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Theorem setsid 14548
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 14531 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( W sSet  <. ( E `  ndx ) ,  C >. )  =  ( ( W  |`  ( _V  \  { ( E `
 ndx ) } ) )  u.  { <. ( E `  ndx ) ,  C >. } ) )
21fveq2d 5876 . 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 5322 . . . . 5  |-  ( W  e.  A  ->  ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  e.  _V )
54adantr 465 . . . 4  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( W  |`  ( _V  \  { ( E `
 ndx ) } ) )  e.  _V )
6 snex 4694 . . . 4  |-  { <. ( E `  ndx ) ,  C >. }  e.  _V
7 unexg 6596 . . . 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 662 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } )  e.  _V )
93, 8strfvnd 14522 . 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 5882 . . . . . 6  |-  ( E `
 ndx )  e. 
_V
1110snid 4061 . . . . 5  |-  ( E `
 ndx )  e. 
{ ( E `  ndx ) }
12 fvres 5886 . . . . 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 5 . . . 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 5292 . . . . . . . . 9  |-  ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  |`  { ( E `  ndx ) } )  =  ( W  |`  (
( _V  \  {
( E `  ndx ) } )  i^i  {
( E `  ndx ) } ) )
15 incom 3696 . . . . . . . . . . . 12  |-  ( ( _V  \  { ( E `  ndx ) } )  i^i  {
( E `  ndx ) } )  =  ( { ( E `  ndx ) }  i^i  ( _V  \  { ( E `
 ndx ) } ) )
16 disjdif 3905 . . . . . . . . . . . 12  |-  ( { ( E `  ndx ) }  i^i  ( _V  \  { ( E `
 ndx ) } ) )  =  (/)
1715, 16eqtri 2496 . . . . . . . . . . 11  |-  ( ( _V  \  { ( E `  ndx ) } )  i^i  {
( E `  ndx ) } )  =  (/)
1817reseq2i 5276 . . . . . . . . . 10  |-  ( W  |`  ( ( _V  \  { ( E `  ndx ) } )  i^i 
{ ( E `  ndx ) } ) )  =  ( W  |`  (/) )
19 res0 5284 . . . . . . . . . 10  |-  ( W  |`  (/) )  =  (/)
2018, 19eqtri 2496 . . . . . . . . 9  |-  ( W  |`  ( ( _V  \  { ( E `  ndx ) } )  i^i 
{ ( E `  ndx ) } ) )  =  (/)
2114, 20eqtri 2496 . . . . . . . 8  |-  ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  |`  { ( E `  ndx ) } )  =  (/)
2221a1i 11 . . . . . . 7  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  |`  { ( E `  ndx ) } )  =  (/) )
23 elex 3127 . . . . . . . . . . 11  |-  ( C  e.  V  ->  C  e.  _V )
2423adantl 466 . . . . . . . . . 10  |-  ( ( W  e.  A  /\  C  e.  V )  ->  C  e.  _V )
25 opelxpi 5037 . . . . . . . . . 10  |-  ( ( ( E `  ndx )  e.  _V  /\  C  e.  _V )  ->  <. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
2610, 24, 25sylancr 663 . . . . . . . . 9  |-  ( ( W  e.  A  /\  C  e.  V )  -> 
<. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
27 opex 4717 . . . . . . . . . 10  |-  <. ( E `  ndx ) ,  C >.  e.  _V
2827relsn 5112 . . . . . . . . 9  |-  ( Rel 
{ <. ( E `  ndx ) ,  C >. }  <->  <. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
2926, 28sylibr 212 . . . . . . . 8  |-  ( ( W  e.  A  /\  C  e.  V )  ->  Rel  { <. ( E `  ndx ) ,  C >. } )
30 dmsnopss 5486 . . . . . . . 8  |-  dom  { <. ( E `  ndx ) ,  C >. } 
C_  { ( E `
 ndx ) }
31 relssres 5317 . . . . . . . 8  |-  ( ( Rel  { <. ( E `  ndx ) ,  C >. }  /\  dom  {
<. ( E `  ndx ) ,  C >. } 
C_  { ( E `
 ndx ) } )  ->  ( { <. ( E `  ndx ) ,  C >. }  |`  { ( E `  ndx ) } )  =  { <. ( E `  ndx ) ,  C >. } )
3229, 30, 31sylancl 662 . . . . . . 7  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( { <. ( E `  ndx ) ,  C >. }  |`  { ( E `  ndx ) } )  =  { <. ( E `  ndx ) ,  C >. } )
3322, 32uneq12d 3664 . . . . . 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 5294 . . . . . 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 3815 . . . . . . 7  |-  ( {
<. ( E `  ndx ) ,  C >. }  u.  (/) )  =  { <. ( E `  ndx ) ,  C >. }
36 uncom 3653 . . . . . . 7  |-  ( {
<. ( E `  ndx ) ,  C >. }  u.  (/) )  =  (
(/)  u.  { <. ( E `  ndx ) ,  C >. } )
3735, 36eqtr3i 2498 . . . . . 6  |-  { <. ( E `  ndx ) ,  C >. }  =  (
(/)  u.  { <. ( E `  ndx ) ,  C >. } )
3833, 34, 373eqtr4g 2533 . . . . 5  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } )  |`  { ( E `  ndx ) } )  =  { <. ( E `  ndx ) ,  C >. } )
3938fveq1d 5874 . . . 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 2522 . . 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 6106 . . . 4  |-  ( ( ( E `  ndx )  e.  _V  /\  C  e.  V )  ->  ( { <. ( E `  ndx ) ,  C >. } `
 ( E `  ndx ) )  =  C )
4341, 42sylancom 667 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( { <. ( E `  ndx ) ,  C >. } `  ( E `  ndx ) )  =  C )
4440, 43eqtrd 2508 . 2  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } ) `  ( E `
 ndx ) )  =  C )
452, 9, 443eqtrrd 2513 1  |-  ( ( W  e.  A  /\  C  e.  V )  ->  C  =  ( E `
 ( W sSet  <. ( E `  ndx ) ,  C >. ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    \ cdif 3478    u. cun 3479    i^i cin 3480    C_ wss 3481   (/)c0 3790   {csn 4033   <.cop 4039    X. cxp 5003   dom cdm 5005    |` cres 5007   Rel wrel 5010   ` cfv 5594  (class class class)co 6295   ndxcnx 14504   sSet csts 14505  Slot cslot 14506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-res 5017  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-slot 14511  df-sets 14513
This theorem is referenced by:  ressbas  14562  oppchomfval  14987  oppccofval  14989  reschom  15077  oduleval  15635  oppgplusfval  16255  mgpplusg  17017  opprmulfval  17146  srasca  17698  sravsca  17699  sraip  17700  opsrle  18010  zlmsca  18427  zlmvsca  18428  znle  18442  thloc  18599  matmulr  18809  tuslem  20638  setsmstset  20848  tngds  21030  tngtset  21031  ttgval  23992  resvsca  27636  uhgrepe  32159  hlhilnvl  37106
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