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Theorem strslfv2d 12015
Description: Deduction version of strslfv 12017. (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 110 . . 3  |-  E  = Slot  ( E `  ndx )
3 strfv2d.s . . 3  |-  ( ph  ->  S  e.  V )
41simpri 112 . . . 4  |-  ( E `
 ndx )  e.  NN
54a1i 9 . . 3  |-  ( ph  ->  ( E `  ndx )  e.  NN )
62, 3, 5strnfvnd 11993 . 2  |-  ( ph  ->  ( E `  S
)  =  ( S `
 ( E `  ndx ) ) )
7 cnvcnv2 4992 . . . 4  |-  `' `' S  =  ( S  |` 
_V )
87fveq1i 5422 . . 3  |-  ( `' `' S `  ( E `
 ndx ) )  =  ( ( S  |`  _V ) `  ( E `  ndx ) )
95elexd 2699 . . . 4  |-  ( ph  ->  ( E `  ndx )  e.  _V )
10 fvres 5445 . . . 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, 11syl5eq 2184 . 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 2699 . . . . . 6  |-  ( ph  ->  C  e.  _V )
179, 16opelxpd 4572 . . . . 5  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
1814, 17elind 3261 . . . 4  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  ( S  i^i  ( _V  X.  _V ) ) )
19 cnvcnv 4991 . . . 4  |-  `' `' S  =  ( S  i^i  ( _V  X.  _V ) )
2018, 19eleqtrrdi 2233 . . 3  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  `' `' S )
21 funopfv 5461 . . 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 2179 1  |-  ( ph  ->  C  =  ( E `
 S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   _Vcvv 2686    i^i cin 3070   <.cop 3530    X. cxp 4537   `'ccnv 4538    |` cres 4541   Fun wfun 5117   ` cfv 5123   NNcn 8732   ndxcnx 11970  Slot cslot 11972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fv 5131  df-slot 11977
This theorem is referenced by:  strslfv2  12016  strslfv  12017  opelstrsl  12069
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