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Theorem strssd 14757
Description: Deduction version of strss 14758. (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 3490 . . 3  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  T )
71, 2, 3, 6strfvd 14752 . 2  |-  ( ph  ->  C  =  ( E `
 T ) )
82, 4ssexd 4584 . . 3  |-  ( ph  ->  S  e.  _V )
9 funss 5588 . . . 4  |-  ( S 
C_  T  ->  ( Fun  T  ->  Fun  S ) )
104, 3, 9sylc 60 . . 3  |-  ( ph  ->  Fun  S )
111, 8, 10, 5strfvd 14752 . 2  |-  ( ph  ->  C  =  ( E `
 S ) )
127, 11eqtr3d 2497 1  |-  ( ph  ->  ( E `  T
)  =  ( E `
 S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461   <.cop 4022   Fun wfun 5564   ` cfv 5570   ndxcnx 14716  Slot cslot 14718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-slot 14723
This theorem is referenced by:  strss  14758
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