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Theorem setsnid 14532
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 14505 . . 3  |-  ( W  e.  _V  ->  ( E `  W )  =  ( W `  ( E `  ndx )
) )
4 ovex 6309 . . . . 5  |-  ( W sSet  <. D ,  C >. )  e.  _V
54, 1strfvn 14507 . . . 4  |-  ( E `
 ( W sSet  <. D ,  C >. )
)  =  ( ( W sSet  <. D ,  C >. ) `  ( E `
 ndx ) )
6 setsres 14518 . . . . . 6  |-  ( W  e.  _V  ->  (
( W sSet  <. D ,  C >. )  |`  ( _V  \  { D }
) )  =  ( W  |`  ( _V  \  { D } ) ) )
76fveq1d 5868 . . . . 5  |-  ( W  e.  _V  ->  (
( ( W sSet  <. D ,  C >. )  |`  ( _V  \  { D } ) ) `  ( E `  ndx )
)  =  ( ( W  |`  ( _V  \  { D } ) ) `  ( E `
 ndx ) ) )
8 fvex 5876 . . . . . . 7  |-  ( E `
 ndx )  e. 
_V
9 setsnid.n . . . . . . 7  |-  ( E `
 ndx )  =/= 
D
10 eldifsn 4152 . . . . . . 7  |-  ( ( E `  ndx )  e.  ( _V  \  { D } )  <->  ( ( E `  ndx )  e. 
_V  /\  ( E `  ndx )  =/=  D
) )
118, 9, 10mpbir2an 918 . . . . . 6  |-  ( E `
 ndx )  e.  ( _V  \  { D } )
12 fvres 5880 . . . . . 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 5880 . . . . . 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 2531 . . . 4  |-  ( W  e.  _V  ->  (
( W sSet  <. D ,  C >. ) `  ( E `  ndx ) )  =  ( W `  ( E `  ndx )
) )
175, 16syl5eq 2520 . . 3  |-  ( W  e.  _V  ->  ( E `  ( W sSet  <. D ,  C >. ) )  =  ( W `
 ( E `  ndx ) ) )
183, 17eqtr4d 2511 . 2  |-  ( W  e.  _V  ->  ( E `  W )  =  ( E `  ( W sSet  <. D ,  C >. ) ) )
191str0 14528 . . 3  |-  (/)  =  ( E `  (/) )
20 fvprc 5860 . . 3  |-  ( -.  W  e.  _V  ->  ( E `  W )  =  (/) )
21 reldmsets 14512 . . . . 5  |-  Rel  dom sSet
2221ovprc1 6312 . . . 4  |-  ( -.  W  e.  _V  ->  ( W sSet  <. D ,  C >. )  =  (/) )
2322fveq2d 5870 . . 3  |-  ( -.  W  e.  _V  ->  ( E `  ( W sSet  <. D ,  C >. ) )  =  ( E `
 (/) ) )
2419, 20, 233eqtr4a 2534 . 2  |-  ( -.  W  e.  _V  ->  ( E `  W )  =  ( E `  ( W sSet  <. D ,  C >. ) ) )
2518, 24pm2.61i 164 1  |-  ( E `
 W )  =  ( E `  ( W sSet  <. D ,  C >. ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    \ cdif 3473   (/)c0 3785   {csn 4027   <.cop 4033    |` cres 5001   ` cfv 5588  (class class class)co 6284   ndxcnx 14487   sSet csts 14488  Slot cslot 14489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-slot 14494  df-sets 14496
This theorem is referenced by:  resslem  14548  oppchomfval  14970  oppcbas  14974  rescbas  15059  rescco  15062  rescabs  15063  odubas  15620  oppglem  16190  mgplem  16948  opprlem  17078  sralem  17623  srasca  17627  sravsca  17628  opsrbaslem  17941  zlmlem  18349  zlmsca  18353  znbaslem  18372  thlbas  18522  thlle  18523  matbas  18710  matplusg  18711  matsca  18712  matvsca  18713  tuslem  20533  setsmsbas  20741  setsmsds  20742  tnglem  20917  tngds  20925  ttgval  23882  ttglem  23883  cchhllem  23894  resvlem  27512  zlmds  27609  zlmtset  27610  uhgrepe  31873  hlhilslem  36756
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