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Theorem strfvn 13373
Description: Value of a structure component extractor  E. Normally,  E is a defined constant symbol such as  Base (df-base 13361) and  N is a fixed integer such as  1.  S is a structure, i.e. a specific member of a class of structures such as  Poset (df-poset 14290) where  S  e.  Poset.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strfv 13388. (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2013.) (New usage is discouraged.)

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

Proof of Theorem strfvn
StepHypRef Expression
1 strfvn.c . . 3  |-  E  = Slot 
N
2 strfvn.f . . . 4  |-  S  e. 
_V
32a1i 10 . . 3  |-  (  T. 
->  S  e.  _V )
41, 3strfvnd 13371 . 2  |-  (  T. 
->  ( E `  S
)  =  ( S `
 N ) )
54trud 1328 1  |-  ( E `
 S )  =  ( S `  N
)
Colors of variables: wff set class
Syntax hints:    T. wtru 1321    = wceq 1647    e. wcel 1715   _Vcvv 2873   ` cfv 5358  Slot cslot 13355
This theorem is referenced by:  ndxarg  13376  str0  13392  setsnid  13396  baseval  13397  ressbas  13406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-slot 13360
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