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Theorem strfv2d 14875
Description: Deduction version of strfv 14877. (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 14856 . 2  |-  ( ph  ->  ( E `  S
)  =  ( S `
 ( E `  ndx ) ) )
4 cnvcnv2 5277 . . . . 5  |-  `' `' S  =  ( S  |` 
_V )
54fveq1i 5850 . . . 4  |-  ( `' `' S `  ( E `
 ndx ) )  =  ( ( S  |`  _V ) `  ( E `  ndx ) )
6 fvex 5859 . . . . 5  |-  ( E `
 ndx )  e. 
_V
7 fvres 5863 . . . . 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 2431 . . 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 3068 . . . . . . . 8  |-  ( C  e.  W  ->  C  e.  _V )
1412, 13syl 17 . . . . . . 7  |-  ( ph  ->  C  e.  _V )
15 opelxpi 4855 . . . . . . 7  |-  ( ( ( E `  ndx )  e.  _V  /\  C  e.  _V )  ->  <. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
166, 14, 15sylancr 661 . . . . . 6  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
1711, 16elind 3627 . . . . 5  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  ( S  i^i  ( _V  X.  _V ) ) )
18 cnvcnv 5276 . . . . 5  |-  `' `' S  =  ( S  i^i  ( _V  X.  _V ) )
1917, 18syl6eleqr 2501 . . . 4  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  `' `' S )
20 funopfv 5888 . . . 4  |-  ( Fun  `' `' S  ->  ( <.
( E `  ndx ) ,  C >.  e.  `' `' S  ->  ( `' `' S `  ( E `
 ndx ) )  =  C ) )
2110, 19, 20sylc 59 . . 3  |-  ( ph  ->  ( `' `' S `  ( E `  ndx ) )  =  C )
229, 21syl5eqr 2457 . 2  |-  ( ph  ->  ( S `  ( E `  ndx ) )  =  C )
233, 22eqtr2d 2444 1  |-  ( ph  ->  C  =  ( E `
 S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   _Vcvv 3059    i^i cin 3413   <.cop 3978    X. cxp 4821   `'ccnv 4822    |` cres 4825   Fun wfun 5563   ` cfv 5569   ndxcnx 14838  Slot cslot 14840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-res 4835  df-iota 5533  df-fun 5571  df-fv 5577  df-slot 14845
This theorem is referenced by:  strfv2  14876  eengbas  24701  ebtwntg  24702  ecgrtg  24703  elntg  24704
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