ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  setsslnid Unicode version

Theorem setsslnid 12036
Description: Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.)
Hypotheses
Ref Expression
setsslid.e  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
setsslnid.n  |-  ( E `
 ndx )  =/= 
D
setsslnid.d  |-  D  e.  NN
Assertion
Ref Expression
setsslnid  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( E `  W
)  =  ( E `
 ( W sSet  <. D ,  C >. )
) )

Proof of Theorem setsslnid
StepHypRef Expression
1 setsslnid.d . . . . 5  |-  D  e.  NN
2 setsresg 12023 . . . . 5  |-  ( ( W  e.  A  /\  D  e.  NN  /\  C  e.  V )  ->  (
( W sSet  <. D ,  C >. )  |`  ( _V  \  { D }
) )  =  ( W  |`  ( _V  \  { D } ) ) )
31, 2mp3an2 1303 . . . 4  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( W sSet  <. D ,  C >. )  |`  ( _V  \  { D } ) )  =  ( W  |`  ( _V  \  { D }
) ) )
43fveq1d 5426 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( W sSet  <. D ,  C >. )  |`  ( _V  \  { D } ) ) `  ( E `  ndx )
)  =  ( ( W  |`  ( _V  \  { D } ) ) `  ( E `
 ndx ) ) )
5 setsslid.e . . . . . . 7  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
65simpri 112 . . . . . 6  |-  ( E `
 ndx )  e.  NN
76elexi 2698 . . . . 5  |-  ( E `
 ndx )  e. 
_V
8 setsslnid.n . . . . 5  |-  ( E `
 ndx )  =/= 
D
9 eldifsn 3653 . . . . 5  |-  ( ( E `  ndx )  e.  ( _V  \  { D } )  <->  ( ( E `  ndx )  e. 
_V  /\  ( E `  ndx )  =/=  D
) )
107, 8, 9mpbir2an 926 . . . 4  |-  ( E `
 ndx )  e.  ( _V  \  { D } )
11 fvres 5448 . . . 4  |-  ( ( E `  ndx )  e.  ( _V  \  { D } )  ->  (
( ( W sSet  <. D ,  C >. )  |`  ( _V  \  { D } ) ) `  ( E `  ndx )
)  =  ( ( W sSet  <. D ,  C >. ) `  ( E `
 ndx ) ) )
1210, 11ax-mp 5 . . 3  |-  ( ( ( W sSet  <. D ,  C >. )  |`  ( _V  \  { D }
) ) `  ( E `  ndx ) )  =  ( ( W sSet  <. D ,  C >. ) `
 ( E `  ndx ) )
13 fvres 5448 . . . 4  |-  ( ( E `  ndx )  e.  ( _V  \  { D } )  ->  (
( W  |`  ( _V  \  { D }
) ) `  ( E `  ndx ) )  =  ( W `  ( E `  ndx )
) )
1410, 13ax-mp 5 . . 3  |-  ( ( W  |`  ( _V  \  { D } ) ) `  ( E `
 ndx ) )  =  ( W `  ( E `  ndx )
)
154, 12, 143eqtr3g 2195 . 2  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( W sSet  <. D ,  C >. ) `  ( E `  ndx ) )  =  ( W `  ( E `
 ndx ) ) )
165simpli 110 . . 3  |-  E  = Slot  ( E `  ndx )
17 setsex 12017 . . . 4  |-  ( ( W  e.  A  /\  D  e.  NN  /\  C  e.  V )  ->  ( W sSet  <. D ,  C >. )  e.  _V )
181, 17mp3an2 1303 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( W sSet  <. D ,  C >. )  e.  _V )
196a1i 9 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( E `  ndx )  e.  NN )
2016, 18, 19strnfvnd 12005 . 2  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( E `  ( W sSet  <. D ,  C >. ) )  =  ( ( W sSet  <. D ,  C >. ) `  ( E `  ndx ) ) )
21 simpl 108 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  W  e.  A )
2216, 21, 19strnfvnd 12005 . 2  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( E `  W
)  =  ( W `
 ( E `  ndx ) ) )
2315, 20, 223eqtr4rd 2183 1  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( E `  W
)  =  ( E `
 ( W sSet  <. D ,  C >. )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480    =/= wne 2308   _Vcvv 2686    \ cdif 3068   {csn 3527   <.cop 3530    |` cres 4544   ` cfv 5126  (class class class)co 5777   NNcn 8739   ndxcnx 11982   sSet csts 11983  Slot cslot 11984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-br 3933  df-opab 3993  df-mpt 3994  df-id 4218  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-iota 5091  df-fun 5128  df-fv 5134  df-ov 5780  df-oprab 5781  df-mpo 5782  df-slot 11989  df-sets 11992
This theorem is referenced by:  setsmsbasg  12674  setsmsdsg  12675
  Copyright terms: Public domain W3C validator