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

Theorem setsslnid 12483
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 12470 . . . . 5  |-  ( ( W  e.  A  /\  D  e.  NN  /\  C  e.  V )  ->  (
( W sSet  <. D ,  C >. )  |`  ( _V  \  { D }
) )  =  ( W  |`  ( _V  \  { D } ) ) )
31, 2mp3an2 1325 . . . 4  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( W sSet  <. D ,  C >. )  |`  ( _V  \  { D } ) )  =  ( W  |`  ( _V  \  { D }
) ) )
43fveq1d 5512 . . 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 113 . . . . . 6  |-  ( E `
 ndx )  e.  NN
76elexi 2749 . . . . 5  |-  ( E `
 ndx )  e. 
_V
8 setsslnid.n . . . . 5  |-  ( E `
 ndx )  =/= 
D
9 eldifsn 3718 . . . . 5  |-  ( ( E `  ndx )  e.  ( _V  \  { D } )  <->  ( ( E `  ndx )  e. 
_V  /\  ( E `  ndx )  =/=  D
) )
107, 8, 9mpbir2an 942 . . . 4  |-  ( E `
 ndx )  e.  ( _V  \  { D } )
11 fvres 5534 . . . 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 5534 . . . 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 2233 . 2  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( W sSet  <. D ,  C >. ) `  ( E `  ndx ) )  =  ( W `  ( E `
 ndx ) ) )
165simpli 111 . . 3  |-  E  = Slot  ( E `  ndx )
17 setsex 12464 . . . 4  |-  ( ( W  e.  A  /\  D  e.  NN  /\  C  e.  V )  ->  ( W sSet  <. D ,  C >. )  e.  _V )
181, 17mp3an2 1325 . . 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 12452 . 2  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( E `  ( W sSet  <. D ,  C >. ) )  =  ( ( W sSet  <. D ,  C >. ) `  ( E `  ndx ) ) )
21 simpl 109 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  W  e.  A )
2216, 21, 19strnfvnd 12452 . 2  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( E `  W
)  =  ( W `
 ( E `  ndx ) ) )
2315, 20, 223eqtr4rd 2221 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 104    = wceq 1353    e. wcel 2148    =/= wne 2347   _Vcvv 2737    \ cdif 3126   {csn 3591   <.cop 3594    |` cres 4624   ` cfv 5211  (class class class)co 5868   NNcn 8895   ndxcnx 12429   sSet csts 12430  Slot cslot 12431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-slot 12436  df-sets 12439
This theorem is referenced by:  resseqnbasd  12501  mgpbasg  12950  mgpscag  12951  mgptsetg  12952  mgpdsg  12954  opprsllem  13058  setsmsbasg  13612  setsmsdsg  13613
  Copyright terms: Public domain W3C validator