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Theorem setsslid 11605
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 11586 . . . 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 1262 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( W sSet  <. ( E `  ndx ) ,  C >. )  =  ( ( W  |`  ( _V  \  { ( E `
 ndx ) } ) )  u.  { <. ( E `  ndx ) ,  C >. } ) )
54fveq2d 5322 . 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 4765 . . . 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 4064 . . . . . 6  |-  ( ( ( E `  ndx )  e.  NN  /\  C  e.  V )  ->  <. ( E `  ndx ) ,  C >.  e.  _V )
102, 8, 9sylancr 406 . . . . 5  |-  ( ( W  e.  A  /\  C  e.  V )  -> 
<. ( E `  ndx ) ,  C >.  e. 
_V )
11 snexg 4025 . . . . 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 4278 . . . 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 563 . . 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 11575 . 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 3477 . . . . 5  |-  ( ( E `  ndx )  e.  NN  ->  ( E `  ndx )  e.  {
( E `  ndx ) } )
18 fvres 5342 . . . . 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 4738 . . . . . . . . 9  |-  ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  |`  { ( E `  ndx ) } )  =  ( W  |`  (
( _V  \  {
( E `  ndx ) } )  i^i  {
( E `  ndx ) } ) )
21 incom 3193 . . . . . . . . . . . 12  |-  ( ( _V  \  { ( E `  ndx ) } )  i^i  {
( E `  ndx ) } )  =  ( { ( E `  ndx ) }  i^i  ( _V  \  { ( E `
 ndx ) } ) )
22 disjdif 3359 . . . . . . . . . . . 12  |-  ( { ( E `  ndx ) }  i^i  ( _V  \  { ( E `
 ndx ) } ) )  =  (/)
2321, 22eqtri 2109 . . . . . . . . . . 11  |-  ( ( _V  \  { ( E `  ndx ) } )  i^i  {
( E `  ndx ) } )  =  (/)
2423reseq2i 4723 . . . . . . . . . 10  |-  ( W  |`  ( ( _V  \  { ( E `  ndx ) } )  i^i 
{ ( E `  ndx ) } ) )  =  ( W  |`  (/) )
25 res0 4730 . . . . . . . . . 10  |-  ( W  |`  (/) )  =  (/)
2624, 25eqtri 2109 . . . . . . . . 9  |-  ( W  |`  ( ( _V  \  { ( E `  ndx ) } )  i^i 
{ ( E `  ndx ) } ) )  =  (/)
2720, 26eqtri 2109 . . . . . . . 8  |-  ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  |`  { ( E `  ndx ) } )  =  (/)
2827a1i 9 . . . . . . 7  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  |`  { ( E `  ndx ) } )  =  (/) )
292elexi 2632 . . . . . . . . . 10  |-  ( E `
 ndx )  e. 
_V
308elexd 2633 . . . . . . . . . 10  |-  ( ( W  e.  A  /\  C  e.  V )  ->  C  e.  _V )
31 opelxpi 4483 . . . . . . . . . 10  |-  ( ( ( E `  ndx )  e.  _V  /\  C  e.  _V )  ->  <. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
3229, 30, 31sylancr 406 . . . . . . . . 9  |-  ( ( W  e.  A  /\  C  e.  V )  -> 
<. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
33 relsng 4554 . . . . . . . . . 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 4915 . . . . . . . . . 10  |-  ( C  e.  V  ->  dom  {
<. ( E `  ndx ) ,  C >. }  =  { ( E `
 ndx ) } )
3736adantl 272 . . . . . . . . 9  |-  ( ( W  e.  A  /\  C  e.  V )  ->  dom  { <. ( E `  ndx ) ,  C >. }  =  {
( E `  ndx ) } )
38 eqimss 3079 . . . . . . . . 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 4763 . . . . . . . 8  |-  ( ( Rel  { <. ( E `  ndx ) ,  C >. }  /\  dom  {
<. ( E `  ndx ) ,  C >. } 
C_  { ( E `
 ndx ) } )  ->  ( { <. ( E `  ndx ) ,  C >. }  |`  { ( E `  ndx ) } )  =  { <. ( E `  ndx ) ,  C >. } )
4135, 39, 40syl2anc 404 . . . . . . 7  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( { <. ( E `  ndx ) ,  C >. }  |`  { ( E `  ndx ) } )  =  { <. ( E `  ndx ) ,  C >. } )
4228, 41uneq12d 3156 . . . . . 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 4740 . . . . . 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 3320 . . . . . . 7  |-  ( {
<. ( E `  ndx ) ,  C >. }  u.  (/) )  =  { <. ( E `  ndx ) ,  C >. }
45 uncom 3145 . . . . . . 7  |-  ( {
<. ( E `  ndx ) ,  C >. }  u.  (/) )  =  (
(/)  u.  { <. ( E `  ndx ) ,  C >. } )
4644, 45eqtr3i 2111 . . . . . 6  |-  { <. ( E `  ndx ) ,  C >. }  =  (
(/)  u.  { <. ( E `  ndx ) ,  C >. } )
4742, 43, 463eqtr4g 2146 . . . . 5  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } )  |`  { ( E `  ndx ) } )  =  { <. ( E `  ndx ) ,  C >. } )
4847fveq1d 5320 . . . 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, 48syl5eqr 2135 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } ) `  ( E `
 ndx ) )  =  ( { <. ( E `  ndx ) ,  C >. } `  ( E `  ndx ) ) )
50 fvsng 5507 . . . 4  |-  ( ( ( E `  ndx )  e.  NN  /\  C  e.  V )  ->  ( { <. ( E `  ndx ) ,  C >. } `
 ( E `  ndx ) )  =  C )
512, 8, 50sylancr 406 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( { <. ( E `  ndx ) ,  C >. } `  ( E `  ndx ) )  =  C )
5249, 51eqtrd 2121 . 2  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } ) `  ( E `
 ndx ) )  =  C )
535, 16, 523eqtrrd 2126 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 1290    e. wcel 1439   _Vcvv 2620    \ cdif 2997    u. cun 2998    i^i cin 2999    C_ wss 3000   (/)c0 3287   {csn 3450   <.cop 3453    X. cxp 4450   dom cdm 4452    |` cres 4454   Rel wrel 4457   ` cfv 5028  (class class class)co 5666   NNcn 8483   ndxcnx 11552   sSet csts 11553  Slot cslot 11554
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-iota 4993  df-fun 5030  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-slot 11559  df-sets 11562
This theorem is referenced by: (None)
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