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Theorem strnfvn 12501
Description: Value of a structure component extractor  E. Normally,  E is a defined constant symbol such as  Base (df-base 12486) and  N is a fixed integer such as  1.  S is a structure, i.e. a specific member of a class of structures.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strslfv 12525. (Contributed by NM, 9-Sep-2011.) (Revised by Jim Kingdon, 19-Jan-2023.) (New usage is discouraged.)

Hypotheses
Ref Expression
strnfvn.f  |-  S  e. 
_V
strnfvn.c  |-  E  = Slot 
N
strnfvn.n  |-  N  e.  NN
Assertion
Ref Expression
strnfvn  |-  ( E `
 S )  =  ( S `  N
)

Proof of Theorem strnfvn
StepHypRef Expression
1 strnfvn.c . . 3  |-  E  = Slot 
N
2 strnfvn.f . . . 4  |-  S  e. 
_V
32a1i 9 . . 3  |-  ( T. 
->  S  e.  _V )
4 strnfvn.n . . . 4  |-  N  e.  NN
54a1i 9 . . 3  |-  ( T. 
->  N  e.  NN )
61, 3, 5strnfvnd 12500 . 2  |-  ( T. 
->  ( E `  S
)  =  ( S `
 N ) )
76mptru 1373 1  |-  ( E `
 S )  =  ( S `  N
)
Colors of variables: wff set class
Syntax hints:    = wceq 1364   T. wtru 1365    e. wcel 2160   _Vcvv 2752   ` cfv 5231   NNcn 8937  Slot cslot 12479
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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-iota 5193  df-fun 5233  df-fv 5239  df-slot 12484
This theorem is referenced by:  ndxarg  12503  strsl0  12529  baseval  12533
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