MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  strfvn Structured version   Unicode version

Theorem strfvn 13491
Description: Value of a structure component extractor  E. Normally,  E is a defined constant symbol such as  Base (df-base 13479) 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 14408) 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 13506. (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 11 . . 3  |-  (  T. 
->  S  e.  _V )
41, 3strfvnd 13489 . 2  |-  (  T. 
->  ( E `  S
)  =  ( S `
 N ) )
54trud 1333 1  |-  ( E `
 S )  =  ( S `  N
)
Colors of variables: wff set class
Syntax hints:    T. wtru 1326    = wceq 1653    e. wcel 1726   _Vcvv 2958   ` cfv 5457  Slot cslot 13473
This theorem is referenced by:  ndxarg  13494  str0  13510  setsnid  13514  baseval  13515  ressbas  13524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-slot 13478
  Copyright terms: Public domain W3C validator