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

Theorem setsslid 12310
Description: Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.)
Hypothesis
Ref Expression
setsslid.e  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
Assertion
Ref Expression
setsslid  |-  ( ( W  e.  A  /\  C  e.  V )  ->  C  =  ( E `
 ( W sSet  <. ( E `  ndx ) ,  C >. ) ) )

Proof of Theorem setsslid
StepHypRef Expression
1 setsslid.e . . . . 5  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
21simpri 112 . . . 4  |-  ( E `
 ndx )  e.  NN
3 setsvala 12291 . . . 4  |-  ( ( W  e.  A  /\  ( E `  ndx )  e.  NN  /\  C  e.  V )  ->  ( W sSet  <. ( E `  ndx ) ,  C >. )  =  ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } ) )
42, 3mp3an2 1307 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( W sSet  <. ( E `  ndx ) ,  C >. )  =  ( ( W  |`  ( _V  \  { ( E `
 ndx ) } ) )  u.  { <. ( E `  ndx ) ,  C >. } ) )
54fveq2d 5474 . 2  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( E `  ( W sSet  <. ( E `  ndx ) ,  C >. ) )  =  ( E `
 ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } ) ) )
61simpli 110 . . 3  |-  E  = Slot  ( E `  ndx )
7 resexg 4908 . . . 4  |-  ( W  e.  A  ->  ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  e.  _V )
8 simpr 109 . . . . . 6  |-  ( ( W  e.  A  /\  C  e.  V )  ->  C  e.  V )
9 opexg 4190 . . . . . 6  |-  ( ( ( E `  ndx )  e.  NN  /\  C  e.  V )  ->  <. ( E `  ndx ) ,  C >.  e.  _V )
102, 8, 9sylancr 411 . . . . 5  |-  ( ( W  e.  A  /\  C  e.  V )  -> 
<. ( E `  ndx ) ,  C >.  e. 
_V )
11 snexg 4147 . . . . 5  |-  ( <.
( E `  ndx ) ,  C >.  e. 
_V  ->  { <. ( E `  ndx ) ,  C >. }  e.  _V )
1210, 11syl 14 . . . 4  |-  ( ( W  e.  A  /\  C  e.  V )  ->  { <. ( E `  ndx ) ,  C >. }  e.  _V )
13 unexg 4405 . . . 4  |-  ( ( ( W  |`  ( _V  \  { ( E `
 ndx ) } ) )  e.  _V  /\ 
{ <. ( E `  ndx ) ,  C >. }  e.  _V )  -> 
( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } )  e.  _V )
147, 12, 13syl2an2r 585 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } )  e.  _V )
152a1i 9 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( E `  ndx )  e.  NN )
166, 14, 15strnfvnd 12280 . 2  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( E `  (
( W  |`  ( _V  \  { ( E `
 ndx ) } ) )  u.  { <. ( E `  ndx ) ,  C >. } ) )  =  ( ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } ) `  ( E `
 ndx ) ) )
17 snidg 3590 . . . . 5  |-  ( ( E `  ndx )  e.  NN  ->  ( E `  ndx )  e.  {
( E `  ndx ) } )
18 fvres 5494 . . . . 5  |-  ( ( E `  ndx )  e.  { ( E `  ndx ) }  ->  (
( ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } )  |`  { ( E `  ndx ) } ) `  ( E `
 ndx ) )  =  ( ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  u.  { <. ( E `  ndx ) ,  C >. } ) `  ( E `  ndx )
) )
192, 17, 18mp2b 8 . . . 4  |-  ( ( ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } )  |`  { ( E `  ndx ) } ) `  ( E `
 ndx ) )  =  ( ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  u.  { <. ( E `  ndx ) ,  C >. } ) `  ( E `  ndx )
)
20 resres 4880 . . . . . . . . 9  |-  ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  |`  { ( E `  ndx ) } )  =  ( W  |`  (
( _V  \  {
( E `  ndx ) } )  i^i  {
( E `  ndx ) } ) )
21 incom 3300 . . . . . . . . . . . 12  |-  ( ( _V  \  { ( E `  ndx ) } )  i^i  {
( E `  ndx ) } )  =  ( { ( E `  ndx ) }  i^i  ( _V  \  { ( E `
 ndx ) } ) )
22 disjdif 3467 . . . . . . . . . . . 12  |-  ( { ( E `  ndx ) }  i^i  ( _V  \  { ( E `
 ndx ) } ) )  =  (/)
2321, 22eqtri 2178 . . . . . . . . . . 11  |-  ( ( _V  \  { ( E `  ndx ) } )  i^i  {
( E `  ndx ) } )  =  (/)
2423reseq2i 4865 . . . . . . . . . 10  |-  ( W  |`  ( ( _V  \  { ( E `  ndx ) } )  i^i 
{ ( E `  ndx ) } ) )  =  ( W  |`  (/) )
25 res0 4872 . . . . . . . . . 10  |-  ( W  |`  (/) )  =  (/)
2624, 25eqtri 2178 . . . . . . . . 9  |-  ( W  |`  ( ( _V  \  { ( E `  ndx ) } )  i^i 
{ ( E `  ndx ) } ) )  =  (/)
2720, 26eqtri 2178 . . . . . . . 8  |-  ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  |`  { ( E `  ndx ) } )  =  (/)
2827a1i 9 . . . . . . 7  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  |`  { ( E `  ndx ) } )  =  (/) )
292elexi 2724 . . . . . . . . . 10  |-  ( E `
 ndx )  e. 
_V
308elexd 2725 . . . . . . . . . 10  |-  ( ( W  e.  A  /\  C  e.  V )  ->  C  e.  _V )
31 opelxpi 4620 . . . . . . . . . 10  |-  ( ( ( E `  ndx )  e.  _V  /\  C  e.  _V )  ->  <. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
3229, 30, 31sylancr 411 . . . . . . . . 9  |-  ( ( W  e.  A  /\  C  e.  V )  -> 
<. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
33 relsng 4691 . . . . . . . . . 10  |-  ( <.
( E `  ndx ) ,  C >.  e. 
_V  ->  ( Rel  { <. ( E `  ndx ) ,  C >. }  <->  <. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) ) )
3410, 33syl 14 . . . . . . . . 9  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( Rel  { <. ( E `  ndx ) ,  C >. }  <->  <. ( E `
 ndx ) ,  C >.  e.  ( _V  X.  _V ) ) )
3532, 34mpbird 166 . . . . . . . 8  |-  ( ( W  e.  A  /\  C  e.  V )  ->  Rel  { <. ( E `  ndx ) ,  C >. } )
36 dmsnopg 5059 . . . . . . . . . 10  |-  ( C  e.  V  ->  dom  {
<. ( E `  ndx ) ,  C >. }  =  { ( E `
 ndx ) } )
3736adantl 275 . . . . . . . . 9  |-  ( ( W  e.  A  /\  C  e.  V )  ->  dom  { <. ( E `  ndx ) ,  C >. }  =  {
( E `  ndx ) } )
38 eqimss 3182 . . . . . . . . 9  |-  ( dom 
{ <. ( E `  ndx ) ,  C >. }  =  { ( E `
 ndx ) }  ->  dom  { <. ( E `  ndx ) ,  C >. }  C_  { ( E `  ndx ) } )
3937, 38syl 14 . . . . . . . 8  |-  ( ( W  e.  A  /\  C  e.  V )  ->  dom  { <. ( E `  ndx ) ,  C >. }  C_  { ( E `  ndx ) } )
40 relssres 4906 . . . . . . . 8  |-  ( ( Rel  { <. ( E `  ndx ) ,  C >. }  /\  dom  {
<. ( E `  ndx ) ,  C >. } 
C_  { ( E `
 ndx ) } )  ->  ( { <. ( E `  ndx ) ,  C >. }  |`  { ( E `  ndx ) } )  =  { <. ( E `  ndx ) ,  C >. } )
4135, 39, 40syl2anc 409 . . . . . . 7  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( { <. ( E `  ndx ) ,  C >. }  |`  { ( E `  ndx ) } )  =  { <. ( E `  ndx ) ,  C >. } )
4228, 41uneq12d 3263 . . . . . 6  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  |`  { ( E `  ndx ) } )  u.  ( { <. ( E `  ndx ) ,  C >. }  |`  { ( E `  ndx ) } ) )  =  ( (/)  u.  { <. ( E `  ndx ) ,  C >. } ) )
43 resundir 4882 . . . . . 6  |-  ( ( ( W  |`  ( _V  \  { ( E `
 ndx ) } ) )  u.  { <. ( E `  ndx ) ,  C >. } )  |`  { ( E `  ndx ) } )  =  ( ( ( W  |`  ( _V  \  { ( E `
 ndx ) } ) )  |`  { ( E `  ndx ) } )  u.  ( { <. ( E `  ndx ) ,  C >. }  |`  { ( E `  ndx ) } ) )
44 un0 3428 . . . . . . 7  |-  ( {
<. ( E `  ndx ) ,  C >. }  u.  (/) )  =  { <. ( E `  ndx ) ,  C >. }
45 uncom 3252 . . . . . . 7  |-  ( {
<. ( E `  ndx ) ,  C >. }  u.  (/) )  =  (
(/)  u.  { <. ( E `  ndx ) ,  C >. } )
4644, 45eqtr3i 2180 . . . . . 6  |-  { <. ( E `  ndx ) ,  C >. }  =  (
(/)  u.  { <. ( E `  ndx ) ,  C >. } )
4742, 43, 463eqtr4g 2215 . . . . 5  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } )  |`  { ( E `  ndx ) } )  =  { <. ( E `  ndx ) ,  C >. } )
4847fveq1d 5472 . . . 4  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  u.  { <. ( E `  ndx ) ,  C >. } )  |`  { ( E `  ndx ) } ) `  ( E `  ndx )
)  =  ( {
<. ( E `  ndx ) ,  C >. } `
 ( E `  ndx ) ) )
4919, 48eqtr3id 2204 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } ) `  ( E `
 ndx ) )  =  ( { <. ( E `  ndx ) ,  C >. } `  ( E `  ndx ) ) )
50 fvsng 5665 . . . 4  |-  ( ( ( E `  ndx )  e.  NN  /\  C  e.  V )  ->  ( { <. ( E `  ndx ) ,  C >. } `
 ( E `  ndx ) )  =  C )
512, 8, 50sylancr 411 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( { <. ( E `  ndx ) ,  C >. } `  ( E `  ndx ) )  =  C )
5249, 51eqtrd 2190 . 2  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } ) `  ( E `
 ndx ) )  =  C )
535, 16, 523eqtrrd 2195 1  |-  ( ( W  e.  A  /\  C  e.  V )  ->  C  =  ( E `
 ( W sSet  <. ( E `  ndx ) ,  C >. ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   _Vcvv 2712    \ cdif 3099    u. cun 3100    i^i cin 3101    C_ wss 3102   (/)c0 3395   {csn 3561   <.cop 3564    X. cxp 4586   dom cdm 4588    |` cres 4590   Rel wrel 4593   ` cfv 5172  (class class class)co 5826   NNcn 8838   ndxcnx 12257   sSet csts 12258  Slot cslot 12259
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-mpt 4029  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-iota 5137  df-fun 5174  df-fv 5180  df-ov 5829  df-oprab 5830  df-mpo 5831  df-slot 12264  df-sets 12267
This theorem is referenced by:  setsmstsetg  12951
  Copyright terms: Public domain W3C validator