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Theorem strsetsid 12489
Description: Value of the structure replacement function. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 30-Jan-2023.)
Hypotheses
Ref Expression
strsetsid.e  |-  E  = Slot  ( E `  ndx )
strsetsid.s  |-  ( ph  ->  S Struct  <. M ,  N >. )
strsetsid.f  |-  ( ph  ->  Fun  S )
strsetsid.d  |-  ( ph  ->  ( E `  ndx )  e.  dom  S )
Assertion
Ref Expression
strsetsid  |-  ( ph  ->  S  =  ( S sSet  <. ( E `  ndx ) ,  ( E `  S ) >. )
)

Proof of Theorem strsetsid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 strsetsid.s . . . 4  |-  ( ph  ->  S Struct  <. M ,  N >. )
2 structex 12468 . . . 4  |-  ( S Struct  <. M ,  N >.  ->  S  e.  _V )
31, 2syl 14 . . 3  |-  ( ph  ->  S  e.  _V )
4 strsetsid.d . . 3  |-  ( ph  ->  ( E `  ndx )  e.  dom  S )
5 strsetsid.e . . . . 5  |-  E  = Slot  ( E `  ndx )
6 isstructim 12470 . . . . . . . . 9  |-  ( S Struct  <. M ,  N >.  -> 
( ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N )  /\  Fun  ( S  \  { (/) } )  /\  dom  S  C_  ( M ... N
) ) )
71, 6syl 14 . . . . . . . 8  |-  ( ph  ->  ( ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N )  /\  Fun  ( S  \  { (/) } )  /\  dom  S  C_  ( M ... N
) ) )
87simp3d 1011 . . . . . . 7  |-  ( ph  ->  dom  S  C_  ( M ... N ) )
97simp1d 1009 . . . . . . . . 9  |-  ( ph  ->  ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N ) )
109simp1d 1009 . . . . . . . 8  |-  ( ph  ->  M  e.  NN )
11 fzssnn 10065 . . . . . . . 8  |-  ( M  e.  NN  ->  ( M ... N )  C_  NN )
1210, 11syl 14 . . . . . . 7  |-  ( ph  ->  ( M ... N
)  C_  NN )
138, 12sstrd 3165 . . . . . 6  |-  ( ph  ->  dom  S  C_  NN )
1413, 4sseldd 3156 . . . . 5  |-  ( ph  ->  ( E `  ndx )  e.  NN )
155, 3, 14strnfvnd 12476 . . . 4  |-  ( ph  ->  ( E `  S
)  =  ( S `
 ( E `  ndx ) ) )
16 strsetsid.f . . . . 5  |-  ( ph  ->  Fun  S )
17 funfvex 5532 . . . . 5  |-  ( ( Fun  S  /\  ( E `  ndx )  e. 
dom  S )  -> 
( S `  ( E `  ndx ) )  e.  _V )
1816, 4, 17syl2anc 411 . . . 4  |-  ( ph  ->  ( S `  ( E `  ndx ) )  e.  _V )
1915, 18eqeltrd 2254 . . 3  |-  ( ph  ->  ( E `  S
)  e.  _V )
20 setsvala 12487 . . 3  |-  ( ( S  e.  _V  /\  ( E `  ndx )  e.  dom  S  /\  ( E `  S )  e.  _V )  ->  ( S sSet  <. ( E `  ndx ) ,  ( E `
 S ) >.
)  =  ( ( S  |`  ( _V  \  { ( E `  ndx ) } ) )  u.  { <. ( E `  ndx ) ,  ( E `  S
) >. } ) )
213, 4, 19, 20syl3anc 1238 . 2  |-  ( ph  ->  ( S sSet  <. ( E `  ndx ) ,  ( E `  S
) >. )  =  ( ( S  |`  ( _V  \  { ( E `
 ndx ) } ) )  u.  { <. ( E `  ndx ) ,  ( E `  S ) >. } ) )
2215opeq2d 3785 . . . 4  |-  ( ph  -> 
<. ( E `  ndx ) ,  ( E `  S ) >.  =  <. ( E `  ndx ) ,  ( S `  ( E `  ndx )
) >. )
2322sneqd 3605 . . 3  |-  ( ph  ->  { <. ( E `  ndx ) ,  ( E `
 S ) >. }  =  { <. ( E `  ndx ) ,  ( S `  ( E `  ndx ) )
>. } )
2423uneq2d 3289 . 2  |-  ( ph  ->  ( ( S  |`  ( _V  \  { ( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  ( E `
 S ) >. } )  =  ( ( S  |`  ( _V  \  { ( E `
 ndx ) } ) )  u.  { <. ( E `  ndx ) ,  ( S `  ( E `  ndx ) ) >. } ) )
25 nnssz 9268 . . . . 5  |-  NN  C_  ZZ
2613, 25sstrdi 3167 . . . 4  |-  ( ph  ->  dom  S  C_  ZZ )
27 zdceq 9326 . . . . 5  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  -> DECID  x  =  y )
2827rgen2a 2531 . . . 4  |-  A. x  e.  ZZ  A. y  e.  ZZ DECID  x  =  y
29 ssralv 3219 . . . . . 6  |-  ( dom 
S  C_  ZZ  ->  ( A. y  e.  ZZ DECID  x  =  y  ->  A. y  e.  dom  SDECID  x  =  y ) )
3029ralimdv 2545 . . . . 5  |-  ( dom 
S  C_  ZZ  ->  ( A. x  e.  ZZ  A. y  e.  ZZ DECID  x  =  y  ->  A. x  e.  ZZ  A. y  e.  dom  SDECID  x  =  y ) )
31 ssralv 3219 . . . . 5  |-  ( dom 
S  C_  ZZ  ->  ( A. x  e.  ZZ  A. y  e.  dom  SDECID  x  =  y  ->  A. x  e.  dom  S A. y  e.  dom  SDECID  x  =  y ) )
3230, 31syld 45 . . . 4  |-  ( dom 
S  C_  ZZ  ->  ( A. x  e.  ZZ  A. y  e.  ZZ DECID  x  =  y  ->  A. x  e.  dom  S A. y  e.  dom  SDECID  x  =  y ) )
3326, 28, 32mpisyl 1446 . . 3  |-  ( ph  ->  A. x  e.  dom  S A. y  e.  dom  SDECID  x  =  y )
34 funresdfunsndc 6506 . . 3  |-  ( ( A. x  e.  dom  S A. y  e.  dom  SDECID  x  =  y  /\  Fun  S  /\  ( E `  ndx )  e.  dom  S )  ->  ( ( S  |`  ( _V  \  { ( E `  ndx ) } ) )  u.  { <. ( E `  ndx ) ,  ( S `  ( E `  ndx ) )
>. } )  =  S )
3533, 16, 4, 34syl3anc 1238 . 2  |-  ( ph  ->  ( ( S  |`  ( _V  \  { ( E `  ndx ) } ) )  u. 
{ <. ( E `  ndx ) ,  ( S `
 ( E `  ndx ) ) >. } )  =  S )
3621, 24, 353eqtrrd 2215 1  |-  ( ph  ->  S  =  ( S sSet  <. ( E `  ndx ) ,  ( E `  S ) >. )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4  DECID wdc 834    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   _Vcvv 2737    \ cdif 3126    u. cun 3127    C_ wss 3129   (/)c0 3422   {csn 3592   <.cop 3595   class class class wbr 4003   dom cdm 4626    |` cres 4628   Fun wfun 5210   ` cfv 5216  (class class class)co 5874    <_ cle 7991   NNcn 8917   ZZcz 9251   ...cfz 10006   Struct cstr 12452   ndxcnx 12453   sSet csts 12454  Slot cslot 12455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-inn 8918  df-n0 9175  df-z 9252  df-uz 9527  df-fz 10007  df-struct 12458  df-slot 12460  df-sets 12463
This theorem is referenced by:  strressid  12524
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