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Theorem strslfv 11592
Description: Extract a structure component  C (such as the base set) from a structure  S with a component extractor  E (such as the base set extractor df-base 11554). By virtue of ndxslid 11573, this can be done without having to refer to the hard-coded numeric index of  E. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Jim Kingdon, 30-Jan-2023.)
Hypotheses
Ref Expression
strfv.s  |-  S Struct  X
strslfv.e  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
strfv.n  |-  { <. ( E `  ndx ) ,  C >. }  C_  S
Assertion
Ref Expression
strslfv  |-  ( C  e.  V  ->  C  =  ( E `  S ) )

Proof of Theorem strslfv
StepHypRef Expression
1 strslfv.e . 2  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
2 strfv.s . . 3  |-  S Struct  X
3 structex 11560 . . 3  |-  ( S Struct  X  ->  S  e.  _V )
42, 3mp1i 10 . 2  |-  ( C  e.  V  ->  S  e.  _V )
52structfun 11566 . . 3  |-  Fun  `' `' S
65a1i 9 . 2  |-  ( C  e.  V  ->  Fun  `' `' S )
7 strfv.n . . 3  |-  { <. ( E `  ndx ) ,  C >. }  C_  S
81simpri 112 . . . . 5  |-  ( E `
 ndx )  e.  NN
9 opexg 4064 . . . . 5  |-  ( ( ( E `  ndx )  e.  NN  /\  C  e.  V )  ->  <. ( E `  ndx ) ,  C >.  e.  _V )
108, 9mpan 416 . . . 4  |-  ( C  e.  V  ->  <. ( E `  ndx ) ,  C >.  e.  _V )
11 snssg 3579 . . . 4  |-  ( <.
( E `  ndx ) ,  C >.  e. 
_V  ->  ( <. ( E `  ndx ) ,  C >.  e.  S  <->  {
<. ( E `  ndx ) ,  C >. } 
C_  S ) )
1210, 11syl 14 . . 3  |-  ( C  e.  V  ->  ( <. ( E `  ndx ) ,  C >.  e.  S  <->  { <. ( E `  ndx ) ,  C >. } 
C_  S ) )
137, 12mpbiri 167 . 2  |-  ( C  e.  V  ->  <. ( E `  ndx ) ,  C >.  e.  S
)
14 id 19 . 2  |-  ( C  e.  V  ->  C  e.  V )
151, 4, 6, 13, 14strslfv2d 11590 1  |-  ( C  e.  V  ->  C  =  ( E `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   _Vcvv 2620    C_ wss 3000   {csn 3450   <.cop 3453   class class class wbr 3851   `'ccnv 4450   Fun wfun 5022   ` cfv 5028   NNcn 8476   Struct cstr 11544   ndxcnx 11545  Slot cslot 11547
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
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 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-iota 4993  df-fun 5030  df-fv 5036  df-struct 11550  df-slot 11552
This theorem is referenced by:  strslfv3  11593
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