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Theorem setsnid 13509
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 20 . . . 4  |-  ( W  e.  _V  ->  W  e.  _V )
31, 2strfvnd 13484 . . 3  |-  ( W  e.  _V  ->  ( E `  W )  =  ( W `  ( E `  ndx )
) )
4 ovex 6106 . . . . 5  |-  ( W sSet  <. D ,  C >. )  e.  _V
54, 1strfvn 13486 . . . 4  |-  ( E `
 ( W sSet  <. D ,  C >. )
)  =  ( ( W sSet  <. D ,  C >. ) `  ( E `
 ndx ) )
6 setsres 13495 . . . . . 6  |-  ( W  e.  _V  ->  (
( W sSet  <. D ,  C >. )  |`  ( _V  \  { D }
) )  =  ( W  |`  ( _V  \  { D } ) ) )
76fveq1d 5730 . . . . 5  |-  ( W  e.  _V  ->  (
( ( W sSet  <. D ,  C >. )  |`  ( _V  \  { D } ) ) `  ( E `  ndx )
)  =  ( ( W  |`  ( _V  \  { D } ) ) `  ( E `
 ndx ) ) )
8 fvex 5742 . . . . . . 7  |-  ( E `
 ndx )  e. 
_V
9 setsnid.n . . . . . . 7  |-  ( E `
 ndx )  =/= 
D
10 eldifsn 3927 . . . . . . 7  |-  ( ( E `  ndx )  e.  ( _V  \  { D } )  <->  ( ( E `  ndx )  e. 
_V  /\  ( E `  ndx )  =/=  D
) )
118, 9, 10mpbir2an 887 . . . . . 6  |-  ( E `
 ndx )  e.  ( _V  \  { D } )
12 fvres 5745 . . . . . 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 5745 . . . . . 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 2491 . . . 4  |-  ( W  e.  _V  ->  (
( W sSet  <. D ,  C >. ) `  ( E `  ndx ) )  =  ( W `  ( E `  ndx )
) )
175, 16syl5eq 2480 . . 3  |-  ( W  e.  _V  ->  ( E `  ( W sSet  <. D ,  C >. ) )  =  ( W `
 ( E `  ndx ) ) )
183, 17eqtr4d 2471 . 2  |-  ( W  e.  _V  ->  ( E `  W )  =  ( E `  ( W sSet  <. D ,  C >. ) ) )
191str0 13505 . . 3  |-  (/)  =  ( E `  (/) )
20 fvprc 5722 . . 3  |-  ( -.  W  e.  _V  ->  ( E `  W )  =  (/) )
21 reldmsets 13491 . . . . 5  |-  Rel  dom sSet
2221ovprc1 6109 . . . 4  |-  ( -.  W  e.  _V  ->  ( W sSet  <. D ,  C >. )  =  (/) )
2322fveq2d 5732 . . 3  |-  ( -.  W  e.  _V  ->  ( E `  ( W sSet  <. D ,  C >. ) )  =  ( E `
 (/) ) )
2419, 20, 233eqtr4a 2494 . 2  |-  ( -.  W  e.  _V  ->  ( E `  W )  =  ( E `  ( W sSet  <. D ,  C >. ) ) )
2518, 24pm2.61i 158 1  |-  ( E `
 W )  =  ( E `  ( W sSet  <. D ,  C >. ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956    \ cdif 3317   (/)c0 3628   {csn 3814   <.cop 3817    |` cres 4880   ` cfv 5454  (class class class)co 6081   ndxcnx 13466   sSet csts 13467  Slot cslot 13468
This theorem is referenced by:  resslem  13522  oppchomfval  13940  oppcbas  13944  rescbas  14029  rescco  14032  rescabs  14033  odubas  14560  oppglem  15146  mgplem  15653  opprlem  15733  sralem  16249  srasca  16253  opsrbaslem  16538  zlmlem  16798  zlmsca  16802  znbaslem  16819  thlbas  16923  thlle  16924  tuslem  18297  setsmsbas  18505  setsmsds  18506  tnglem  18681  tngds  18689  zlmds  24348  zlmtset  24349  matbas  27445  matplusg  27446  matsca  27447  matvsca  27448  hlhilslem  32739
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-res 4890  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-slot 13473  df-sets 13475
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