MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  setsnid Unicode version

Theorem setsnid 13204
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 13179 . . 3  |-  ( W  e.  _V  ->  ( E `  W )  =  ( W `  ( E `  ndx )
) )
4 ovex 5899 . . . . 5  |-  ( W sSet  <. D ,  C >. )  e.  _V
54, 1strfvn 13181 . . . 4  |-  ( E `
 ( W sSet  <. D ,  C >. )
)  =  ( ( W sSet  <. D ,  C >. ) `  ( E `
 ndx ) )
6 setsres 13190 . . . . . 6  |-  ( W  e.  _V  ->  (
( W sSet  <. D ,  C >. )  |`  ( _V  \  { D }
) )  =  ( W  |`  ( _V  \  { D } ) ) )
76fveq1d 5543 . . . . 5  |-  ( W  e.  _V  ->  (
( ( W sSet  <. D ,  C >. )  |`  ( _V  \  { D } ) ) `  ( E `  ndx )
)  =  ( ( W  |`  ( _V  \  { D } ) ) `  ( E `
 ndx ) ) )
8 fvex 5555 . . . . . . 7  |-  ( E `
 ndx )  e. 
_V
9 setsnid.n . . . . . . 7  |-  ( E `
 ndx )  =/= 
D
10 eldifsn 3762 . . . . . . 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 5558 . . . . . 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 5558 . . . . . 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 2351 . . . 4  |-  ( W  e.  _V  ->  (
( W sSet  <. D ,  C >. ) `  ( E `  ndx ) )  =  ( W `  ( E `  ndx )
) )
175, 16syl5eq 2340 . . 3  |-  ( W  e.  _V  ->  ( E `  ( W sSet  <. D ,  C >. ) )  =  ( W `
 ( E `  ndx ) ) )
183, 17eqtr4d 2331 . 2  |-  ( W  e.  _V  ->  ( E `  W )  =  ( E `  ( W sSet  <. D ,  C >. ) ) )
191str0 13200 . . 3  |-  (/)  =  ( E `  (/) )
20 fvprc 5535 . . 3  |-  ( -.  W  e.  _V  ->  ( E `  W )  =  (/) )
21 reldmsets 13186 . . . . 5  |-  Rel  dom sSet
2221ovprc1 5902 . . . 4  |-  ( -.  W  e.  _V  ->  ( W sSet  <. D ,  C >. )  =  (/) )
2322fveq2d 5545 . . 3  |-  ( -.  W  e.  _V  ->  ( E `  ( W sSet  <. D ,  C >. ) )  =  ( E `
 (/) ) )
2419, 20, 233eqtr4a 2354 . 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 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    \ cdif 3162   (/)c0 3468   {csn 3653   <.cop 3656    |` cres 4707   ` cfv 5271  (class class class)co 5874   ndxcnx 13161   sSet csts 13162  Slot cslot 13163
This theorem is referenced by:  resslem  13217  oppchomfval  13633  oppcbas  13637  rescbas  13722  rescco  13725  rescabs  13726  odubas  14253  oppglem  14839  mgplem  15346  opprlem  15426  sralem  15946  srasca  15950  opsrbaslem  16235  zlmlem  16487  zlmsca  16491  znbaslem  16508  thlbas  16612  thlle  16613  setsmsbas  18037  setsmsds  18038  tnglem  18172  tngds  18180  matbas  27571  matplusg  27572  matsca  27573  matvsca  27574  hlhilslem  32753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-slot 13168  df-sets 13170
  Copyright terms: Public domain W3C validator