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

Theorem ndxid 14500
Description: A structure component extractor is defined by its own index. This theorem, together with strfv 14513 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the  1 in df-base 14484 and the  10 in df-ple 14564, making it easier to change should the need arise. For example, we can refer to a specific poset with base set  B and order relation  L using  { <. ( Base `  ndx ) ,  B >. ,  <. ( le `  ndx ) ,  L >. } rather than  { <. 1 ,  B >. ,  <. 10 ,  L >. }. The latter, while shorter to state, requires revision if we later change  10 to some other number, and it may also be harder to remember. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.)
Hypotheses
Ref Expression
ndxarg.1  |-  E  = Slot 
N
ndxarg.2  |-  N  e.  NN
Assertion
Ref Expression
ndxid  |-  E  = Slot  ( E `  ndx )

Proof of Theorem ndxid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ndxarg.1 . 2  |-  E  = Slot 
N
2 df-slot 14483 . . 3  |- Slot  ( E `
 ndx )  =  ( x  e.  _V  |->  ( x `  ( E `  ndx ) ) )
3 df-slot 14483 . . . 4  |- Slot  N  =  ( x  e.  _V  |->  ( x `  N
) )
4 ndxarg.2 . . . . . . 7  |-  N  e.  NN
51, 4ndxarg 14499 . . . . . 6  |-  ( E `
 ndx )  =  N
65fveq2i 5860 . . . . 5  |-  ( x `
 ( E `  ndx ) )  =  ( x `  N )
76mpteq2i 4523 . . . 4  |-  ( x  e.  _V  |->  ( x `
 ( E `  ndx ) ) )  =  ( x  e.  _V  |->  ( x `  N
) )
83, 7eqtr4i 2492 . . 3  |- Slot  N  =  ( x  e.  _V  |->  ( x `  ( E `  ndx ) ) )
92, 8eqtr4i 2492 . 2  |- Slot  ( E `
 ndx )  = Slot 
N
101, 9eqtr4i 2492 1  |-  E  = Slot  ( E `  ndx )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    e. wcel 1762   _Vcvv 3106    |-> cmpt 4498   ` cfv 5579   NNcn 10525   ndxcnx 14476  Slot cslot 14478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-i2m1 9549  ax-1ne0 9550  ax-rrecex 9553  ax-cnre 9554
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-om 6672  df-recs 7032  df-rdg 7066  df-nn 10526  df-ndx 14482  df-slot 14483
This theorem is referenced by:  baseid  14525  resslem  14537  plusgid  14579  2strop  14582  mulrid  14590  starvid  14596  scaid  14605  vscaid  14607  ipid  14614  tsetid  14632  pleid  14639  ocid  14646  dsid  14648  unifid  14650  homid  14660  ccoid  14662  oppglem  16173  mgplem  16929  opprlem  17054  sralem  17599  opsrbaslem  17906  zlmlem  18314  znbaslem  18337  tnglem  20882  itvid  23559  lngid  23560  ttglem  23848  cchhllem  23859  resvlem  27334  hlhilslem  36613
  Copyright terms: Public domain W3C validator