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Theorem ndxid 14283
Description: A structure component extractor is defined by its own index. This theorem, together with strfv 14296 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the  1 in df-base 14267 and the  10 in df-ple 14346, 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 14266 . . 3  |- Slot  ( E `
 ndx )  =  ( x  e.  _V  |->  ( x `  ( E `  ndx ) ) )
3 df-slot 14266 . . . 4  |- Slot  N  =  ( x  e.  _V  |->  ( x `  N
) )
4 ndxarg.2 . . . . . . 7  |-  N  e.  NN
51, 4ndxarg 14282 . . . . . 6  |-  ( E `
 ndx )  =  N
65fveq2i 5778 . . . . 5  |-  ( x `
 ( E `  ndx ) )  =  ( x `  N )
76mpteq2i 4459 . . . 4  |-  ( x  e.  _V  |->  ( x `
 ( E `  ndx ) ) )  =  ( x  e.  _V  |->  ( x `  N
) )
83, 7eqtr4i 2481 . . 3  |- Slot  N  =  ( x  e.  _V  |->  ( x `  ( E `  ndx ) ) )
92, 8eqtr4i 2481 . 2  |- Slot  ( E `
 ndx )  = Slot 
N
101, 9eqtr4i 2481 1  |-  E  = Slot  ( E `  ndx )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1757   _Vcvv 3054    |-> cmpt 4434   ` cfv 5502   NNcn 10409   ndxcnx 14259  Slot cslot 14261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-i2m1 9437  ax-1ne0 9438  ax-rrecex 9441  ax-cnre 9442
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-ov 6179  df-om 6563  df-recs 6918  df-rdg 6952  df-nn 10410  df-ndx 14265  df-slot 14266
This theorem is referenced by:  baseid  14308  resslem  14319  plusgid  14361  2strop  14364  mulrid  14372  starvid  14378  scaid  14387  vscaid  14389  ipid  14396  tsetid  14414  pleid  14421  ocid  14428  dsid  14430  unifid  14432  homid  14442  ccoid  14444  oppglem  15953  mgplem  16687  opprlem  16812  sralem  17350  opsrbaslem  17652  zlmlem  18043  znbaslem  18066  tnglem  20328  itvid  23004  lngid  23005  ttglem  23243  cchhllem  23254  resvlem  26419  hlhilslem  35868
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