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Theorem strslfv2d 11701
Description: Deduction version of strslfv 11703. (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 11679 . 2  |-  ( ph  ->  ( E `  S
)  =  ( S `
 ( E `  ndx ) ) )
7 cnvcnv2 4918 . . . 4  |-  `' `' S  =  ( S  |` 
_V )
87fveq1i 5341 . . 3  |-  ( `' `' S `  ( E `
 ndx ) )  =  ( ( S  |`  _V ) `  ( E `  ndx ) )
95elexd 2646 . . . 4  |-  ( ph  ->  ( E `  ndx )  e.  _V )
10 fvres 5364 . . . 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 2139 . 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 2646 . . . . . 6  |-  ( ph  ->  C  e.  _V )
179, 16opelxpd 4500 . . . . 5  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
1814, 17elind 3200 . . . 4  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  ( S  i^i  ( _V  X.  _V ) ) )
19 cnvcnv 4917 . . . 4  |-  `' `' S  =  ( S  i^i  ( _V  X.  _V ) )
2018, 19syl6eleqr 2188 . . 3  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  `' `' S )
21 funopfv 5379 . . 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 2134 1  |-  ( ph  ->  C  =  ( E `
 S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1296    e. wcel 1445   _Vcvv 2633    i^i cin 3012   <.cop 3469    X. cxp 4465   `'ccnv 4466    |` cres 4469   Fun wfun 5043   ` cfv 5049   NNcn 8520   ndxcnx 11656  Slot cslot 11658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-iota 5014  df-fun 5051  df-fv 5057  df-slot 11663
This theorem is referenced by:  strslfv2  11702  strslfv  11703  opelstrsl  11755
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