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

Theorem strfvn 13445
Description: Value of a structure component extractor  E. Normally,  E is a defined constant symbol such as  Base (df-base 13433) 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 14362) 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 13460. (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 13443 . 2  |-  (  T. 
->  ( E `  S
)  =  ( S `
 N ) )
54trud 1329 1  |-  ( E `
 S )  =  ( S `  N
)
Colors of variables: wff set class
Syntax hints:    T. wtru 1322    = wceq 1649    e. wcel 1721   _Vcvv 2920   ` cfv 5417  Slot cslot 13427
This theorem is referenced by:  ndxarg  13448  str0  13464  setsnid  13468  baseval  13469  ressbas  13478
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-slot 13432
  Copyright terms: Public domain W3C validator