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Theorem strslssd 12035
Description: Deduction version of strslss 12036. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 31-Jan-2023.)
Hypotheses
Ref Expression
strslssd.e  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
strssd.t  |-  ( ph  ->  T  e.  V )
strssd.f  |-  ( ph  ->  Fun  T )
strssd.s  |-  ( ph  ->  S  C_  T )
strssd.n  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  S )
Assertion
Ref Expression
strslssd  |-  ( ph  ->  ( E `  T
)  =  ( E `
 S ) )

Proof of Theorem strslssd
StepHypRef Expression
1 strslssd.e . . 3  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
2 strssd.t . . 3  |-  ( ph  ->  T  e.  V )
3 strssd.f . . 3  |-  ( ph  ->  Fun  T )
4 strssd.s . . . 4  |-  ( ph  ->  S  C_  T )
5 strssd.n . . . 4  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  S )
64, 5sseldd 3099 . . 3  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  T )
71, 2, 3, 6strslfvd 12030 . 2  |-  ( ph  ->  C  =  ( E `
 T ) )
82, 4ssexd 4072 . . 3  |-  ( ph  ->  S  e.  _V )
9 funss 5146 . . . 4  |-  ( S 
C_  T  ->  ( Fun  T  ->  Fun  S ) )
104, 3, 9sylc 62 . . 3  |-  ( ph  ->  Fun  S )
111, 8, 10, 5strslfvd 12030 . 2  |-  ( ph  ->  C  =  ( E `
 S ) )
127, 11eqtr3d 2175 1  |-  ( ph  ->  ( E `  T
)  =  ( E `
 S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481   _Vcvv 2687    C_ wss 3072   <.cop 3531   Fun wfun 5121   ` cfv 5127   NNcn 8740   ndxcnx 11986  Slot cslot 11988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135  ax-un 4359
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-sbc 2911  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-br 3934  df-opab 3994  df-mpt 3995  df-id 4219  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-iota 5092  df-fun 5129  df-fv 5135  df-slot 11993
This theorem is referenced by:  strslss  12036
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