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Theorem strfv2d 13178
Description: Deduction version of strfv 13180. (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 13163 . 2  |-  ( ph  ->  ( E `  S
)  =  ( S `
 ( E `  ndx ) ) )
4 cnvcnv2 5127 . . . . 5  |-  `' `' S  =  ( S  |` 
_V )
54fveq1i 5526 . . . 4  |-  ( `' `' S `  ( E `
 ndx ) )  =  ( ( S  |`  _V ) `  ( E `  ndx ) )
6 fvex 5539 . . . . 5  |-  ( E `
 ndx )  e. 
_V
7 fvres 5542 . . . . 5  |-  ( ( E `  ndx )  e.  _V  ->  ( ( S  |`  _V ) `  ( E `  ndx )
)  =  ( S `
 ( E `  ndx ) ) )
86, 7ax-mp 8 . . . 4  |-  ( ( S  |`  _V ) `  ( E `  ndx ) )  =  ( S `  ( E `
 ndx ) )
95, 8eqtri 2303 . . 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 2796 . . . . . . . 8  |-  ( C  e.  W  ->  C  e.  _V )
1412, 13syl 15 . . . . . . 7  |-  ( ph  ->  C  e.  _V )
15 opelxpi 4721 . . . . . . 7  |-  ( ( ( E `  ndx )  e.  _V  /\  C  e.  _V )  ->  <. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
166, 14, 15sylancr 644 . . . . . 6  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
17 elin 3358 . . . . . 6  |-  ( <.
( E `  ndx ) ,  C >.  e.  ( S  i^i  ( _V  X.  _V ) )  <-> 
( <. ( E `  ndx ) ,  C >.  e.  S  /\  <. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) ) )
1811, 16, 17sylanbrc 645 . . . . 5  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  ( S  i^i  ( _V  X.  _V ) ) )
19 cnvcnv 5126 . . . . 5  |-  `' `' S  =  ( S  i^i  ( _V  X.  _V ) )
2018, 19syl6eleqr 2374 . . . 4  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  `' `' S )
21 funopfv 5562 . . . 4  |-  ( Fun  `' `' S  ->  ( <.
( E `  ndx ) ,  C >.  e.  `' `' S  ->  ( `' `' S `  ( E `
 ndx ) )  =  C ) )
2210, 20, 21sylc 56 . . 3  |-  ( ph  ->  ( `' `' S `  ( E `  ndx ) )  =  C )
239, 22syl5eqr 2329 . 2  |-  ( ph  ->  ( S `  ( E `  ndx ) )  =  C )
243, 23eqtr2d 2316 1  |-  ( ph  ->  C  =  ( E `
 S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   <.cop 3643    X. cxp 4687   `'ccnv 4688    |` cres 4691   Fun wfun 5249   ` cfv 5255   ndxcnx 13145  Slot cslot 13147
This theorem is referenced by:  strfv2  13179
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-slot 13152
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