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Theorem strslfv 12520
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 12481). By virtue of ndxslid 12500, 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 12487 . . 3  |-  ( S Struct  X  ->  S  e.  _V )
42, 3mp1i 10 . 2  |-  ( C  e.  V  ->  S  e.  _V )
52structfun 12493 . . 3  |-  Fun  `' `' S
65a1i 9 . 2  |-  ( C  e.  V  ->  Fun  `' `' S )
7 strfv.n . . 3  |-  { <. ( E `  ndx ) ,  C >. }  C_  S
81simpri 113 . . . . 5  |-  ( E `
 ndx )  e.  NN
9 opexg 4240 . . . . 5  |-  ( ( ( E `  ndx )  e.  NN  /\  C  e.  V )  ->  <. ( E `  ndx ) ,  C >.  e.  _V )
108, 9mpan 424 . . . 4  |-  ( C  e.  V  ->  <. ( E `  ndx ) ,  C >.  e.  _V )
11 snssg 3738 . . . 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 168 . 2  |-  ( C  e.  V  ->  <. ( E `  ndx ) ,  C >.  e.  S
)
14 id 19 . 2  |-  ( C  e.  V  ->  C  e.  V )
151, 4, 6, 13, 14strslfv2d 12518 1  |-  ( C  e.  V  ->  C  =  ( E `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1363    e. wcel 2158   _Vcvv 2749    C_ wss 3141   {csn 3604   <.cop 3607   class class class wbr 4015   `'ccnv 4637   Fun wfun 5222   ` cfv 5228   NNcn 8932   Struct cstr 12471   ndxcnx 12472  Slot cslot 12474
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fv 5236  df-struct 12477  df-slot 12479
This theorem is referenced by:  strslfv3  12521  cnfldbas  13672  cnfldadd  13673  cnfldmul  13674  cnfldcj  13675
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