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

Theorem strfvn 13165
Description: Value of a structure component extractor  E. Normally,  E is a defined constant symbol such as  Base (df-base 13153) 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 14080) 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 13180. (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 13163 . 2  |-  (  T. 
->  ( E `  S
)  =  ( S `
 N ) )
54trud 1314 1  |-  ( E `
 S )  =  ( S `  N
)
Colors of variables: wff set class
Syntax hints:    T. wtru 1307    = wceq 1623    e. wcel 1684   _Vcvv 2788   ` cfv 5255  Slot cslot 13147
This theorem is referenced by:  ndxarg  13168  str0  13184  setsnid  13188  baseval  13189  ressbas  13198
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-slot 13152
  Copyright terms: Public domain W3C validator