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Theorem strfv2d 13194
Description: Deduction version of strfv 13196. (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 13179 . 2  |-  ( ph  ->  ( E `  S
)  =  ( S `
 ( E `  ndx ) ) )
4 cnvcnv2 5143 . . . . 5  |-  `' `' S  =  ( S  |` 
_V )
54fveq1i 5542 . . . 4  |-  ( `' `' S `  ( E `
 ndx ) )  =  ( ( S  |`  _V ) `  ( E `  ndx ) )
6 fvex 5555 . . . . 5  |-  ( E `
 ndx )  e. 
_V
7 fvres 5558 . . . . 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 2316 . . 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 2809 . . . . . . . 8  |-  ( C  e.  W  ->  C  e.  _V )
1412, 13syl 15 . . . . . . 7  |-  ( ph  ->  C  e.  _V )
15 opelxpi 4737 . . . . . . 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 3371 . . . . . 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 5142 . . . . 5  |-  `' `' S  =  ( S  i^i  ( _V  X.  _V ) )
2018, 19syl6eleqr 2387 . . . 4  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  `' `' S )
21 funopfv 5578 . . . 4  |-  ( Fun  `' `' S  ->  ( <.
( E `  ndx ) ,  C >.  e.  `' `' S  ->  ( `' `' S `  ( E `
 ndx ) )  =  C ) )
2210, 20, 21sylc 56 . . 3  |-  ( ph  ->  ( `' `' S `  ( E `  ndx ) )  =  C )
239, 22syl5eqr 2342 . 2  |-  ( ph  ->  ( S `  ( E `  ndx ) )  =  C )
243, 23eqtr2d 2329 1  |-  ( ph  ->  C  =  ( E `
 S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164   <.cop 3656    X. cxp 4703   `'ccnv 4704    |` cres 4707   Fun wfun 5265   ` cfv 5271   ndxcnx 13161  Slot cslot 13163
This theorem is referenced by:  strfv2  13195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279  df-slot 13168
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