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Theorem strslfv2d 11928
Description: Deduction version of strslfv 11930. (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 11906 . 2  |-  ( ph  ->  ( E `  S
)  =  ( S `
 ( E `  ndx ) ) )
7 cnvcnv2 4962 . . . 4  |-  `' `' S  =  ( S  |` 
_V )
87fveq1i 5390 . . 3  |-  ( `' `' S `  ( E `
 ndx ) )  =  ( ( S  |`  _V ) `  ( E `  ndx ) )
95elexd 2673 . . . 4  |-  ( ph  ->  ( E `  ndx )  e.  _V )
10 fvres 5413 . . . 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 2162 . 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 2673 . . . . . 6  |-  ( ph  ->  C  e.  _V )
179, 16opelxpd 4542 . . . . 5  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
1814, 17elind 3231 . . . 4  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  ( S  i^i  ( _V  X.  _V ) ) )
19 cnvcnv 4961 . . . 4  |-  `' `' S  =  ( S  i^i  ( _V  X.  _V ) )
2018, 19eleqtrrdi 2211 . . 3  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  `' `' S )
21 funopfv 5429 . . 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 2157 1  |-  ( ph  ->  C  =  ( E `
 S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   _Vcvv 2660    i^i cin 3040   <.cop 3500    X. cxp 4507   `'ccnv 4508    |` cres 4511   Fun wfun 5087   ` cfv 5093   NNcn 8688   ndxcnx 11883  Slot cslot 11885
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-iota 5058  df-fun 5095  df-fv 5101  df-slot 11890
This theorem is referenced by:  strslfv2  11929  strslfv  11930  opelstrsl  11982
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