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Theorem setsnid 15213
Description: Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
setsid.e  |-  E  = Slot  ( E `  ndx )
setsnid.n  |-  ( E `
 ndx )  =/= 
D
Assertion
Ref Expression
setsnid  |-  ( E `
 W )  =  ( E `  ( W sSet  <. D ,  C >. ) )

Proof of Theorem setsnid
StepHypRef Expression
1 setsid.e . . . 4  |-  E  = Slot  ( E `  ndx )
2 id 22 . . . 4  |-  ( W  e.  _V  ->  W  e.  _V )
31, 2strfvnd 15184 . . 3  |-  ( W  e.  _V  ->  ( E `  W )  =  ( W `  ( E `  ndx )
) )
4 ovex 6342 . . . . 5  |-  ( W sSet  <. D ,  C >. )  e.  _V
54, 1strfvn 15186 . . . 4  |-  ( E `
 ( W sSet  <. D ,  C >. )
)  =  ( ( W sSet  <. D ,  C >. ) `  ( E `
 ndx ) )
6 setsres 15199 . . . . . 6  |-  ( W  e.  _V  ->  (
( W sSet  <. D ,  C >. )  |`  ( _V  \  { D }
) )  =  ( W  |`  ( _V  \  { D } ) ) )
76fveq1d 5889 . . . . 5  |-  ( W  e.  _V  ->  (
( ( W sSet  <. D ,  C >. )  |`  ( _V  \  { D } ) ) `  ( E `  ndx )
)  =  ( ( W  |`  ( _V  \  { D } ) ) `  ( E `
 ndx ) ) )
8 fvex 5897 . . . . . . 7  |-  ( E `
 ndx )  e. 
_V
9 setsnid.n . . . . . . 7  |-  ( E `
 ndx )  =/= 
D
10 eldifsn 4109 . . . . . . 7  |-  ( ( E `  ndx )  e.  ( _V  \  { D } )  <->  ( ( E `  ndx )  e. 
_V  /\  ( E `  ndx )  =/=  D
) )
118, 9, 10mpbir2an 936 . . . . . 6  |-  ( E `
 ndx )  e.  ( _V  \  { D } )
12 fvres 5901 . . . . . 6  |-  ( ( E `  ndx )  e.  ( _V  \  { D } )  ->  (
( ( W sSet  <. D ,  C >. )  |`  ( _V  \  { D } ) ) `  ( E `  ndx )
)  =  ( ( W sSet  <. D ,  C >. ) `  ( E `
 ndx ) ) )
1311, 12ax-mp 5 . . . . 5  |-  ( ( ( W sSet  <. D ,  C >. )  |`  ( _V  \  { D }
) ) `  ( E `  ndx ) )  =  ( ( W sSet  <. D ,  C >. ) `
 ( E `  ndx ) )
14 fvres 5901 . . . . . 6  |-  ( ( E `  ndx )  e.  ( _V  \  { D } )  ->  (
( W  |`  ( _V  \  { D }
) ) `  ( E `  ndx ) )  =  ( W `  ( E `  ndx )
) )
1511, 14ax-mp 5 . . . . 5  |-  ( ( W  |`  ( _V  \  { D } ) ) `  ( E `
 ndx ) )  =  ( W `  ( E `  ndx )
)
167, 13, 153eqtr3g 2518 . . . 4  |-  ( W  e.  _V  ->  (
( W sSet  <. D ,  C >. ) `  ( E `  ndx ) )  =  ( W `  ( E `  ndx )
) )
175, 16syl5eq 2507 . . 3  |-  ( W  e.  _V  ->  ( E `  ( W sSet  <. D ,  C >. ) )  =  ( W `
 ( E `  ndx ) ) )
183, 17eqtr4d 2498 . 2  |-  ( W  e.  _V  ->  ( E `  W )  =  ( E `  ( W sSet  <. D ,  C >. ) ) )
191str0 15209 . . 3  |-  (/)  =  ( E `  (/) )
20 fvprc 5881 . . 3  |-  ( -.  W  e.  _V  ->  ( E `  W )  =  (/) )
21 reldmsets 15192 . . . . 5  |-  Rel  dom sSet
2221ovprc1 6345 . . . 4  |-  ( -.  W  e.  _V  ->  ( W sSet  <. D ,  C >. )  =  (/) )
2322fveq2d 5891 . . 3  |-  ( -.  W  e.  _V  ->  ( E `  ( W sSet  <. D ,  C >. ) )  =  ( E `
 (/) ) )
2419, 20, 233eqtr4a 2521 . 2  |-  ( -.  W  e.  _V  ->  ( E `  W )  =  ( E `  ( W sSet  <. D ,  C >. ) ) )
2518, 24pm2.61i 169 1  |-  ( E `
 W )  =  ( E `  ( W sSet  <. D ,  C >. ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1454    e. wcel 1897    =/= wne 2632   _Vcvv 3056    \ cdif 3412   (/)c0 3742   {csn 3979   <.cop 3985    |` cres 4854   ` cfv 5600  (class class class)co 6314   ndxcnx 15166   sSet csts 15167  Slot cslot 15168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-res 4864  df-iota 5564  df-fun 5602  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-slot 15173  df-sets 15175
This theorem is referenced by:  resslem  15230  oppchomfval  15667  oppcbas  15671  rescbas  15782  rescco  15785  rescabs  15786  odubas  16427  oppglem  17049  mgplem  17776  opprlem  17904  sralem  18448  srasca  18452  sravsca  18453  opsrbaslem  18749  zlmlem  19136  zlmsca  19140  znbaslem  19157  thlbas  19307  thlle  19308  matbas  19486  matplusg  19487  matsca  19488  matvsca  19489  tuslem  21330  setsmsbas  21538  setsmsds  21539  tnglem  21696  tngds  21704  ttgval  24953  ttglem  24954  cchhllem  24965  resvlem  28642  zlmds  28816  zlmtset  28817  hlhilslem  35553  uhgrepe  39962  cznrnglem  40227  cznabel  40228  cznrng  40229
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