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Theorem ndxid 14862
Description: A structure component extractor is defined by its own index. This theorem, together with strfv 14877 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the  1 in df-base 14846 and the  10 in df-ple 14929, 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 14845 . . 3  |- Slot  ( E `
 ndx )  =  ( x  e.  _V  |->  ( x `  ( E `  ndx ) ) )
3 df-slot 14845 . . . 4  |- Slot  N  =  ( x  e.  _V  |->  ( x `  N
) )
4 ndxarg.2 . . . . . . 7  |-  N  e.  NN
51, 4ndxarg 14861 . . . . . 6  |-  ( E `
 ndx )  =  N
65fveq2i 5852 . . . . 5  |-  ( x `
 ( E `  ndx ) )  =  ( x `  N )
76mpteq2i 4478 . . . 4  |-  ( x  e.  _V  |->  ( x `
 ( E `  ndx ) ) )  =  ( x  e.  _V  |->  ( x `  N
) )
83, 7eqtr4i 2434 . . 3  |- Slot  N  =  ( x  e.  _V  |->  ( x `  ( E `  ndx ) ) )
92, 8eqtr4i 2434 . 2  |- Slot  ( E `
 ndx )  = Slot 
N
101, 9eqtr4i 2434 1  |-  E  = Slot  ( E `  ndx )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    e. wcel 1842   _Vcvv 3059    |-> cmpt 4453   ` cfv 5569   NNcn 10576   ndxcnx 14838  Slot cslot 14840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-i2m1 9590  ax-1ne0 9591  ax-rrecex 9594  ax-cnre 9595
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-om 6684  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-nn 10577  df-ndx 14844  df-slot 14845
This theorem is referenced by:  strndxid  14863  setsidvald  14867  baseid  14889  resslem  14901  plusgid  14944  2strop  14951  mulrid  14959  starvid  14965  scaid  14974  vscaid  14976  ipid  14983  tsetid  15001  pleid  15008  ocid  15015  dsid  15017  unifid  15019  homid  15029  ccoid  15031  oppglem  16709  mgplem  17466  opprlem  17597  sralem  18143  opsrbaslem  18462  zlmlem  18854  znbaslem  18875  tnglem  21446  itvid  24218  lngid  24219  ttglem  24596  cchhllem  24607  resvlem  28274  hlhilslem  34961  uhgrepe  38007
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