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Theorem strfv2d 15148
Description: Deduction version of strfv 15150. (Contributed by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
strfv2d.e  |-  E  = Slot  ( E `  ndx )
strfv2d.s  |-  ( ph  ->  S  e.  V )
strfv2d.f  |-  ( ph  ->  Fun  `' `' S
)
strfv2d.n  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  S )
strfv2d.c  |-  ( ph  ->  C  e.  W )
Assertion
Ref Expression
strfv2d  |-  ( ph  ->  C  =  ( E `
 S ) )

Proof of Theorem strfv2d
StepHypRef Expression
1 strfv2d.e . . 3  |-  E  = Slot  ( E `  ndx )
2 strfv2d.s . . 3  |-  ( ph  ->  S  e.  V )
31, 2strfvnd 15129 . 2  |-  ( ph  ->  ( E `  S
)  =  ( S `
 ( E `  ndx ) ) )
4 cnvcnv2 5288 . . . . 5  |-  `' `' S  =  ( S  |` 
_V )
54fveq1i 5864 . . . 4  |-  ( `' `' S `  ( E `
 ndx ) )  =  ( ( S  |`  _V ) `  ( E `  ndx ) )
6 fvex 5873 . . . . 5  |-  ( E `
 ndx )  e. 
_V
7 fvres 5877 . . . . 5  |-  ( ( E `  ndx )  e.  _V  ->  ( ( S  |`  _V ) `  ( E `  ndx )
)  =  ( S `
 ( E `  ndx ) ) )
86, 7ax-mp 5 . . . 4  |-  ( ( S  |`  _V ) `  ( E `  ndx ) )  =  ( S `  ( E `
 ndx ) )
95, 8eqtri 2472 . . 3  |-  ( `' `' S `  ( E `
 ndx ) )  =  ( S `  ( E `  ndx )
)
10 strfv2d.f . . . 4  |-  ( ph  ->  Fun  `' `' S
)
11 strfv2d.n . . . . . 6  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  S )
12 strfv2d.c . . . . . . . 8  |-  ( ph  ->  C  e.  W )
13 elex 3053 . . . . . . . 8  |-  ( C  e.  W  ->  C  e.  _V )
1412, 13syl 17 . . . . . . 7  |-  ( ph  ->  C  e.  _V )
15 opelxpi 4865 . . . . . . 7  |-  ( ( ( E `  ndx )  e.  _V  /\  C  e.  _V )  ->  <. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
166, 14, 15sylancr 668 . . . . . 6  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
1711, 16elind 3617 . . . . 5  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  ( S  i^i  ( _V  X.  _V ) ) )
18 cnvcnv 5287 . . . . 5  |-  `' `' S  =  ( S  i^i  ( _V  X.  _V ) )
1917, 18syl6eleqr 2539 . . . 4  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  `' `' S )
20 funopfv 5902 . . . 4  |-  ( Fun  `' `' S  ->  ( <.
( E `  ndx ) ,  C >.  e.  `' `' S  ->  ( `' `' S `  ( E `
 ndx ) )  =  C ) )
2110, 19, 20sylc 62 . . 3  |-  ( ph  ->  ( `' `' S `  ( E `  ndx ) )  =  C )
229, 21syl5eqr 2498 . 2  |-  ( ph  ->  ( S `  ( E `  ndx ) )  =  C )
233, 22eqtr2d 2485 1  |-  ( ph  ->  C  =  ( E `
 S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1443    e. wcel 1886   _Vcvv 3044    i^i cin 3402   <.cop 3973    X. cxp 4831   `'ccnv 4832    |` cres 4835   Fun wfun 5575   ` cfv 5581   ndxcnx 15111  Slot cslot 15113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-res 4845  df-iota 5545  df-fun 5583  df-fv 5589  df-slot 15118
This theorem is referenced by:  strfv2  15149  eengbas  25004  ebtwntg  25005  ecgrtg  25006  elntg  25007
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