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Theorem ndxid 16498
Description: A structure component extractor is defined by its own index. This theorem, together with strfv 16520 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 16478 and the 10 in df-ple 16574, 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.) (Proof shortened by BJ, 27-Dec-2021.)

Hypotheses
Ref Expression
ndxarg.1 𝐸 = Slot 𝑁
ndxarg.2 𝑁 ∈ ℕ
Assertion
Ref Expression
ndxid 𝐸 = Slot (𝐸‘ndx)

Proof of Theorem ndxid
StepHypRef Expression
1 ndxarg.1 . . . 4 𝐸 = Slot 𝑁
2 ndxarg.2 . . . 4 𝑁 ∈ ℕ
31, 2ndxarg 16497 . . 3 (𝐸‘ndx) = 𝑁
43eqcomi 2833 . 2 𝑁 = (𝐸‘ndx)
5 sloteq 16477 . . 3 (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx))
61, 5syl5eq 2871 . 2 (𝑁 = (𝐸‘ndx) → 𝐸 = Slot (𝐸‘ndx))
74, 6ax-mp 5 1 𝐸 = Slot (𝐸‘ndx)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2115  cfv 6336  cn 11623  ndxcnx 16469  Slot cslot 16471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-cnex 10578  ax-1cn 10580  ax-addcl 10582
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7141  df-om 7564  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-nn 11624  df-ndx 16475  df-slot 16476
This theorem is referenced by:  strndxid  16499  setsidvald  16503  baseid  16532  resslem  16546  plusgid  16585  2strop  16593  2strop1  16596  mulrid  16605  starvid  16613  scaid  16622  vscaid  16624  ipid  16631  tsetid  16649  pleid  16656  ocid  16663  dsid  16665  unifid  16667  homid  16677  ccoid  16679  oppglem  18467  mgplem  19233  opprlem  19367  sralem  19935  opsrbaslem  20244  zlmlem  20650  znbaslem  20671  tnglem  23235  itvid  26225  lngid  26226  ttglem  26659  cchhllem  26670  edgfid  26773  resvlem  30922  hlhilslem  39134  mnringnmulrd  40758
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