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Theorem setsid 13501
Description: Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
Hypothesis
Ref Expression
setsid.e  |-  E  = Slot  ( E `  ndx )
Assertion
Ref Expression
setsid  |-  ( ( W  e.  A  /\  C  e.  V )  ->  C  =  ( E `
 ( W sSet  <. ( E `  ndx ) ,  C >. ) ) )

Proof of Theorem setsid
StepHypRef Expression
1 setsval 13486 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( W sSet  <. ( E `  ndx ) ,  C >. )  =  ( ( W  |`  ( _V  \  { ( E `
 ndx ) } ) )  u.  { <. ( E `  ndx ) ,  C >. } ) )
21fveq2d 5725 . 2  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( E `  ( W sSet  <. ( E `  ndx ) ,  C >. ) )  =  ( E `
 ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } ) ) )
3 setsid.e . . 3  |-  E  = Slot  ( E `  ndx )
4 resexg 5178 . . . . 5  |-  ( W  e.  A  ->  ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  e.  _V )
54adantr 452 . . . 4  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( W  |`  ( _V  \  { ( E `
 ndx ) } ) )  e.  _V )
6 snex 4398 . . . 4  |-  { <. ( E `  ndx ) ,  C >. }  e.  _V
7 unexg 4703 . . . 4  |-  ( ( ( W  |`  ( _V  \  { ( E `
 ndx ) } ) )  e.  _V  /\ 
{ <. ( E `  ndx ) ,  C >. }  e.  _V )  -> 
( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } )  e.  _V )
85, 6, 7sylancl 644 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } )  e.  _V )
93, 8strfvnd 13477 . 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 ) ) )
10 fvex 5735 . . . . . 6  |-  ( E `
 ndx )  e. 
_V
1110snid 3834 . . . . 5  |-  ( E `
 ndx )  e. 
{ ( E `  ndx ) }
12 fvres 5738 . . . . 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 )
) )
1311, 12ax-mp 8 . . . 4  |-  ( ( ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } )  |`  { ( E `  ndx ) } ) `  ( E `
 ndx ) )  =  ( ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  u.  { <. ( E `  ndx ) ,  C >. } ) `  ( E `  ndx )
)
14 resres 5152 . . . . . . . . 9  |-  ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  |`  { ( E `  ndx ) } )  =  ( W  |`  (
( _V  \  {
( E `  ndx ) } )  i^i  {
( E `  ndx ) } ) )
15 incom 3526 . . . . . . . . . . . 12  |-  ( ( _V  \  { ( E `  ndx ) } )  i^i  {
( E `  ndx ) } )  =  ( { ( E `  ndx ) }  i^i  ( _V  \  { ( E `
 ndx ) } ) )
16 disjdif 3693 . . . . . . . . . . . 12  |-  ( { ( E `  ndx ) }  i^i  ( _V  \  { ( E `
 ndx ) } ) )  =  (/)
1715, 16eqtri 2456 . . . . . . . . . . 11  |-  ( ( _V  \  { ( E `  ndx ) } )  i^i  {
( E `  ndx ) } )  =  (/)
1817reseq2i 5136 . . . . . . . . . 10  |-  ( W  |`  ( ( _V  \  { ( E `  ndx ) } )  i^i 
{ ( E `  ndx ) } ) )  =  ( W  |`  (/) )
19 res0 5143 . . . . . . . . . 10  |-  ( W  |`  (/) )  =  (/)
2018, 19eqtri 2456 . . . . . . . . 9  |-  ( W  |`  ( ( _V  \  { ( E `  ndx ) } )  i^i 
{ ( E `  ndx ) } ) )  =  (/)
2114, 20eqtri 2456 . . . . . . . 8  |-  ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  |`  { ( E `  ndx ) } )  =  (/)
2221a1i 11 . . . . . . 7  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  |`  { ( E `  ndx ) } )  =  (/) )
23 elex 2957 . . . . . . . . . . 11  |-  ( C  e.  V  ->  C  e.  _V )
2423adantl 453 . . . . . . . . . 10  |-  ( ( W  e.  A  /\  C  e.  V )  ->  C  e.  _V )
25 opelxpi 4903 . . . . . . . . . 10  |-  ( ( ( E `  ndx )  e.  _V  /\  C  e.  _V )  ->  <. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
2610, 24, 25sylancr 645 . . . . . . . . 9  |-  ( ( W  e.  A  /\  C  e.  V )  -> 
<. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
27 opex 4420 . . . . . . . . . 10  |-  <. ( E `  ndx ) ,  C >.  e.  _V
2827relsn 4972 . . . . . . . . 9  |-  ( Rel 
{ <. ( E `  ndx ) ,  C >. }  <->  <. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
2926, 28sylibr 204 . . . . . . . 8  |-  ( ( W  e.  A  /\  C  e.  V )  ->  Rel  { <. ( E `  ndx ) ,  C >. } )
30 dmsnopss 5335 . . . . . . . 8  |-  dom  { <. ( E `  ndx ) ,  C >. } 
C_  { ( E `
 ndx ) }
31 relssres 5176 . . . . . . . 8  |-  ( ( Rel  { <. ( E `  ndx ) ,  C >. }  /\  dom  {
<. ( E `  ndx ) ,  C >. } 
C_  { ( E `
 ndx ) } )  ->  ( { <. ( E `  ndx ) ,  C >. }  |`  { ( E `  ndx ) } )  =  { <. ( E `  ndx ) ,  C >. } )
3229, 30, 31sylancl 644 . . . . . . 7  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( { <. ( E `  ndx ) ,  C >. }  |`  { ( E `  ndx ) } )  =  { <. ( E `  ndx ) ,  C >. } )
3322, 32uneq12d 3495 . . . . . 6  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  |`  { ( E `  ndx ) } )  u.  ( { <. ( E `  ndx ) ,  C >. }  |`  { ( E `  ndx ) } ) )  =  ( (/)  u.  { <. ( E `  ndx ) ,  C >. } ) )
34 resundir 5154 . . . . . 6  |-  ( ( ( W  |`  ( _V  \  { ( E `
 ndx ) } ) )  u.  { <. ( E `  ndx ) ,  C >. } )  |`  { ( E `  ndx ) } )  =  ( ( ( W  |`  ( _V  \  { ( E `
 ndx ) } ) )  |`  { ( E `  ndx ) } )  u.  ( { <. ( E `  ndx ) ,  C >. }  |`  { ( E `  ndx ) } ) )
35 un0 3645 . . . . . . 7  |-  ( {
<. ( E `  ndx ) ,  C >. }  u.  (/) )  =  { <. ( E `  ndx ) ,  C >. }
36 uncom 3484 . . . . . . 7  |-  ( {
<. ( E `  ndx ) ,  C >. }  u.  (/) )  =  (
(/)  u.  { <. ( E `  ndx ) ,  C >. } )
3735, 36eqtr3i 2458 . . . . . 6  |-  { <. ( E `  ndx ) ,  C >. }  =  (
(/)  u.  { <. ( E `  ndx ) ,  C >. } )
3833, 34, 373eqtr4g 2493 . . . . 5  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } )  |`  { ( E `  ndx ) } )  =  { <. ( E `  ndx ) ,  C >. } )
3938fveq1d 5723 . . . 4  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  u.  { <. ( E `  ndx ) ,  C >. } )  |`  { ( E `  ndx ) } ) `  ( E `  ndx )
)  =  ( {
<. ( E `  ndx ) ,  C >. } `
 ( E `  ndx ) ) )
4013, 39syl5eqr 2482 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } ) `  ( E `
 ndx ) )  =  ( { <. ( E `  ndx ) ,  C >. } `  ( E `  ndx ) ) )
4110a1i 11 . . . 4  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( E `  ndx )  e.  _V )
42 fvsng 5920 . . . 4  |-  ( ( ( E `  ndx )  e.  _V  /\  C  e.  V )  ->  ( { <. ( E `  ndx ) ,  C >. } `
 ( E `  ndx ) )  =  C )
4341, 42sylancom 649 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( { <. ( E `  ndx ) ,  C >. } `  ( E `  ndx ) )  =  C )
4440, 43eqtrd 2468 . 2  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } ) `  ( E `
 ndx ) )  =  C )
452, 9, 443eqtrrd 2473 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 359    = wceq 1652    e. wcel 1725   _Vcvv 2949    \ cdif 3310    u. cun 3311    i^i cin 3312    C_ wss 3313   (/)c0 3621   {csn 3807   <.cop 3810    X. cxp 4869   dom cdm 4871    |` cres 4873   Rel wrel 4876   ` cfv 5447  (class class class)co 6074   ndxcnx 13459   sSet csts 13460  Slot cslot 13461
This theorem is referenced by:  ressbas  13512  oppchomfval  13933  oppccofval  13935  reschom  14023  oduleval  14551  oppgplusfval  15137  mgpplusg  15645  opprmulfval  15723  srasca  16246  sravsca  16247  opsrle  16529  zlmsca  16795  zlmvsca  16796  znle  16810  thloc  16919  tuslem  18290  setsmstset  18500  tngds  18682  tngtset  18683  matmulr  27436  hlhilnvl  32689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-res 4883  df-iota 5411  df-fun 5449  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-slot 13466  df-sets 13468
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