Theorem ndxid 16898

Description: A structure component extractor is defined by its own index. This theorem, together with strfv 16905 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 16913 and the ;10 in df-ple 16982, 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.e	⊢ 𝐸 = Slot 𝑁
ndxarg.n	⊢ 𝑁 ∈ ℕ

Assertion

Ref	Expression
ndxid	⊢ 𝐸 = Slot (𝐸‘ndx)

Proof of Theorem ndxid

Step	Hyp	Ref	Expression
1		ndxarg.e	. . . 4 ⊢ 𝐸 = Slot 𝑁
2		ndxarg.n	. . . 4 ⊢ 𝑁 ∈ ℕ
3	1, 2	ndxarg 16897	. . 3 ⊢ (𝐸‘ndx) = 𝑁
4	3	eqcomi 2747	. 2 ⊢ 𝑁 = (𝐸‘ndx)
5		sloteq 16884	. . 3 ⊢ (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx))
6	1, 5	eqtrid 2790	. 2 ⊢ (𝑁 = (𝐸‘ndx) → 𝐸 = Slot (𝐸‘ndx))
7	4, 6	ax-mp 5	1 ⊢ 𝐸 = Slot (𝐸‘ndx)

Syntax hints:   = wceq 1539   ∈ wcel 2106  ‘cfv 6433  ℕcn 11973  Slot cslot 16882  ndxcnx 16894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-1cn 10929  ax-addcl 10931

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-nn 11974  df-slot 16883  df-ndx 16895

This theorem is referenced by:  strndxid  16899  setsidvaldOLD  16901  baseid  16915  2strop  16936  2strop1  16940  resslemOLD  16952  plusgid  16989  mulrid  17004  starvid  17013  scaid  17025  vscaid  17030  ipid  17041  tsetid  17063  pleid  17077  ocid  17092  dsid  17096  unifid  17106  homid  17122  ccoid  17124  oppglemOLD  18955  mgplemOLD  19725  opprlemOLD  19868  sralemOLD  20440  zlmlemOLD  20719  znbaslemOLD  20743  opsrbaslemOLD  21251  tnglemOLD  23797  itvid  26800  lngid  26801  ttglemOLD  27239  cchhllemOLD  27255  edgfid  27358  resvlemOLD  31531  hlhilslemOLD  39953  mnringnmulrdOLD  41828