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Theorem setsslid 11904
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 11885 . . . 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 1286 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( W sSet  <. ( E `  ndx ) ,  C >. )  =  ( ( W  |`  ( _V  \  { ( E `
 ndx ) } ) )  u.  { <. ( E `  ndx ) ,  C >. } ) )
54fveq2d 5391 . 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 4827 . . . 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 4118 . . . . . 6  |-  ( ( ( E `  ndx )  e.  NN  /\  C  e.  V )  ->  <. ( E `  ndx ) ,  C >.  e.  _V )
102, 8, 9sylancr 408 . . . . 5  |-  ( ( W  e.  A  /\  C  e.  V )  -> 
<. ( E `  ndx ) ,  C >.  e. 
_V )
11 snexg 4076 . . . . 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 4332 . . . 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 567 . . 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 11874 . 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 3522 . . . . 5  |-  ( ( E `  ndx )  e.  NN  ->  ( E `  ndx )  e.  {
( E `  ndx ) } )
18 fvres 5411 . . . . 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 4799 . . . . . . . . 9  |-  ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  |`  { ( E `  ndx ) } )  =  ( W  |`  (
( _V  \  {
( E `  ndx ) } )  i^i  {
( E `  ndx ) } ) )
21 incom 3236 . . . . . . . . . . . 12  |-  ( ( _V  \  { ( E `  ndx ) } )  i^i  {
( E `  ndx ) } )  =  ( { ( E `  ndx ) }  i^i  ( _V  \  { ( E `
 ndx ) } ) )
22 disjdif 3403 . . . . . . . . . . . 12  |-  ( { ( E `  ndx ) }  i^i  ( _V  \  { ( E `
 ndx ) } ) )  =  (/)
2321, 22eqtri 2136 . . . . . . . . . . 11  |-  ( ( _V  \  { ( E `  ndx ) } )  i^i  {
( E `  ndx ) } )  =  (/)
2423reseq2i 4784 . . . . . . . . . 10  |-  ( W  |`  ( ( _V  \  { ( E `  ndx ) } )  i^i 
{ ( E `  ndx ) } ) )  =  ( W  |`  (/) )
25 res0 4791 . . . . . . . . . 10  |-  ( W  |`  (/) )  =  (/)
2624, 25eqtri 2136 . . . . . . . . 9  |-  ( W  |`  ( ( _V  \  { ( E `  ndx ) } )  i^i 
{ ( E `  ndx ) } ) )  =  (/)
2720, 26eqtri 2136 . . . . . . . 8  |-  ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  |`  { ( E `  ndx ) } )  =  (/)
2827a1i 9 . . . . . . 7  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( W  |`  ( _V  \  { ( E `  ndx ) } ) )  |`  { ( E `  ndx ) } )  =  (/) )
292elexi 2670 . . . . . . . . . 10  |-  ( E `
 ndx )  e. 
_V
308elexd 2671 . . . . . . . . . 10  |-  ( ( W  e.  A  /\  C  e.  V )  ->  C  e.  _V )
31 opelxpi 4539 . . . . . . . . . 10  |-  ( ( ( E `  ndx )  e.  _V  /\  C  e.  _V )  ->  <. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
3229, 30, 31sylancr 408 . . . . . . . . 9  |-  ( ( W  e.  A  /\  C  e.  V )  -> 
<. ( E `  ndx ) ,  C >.  e.  ( _V  X.  _V ) )
33 relsng 4610 . . . . . . . . . 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 4978 . . . . . . . . . 10  |-  ( C  e.  V  ->  dom  {
<. ( E `  ndx ) ,  C >. }  =  { ( E `
 ndx ) } )
3736adantl 273 . . . . . . . . 9  |-  ( ( W  e.  A  /\  C  e.  V )  ->  dom  { <. ( E `  ndx ) ,  C >. }  =  {
( E `  ndx ) } )
38 eqimss 3119 . . . . . . . . 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 4825 . . . . . . . 8  |-  ( ( Rel  { <. ( E `  ndx ) ,  C >. }  /\  dom  {
<. ( E `  ndx ) ,  C >. } 
C_  { ( E `
 ndx ) } )  ->  ( { <. ( E `  ndx ) ,  C >. }  |`  { ( E `  ndx ) } )  =  { <. ( E `  ndx ) ,  C >. } )
4135, 39, 40syl2anc 406 . . . . . . 7  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( { <. ( E `  ndx ) ,  C >. }  |`  { ( E `  ndx ) } )  =  { <. ( E `  ndx ) ,  C >. } )
4228, 41uneq12d 3199 . . . . . 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 4801 . . . . . 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 3364 . . . . . . 7  |-  ( {
<. ( E `  ndx ) ,  C >. }  u.  (/) )  =  { <. ( E `  ndx ) ,  C >. }
45 uncom 3188 . . . . . . 7  |-  ( {
<. ( E `  ndx ) ,  C >. }  u.  (/) )  =  (
(/)  u.  { <. ( E `  ndx ) ,  C >. } )
4644, 45eqtr3i 2138 . . . . . 6  |-  { <. ( E `  ndx ) ,  C >. }  =  (
(/)  u.  { <. ( E `  ndx ) ,  C >. } )
4742, 43, 463eqtr4g 2173 . . . . 5  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } )  |`  { ( E `  ndx ) } )  =  { <. ( E `  ndx ) ,  C >. } )
4847fveq1d 5389 . . . 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 2162 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } ) `  ( E `
 ndx ) )  =  ( { <. ( E `  ndx ) ,  C >. } `  ( E `  ndx ) ) )
50 fvsng 5582 . . . 4  |-  ( ( ( E `  ndx )  e.  NN  /\  C  e.  V )  ->  ( { <. ( E `  ndx ) ,  C >. } `
 ( E `  ndx ) )  =  C )
512, 8, 50sylancr 408 . . 3  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( { <. ( E `  ndx ) ,  C >. } `  ( E `  ndx ) )  =  C )
5249, 51eqtrd 2148 . 2  |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( ( ( W  |`  ( _V  \  {
( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  C >. } ) `  ( E `
 ndx ) )  =  C )
535, 16, 523eqtrrd 2153 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 1314    e. wcel 1463   _Vcvv 2658    \ cdif 3036    u. cun 3037    i^i cin 3038    C_ wss 3039   (/)c0 3331   {csn 3495   <.cop 3498    X. cxp 4505   dom cdm 4507    |` cres 4509   Rel wrel 4512   ` cfv 5091  (class class class)co 5740   NNcn 8677   ndxcnx 11851   sSet csts 11852  Slot cslot 11853
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-iota 5056  df-fun 5093  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-slot 11858  df-sets 11861
This theorem is referenced by:  setsmstsetg  12545
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