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Theorem strslssd 12488
Description: Deduction version of strslss 12489. (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 3156 . . 3  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  T )
71, 2, 3, 6strslfvd 12483 . 2  |-  ( ph  ->  C  =  ( E `
 T ) )
82, 4ssexd 4140 . . 3  |-  ( ph  ->  S  e.  _V )
9 funss 5231 . . . 4  |-  ( S 
C_  T  ->  ( Fun  T  ->  Fun  S ) )
104, 3, 9sylc 62 . . 3  |-  ( ph  ->  Fun  S )
111, 8, 10, 5strslfvd 12483 . 2  |-  ( ph  ->  C  =  ( E `
 S ) )
127, 11eqtr3d 2212 1  |-  ( ph  ->  ( E `  T
)  =  ( E `
 S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   _Vcvv 2737    C_ wss 3129   <.cop 3594   Fun wfun 5206   ` cfv 5212   NNcn 8905   ndxcnx 12439  Slot cslot 12441
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-iota 5174  df-fun 5214  df-fv 5220  df-slot 12446
This theorem is referenced by:  strslss  12489
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