component extraction to a structure  T from a subset structure  S. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Jim Kingdon, 31-Jan-2023.)
Hypotheses
Ref Expression
strss.t  |-  T  e. 
_V
strss.f  |-  Fun  T
strss.s  |-  S  C_  T
strslss.e  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
strss.n  |-  <. ( E `  ndx ) ,  C >.  e.  S
Assertion
Ref Expression
strslss  |-  ( E `
 T )  =  ( E `  S
)

Proof of Theorem strslss
StepHypRef Expression
1 strslss.e . . 3  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
2 strss.t . . . 4  |-  T  e. 
_V
32a1i 9 . . 3  |-  ( T. 
->  T  e.  _V )
4 strss.f . . . 4  |-  Fun  T
54a1i 9 . . 3  |-  ( T. 
->  Fun  T )
6 strss.s . . . 4  |-  S  C_  T
76a1i 9 . . 3  |-  ( T. 
->  S  C_  T )
8 strss.n . . . 4  |-  <. ( E `  ndx ) ,  C >.  e.  S
98a1i 9 . . 3  |-  ( T. 
->  <. ( E `  ndx ) ,  C >.  e.  S )
101, 3, 5, 7, 9strslssd 12522 . 2  |-  ( T. 
->  ( E `  T
)  =  ( E `
 S ) )
1110mptru 1372 1  |-  ( E `
 T )  =  ( E `  S
)
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1363   T. wtru 1364    e. wcel 2158   _Vcvv 2749    C_ wss 3141   <.cop 3607   Fun wfun 5222   ` cfv 5228   NNcn 8932   ndxcnx 12472  Slot cslot 12474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fv 5236  df-slot 12479
This theorem is referenced by: (None)
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