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Theorem setsnid 13188
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 19 . . . 4  |-  ( W  e.  _V  ->  W  e.  _V )
31, 2strfvnd 13163 . . 3  |-  ( W  e.  _V  ->  ( E `  W )  =  ( W `  ( E `  ndx )
) )
4 ovex 5883 . . . . 5  |-  ( W sSet  <. D ,  C >. )  e.  _V
54, 1strfvn 13165 . . . 4  |-  ( E `
 ( W sSet  <. D ,  C >. )
)  =  ( ( W sSet  <. D ,  C >. ) `  ( E `
 ndx ) )
6 setsres 13174 . . . . . 6  |-  ( W  e.  _V  ->  (
( W sSet  <. D ,  C >. )  |`  ( _V  \  { D }
) )  =  ( W  |`  ( _V  \  { D } ) ) )
76fveq1d 5527 . . . . 5  |-  ( W  e.  _V  ->  (
( ( W sSet  <. D ,  C >. )  |`  ( _V  \  { D } ) ) `  ( E `  ndx )
)  =  ( ( W  |`  ( _V  \  { D } ) ) `  ( E `
 ndx ) ) )
8 fvex 5539 . . . . . . 7  |-  ( E `
 ndx )  e. 
_V
9 setsnid.n . . . . . . 7  |-  ( E `
 ndx )  =/= 
D
10 eldifsn 3749 . . . . . . 7  |-  ( ( E `  ndx )  e.  ( _V  \  { D } )  <->  ( ( E `  ndx )  e. 
_V  /\  ( E `  ndx )  =/=  D
) )
118, 9, 10mpbir2an 886 . . . . . 6  |-  ( E `
 ndx )  e.  ( _V  \  { D } )
12 fvres 5542 . . . . . 6  |-  ( ( E `  ndx )  e.  ( _V  \  { D } )  ->  (
( ( W sSet  <. D ,  C >. )  |`  ( _V  \  { D } ) ) `  ( E `  ndx )
)  =  ( ( W sSet  <. D ,  C >. ) `  ( E `
 ndx ) ) )
1311, 12ax-mp 8 . . . . 5  |-  ( ( ( W sSet  <. D ,  C >. )  |`  ( _V  \  { D }
) ) `  ( E `  ndx ) )  =  ( ( W sSet  <. D ,  C >. ) `
 ( E `  ndx ) )
14 fvres 5542 . . . . . 6  |-  ( ( E `  ndx )  e.  ( _V  \  { D } )  ->  (
( W  |`  ( _V  \  { D }
) ) `  ( E `  ndx ) )  =  ( W `  ( E `  ndx )
) )
1511, 14ax-mp 8 . . . . 5  |-  ( ( W  |`  ( _V  \  { D } ) ) `  ( E `
 ndx ) )  =  ( W `  ( E `  ndx )
)
167, 13, 153eqtr3g 2338 . . . 4  |-  ( W  e.  _V  ->  (
( W sSet  <. D ,  C >. ) `  ( E `  ndx ) )  =  ( W `  ( E `  ndx )
) )
175, 16syl5eq 2327 . . 3  |-  ( W  e.  _V  ->  ( E `  ( W sSet  <. D ,  C >. ) )  =  ( W `
 ( E `  ndx ) ) )
183, 17eqtr4d 2318 . 2  |-  ( W  e.  _V  ->  ( E `  W )  =  ( E `  ( W sSet  <. D ,  C >. ) ) )
191str0 13184 . . 3  |-  (/)  =  ( E `  (/) )
20 fvprc 5519 . . 3  |-  ( -.  W  e.  _V  ->  ( E `  W )  =  (/) )
21 reldmsets 13170 . . . . 5  |-  Rel  dom sSet
2221ovprc1 5886 . . . 4  |-  ( -.  W  e.  _V  ->  ( W sSet  <. D ,  C >. )  =  (/) )
2322fveq2d 5529 . . 3  |-  ( -.  W  e.  _V  ->  ( E `  ( W sSet  <. D ,  C >. ) )  =  ( E `
 (/) ) )
2419, 20, 233eqtr4a 2341 . 2  |-  ( -.  W  e.  _V  ->  ( E `  W )  =  ( E `  ( W sSet  <. D ,  C >. ) ) )
2518, 24pm2.61i 156 1  |-  ( E `
 W )  =  ( E `  ( W sSet  <. D ,  C >. ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    \ cdif 3149   (/)c0 3455   {csn 3640   <.cop 3643    |` cres 4691   ` cfv 5255  (class class class)co 5858   ndxcnx 13145   sSet csts 13146  Slot cslot 13147
This theorem is referenced by:  resslem  13201  oppchomfval  13617  oppcbas  13621  rescbas  13706  rescco  13709  rescabs  13710  odubas  14237  oppglem  14823  mgplem  15330  opprlem  15410  sralem  15930  srasca  15934  opsrbaslem  16219  zlmlem  16471  zlmsca  16475  znbaslem  16492  thlbas  16596  thlle  16597  setsmsbas  18021  setsmsds  18022  tnglem  18156  tngds  18164  matbas  27468  matplusg  27469  matsca  27470  matvsca  27471  hlhilslem  32131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-slot 13152  df-sets 13154
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