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Theorem strfv2d 13501
Description: Deduction version of strfv 13503. (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 13486 . 2  |-  ( ph  ->  ( E `  S
)  =  ( S `
 ( E `  ndx ) ) )
4 cnvcnv2 5326 . . . . 5  |-  `' `' S  =  ( S  |` 
_V )
54fveq1i 5731 . . . 4  |-  ( `' `' S `  ( E `
 ndx ) )  =  ( ( S  |`  _V ) `  ( E `  ndx ) )
6 fvex 5744 . . . . 5  |-  ( E `
 ndx )  e. 
_V
7 fvres 5747 . . . . 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 2458 . . 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 2966 . . . . . . . 8  |-  ( C  e.  W  ->  C  e.  _V )
1412, 13syl 16 . . . . . . 7  |-  ( ph  ->  C  e.  _V )
15 opelxpi 4912 . . . . . . 7  |-  ( ( ( E `  ndx )  e.  _V  /\  C  e.  _V )  ->  <. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
166, 14, 15sylancr 646 . . . . . 6  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
17 elin 3532 . . . . . 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 647 . . . . 5  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  ( S  i^i  ( _V  X.  _V ) ) )
19 cnvcnv 5325 . . . . 5  |-  `' `' S  =  ( S  i^i  ( _V  X.  _V ) )
2018, 19syl6eleqr 2529 . . . 4  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  `' `' S )
21 funopfv 5768 . . . 4  |-  ( Fun  `' `' S  ->  ( <.
( E `  ndx ) ,  C >.  e.  `' `' S  ->  ( `' `' S `  ( E `
 ndx ) )  =  C ) )
2210, 20, 21sylc 59 . . 3  |-  ( ph  ->  ( `' `' S `  ( E `  ndx ) )  =  C )
239, 22syl5eqr 2484 . 2  |-  ( ph  ->  ( S `  ( E `  ndx ) )  =  C )
243, 23eqtr2d 2471 1  |-  ( ph  ->  C  =  ( E `
 S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958    i^i cin 3321   <.cop 3819    X. cxp 4878   `'ccnv 4879    |` cres 4882   Fun wfun 5450   ` cfv 5456   ndxcnx 13468  Slot cslot 13470
This theorem is referenced by:  strfv2  13502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-res 4892  df-iota 5420  df-fun 5458  df-fv 5464  df-slot 13475
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