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Theorem strslfv2d 12566
Description: Deduction version of strslfv 12568. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)
Hypotheses
Ref Expression
strslfv2d.e  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
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
strslfv2d  |-  ( ph  ->  C  =  ( E `
 S ) )

Proof of Theorem strslfv2d
StepHypRef Expression
1 strslfv2d.e . . . 4  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
21simpli 111 . . 3  |-  E  = Slot  ( E `  ndx )
3 strfv2d.s . . 3  |-  ( ph  ->  S  e.  V )
41simpri 113 . . . 4  |-  ( E `
 ndx )  e.  NN
54a1i 9 . . 3  |-  ( ph  ->  ( E `  ndx )  e.  NN )
62, 3, 5strnfvnd 12543 . 2  |-  ( ph  ->  ( E `  S
)  =  ( S `
 ( E `  ndx ) ) )
7 cnvcnv2 5103 . . . 4  |-  `' `' S  =  ( S  |` 
_V )
87fveq1i 5538 . . 3  |-  ( `' `' S `  ( E `
 ndx ) )  =  ( ( S  |`  _V ) `  ( E `  ndx ) )
95elexd 2765 . . . 4  |-  ( ph  ->  ( E `  ndx )  e.  _V )
10 fvres 5561 . . . 4  |-  ( ( E `  ndx )  e.  _V  ->  ( ( S  |`  _V ) `  ( E `  ndx )
)  =  ( S `
 ( E `  ndx ) ) )
119, 10syl 14 . . 3  |-  ( ph  ->  ( ( S  |`  _V ) `  ( E `
 ndx ) )  =  ( S `  ( E `  ndx )
) )
128, 11eqtrid 2234 . 2  |-  ( ph  ->  ( `' `' S `  ( E `  ndx ) )  =  ( S `  ( E `
 ndx ) ) )
13 strfv2d.f . . 3  |-  ( ph  ->  Fun  `' `' S
)
14 strfv2d.n . . . . 5  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  S )
15 strfv2d.c . . . . . . 7  |-  ( ph  ->  C  e.  W )
1615elexd 2765 . . . . . 6  |-  ( ph  ->  C  e.  _V )
179, 16opelxpd 4680 . . . . 5  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
1814, 17elind 3335 . . . 4  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  ( S  i^i  ( _V  X.  _V ) ) )
19 cnvcnv 5102 . . . 4  |-  `' `' S  =  ( S  i^i  ( _V  X.  _V ) )
2018, 19eleqtrrdi 2283 . . 3  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  `' `' S )
21 funopfv 5579 . . 3  |-  ( Fun  `' `' S  ->  ( <.
( E `  ndx ) ,  C >.  e.  `' `' S  ->  ( `' `' S `  ( E `
 ndx ) )  =  C ) )
2213, 20, 21sylc 62 . 2  |-  ( ph  ->  ( `' `' S `  ( E `  ndx ) )  =  C )
236, 12, 223eqtr2rd 2229 1  |-  ( ph  ->  C  =  ( E `
 S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   _Vcvv 2752    i^i cin 3143   <.cop 3613    X. cxp 4645   `'ccnv 4646    |` cres 4649   Fun wfun 5232   ` cfv 5238   NNcn 8954   ndxcnx 12520  Slot cslot 12522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-iota 5199  df-fun 5240  df-fv 5246  df-slot 12527
This theorem is referenced by:  strslfv2  12567  strslfv  12568  opelstrsl  12637
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