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Theorem strssd 13430
Description: Deduction version of strss 13431. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
strssd.e  |-  E  = Slot  ( E `  ndx )
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
strssd  |-  ( ph  ->  ( E `  T
)  =  ( E `
 S ) )

Proof of Theorem strssd
StepHypRef Expression
1 strssd.e . . 3  |-  E  = Slot  ( E `  ndx )
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 3292 . . 3  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  T )
71, 2, 3, 6strfvd 13425 . 2  |-  ( ph  ->  C  =  ( E `
 T ) )
82, 4ssexd 4291 . . 3  |-  ( ph  ->  S  e.  _V )
9 funss 5412 . . . 4  |-  ( S 
C_  T  ->  ( Fun  T  ->  Fun  S ) )
104, 3, 9sylc 58 . . 3  |-  ( ph  ->  Fun  S )
111, 8, 10, 5strfvd 13425 . 2  |-  ( ph  ->  C  =  ( E `
 S ) )
127, 11eqtr3d 2421 1  |-  ( ph  ->  ( E `  T
)  =  ( E `
 S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2899    C_ wss 3263   <.cop 3760   Fun wfun 5388   ` cfv 5394   ndxcnx 13393  Slot cslot 13395
This theorem is referenced by:  strss  13431
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-slot 13400
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