Theorem ndxid 16512

Description: A structure component extractor is defined by its own index. This theorem, together with strfv 16534 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 16492 and the ;10 in df-ple 16588, 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

Step	Hyp	Ref	Expression
1		ndxarg.1	. . . 4 ⊢ 𝐸 = Slot 𝑁
2		ndxarg.2	. . . 4 ⊢ 𝑁 ∈ ℕ
3	1, 2	ndxarg 16511	. . 3 ⊢ (𝐸‘ndx) = 𝑁
4	3	eqcomi 2833	. 2 ⊢ 𝑁 = (𝐸‘ndx)
5		sloteq 16491	. . 3 ⊢ (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx))
6	1, 5	syl5eq 2871	. 2 ⊢ (𝑁 = (𝐸‘ndx) → 𝐸 = Slot (𝐸‘ndx))
7	4, 6	ax-mp 5	1 ⊢ 𝐸 = Slot (𝐸‘ndx)

Syntax hints:   = wceq 1536   ∈ wcel 2113  ‘cfv 6358  ℕcn 11641  ndxcnx 16483  Slot cslot 16485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-1cn 10598  ax-addcl 10600

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-om 7584  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-nn 11642  df-ndx 16489  df-slot 16490

This theorem is referenced by:  strndxid  16513  setsidvald  16517  baseid  16546  resslem  16560  plusgid  16599  2strop  16607  2strop1  16610  mulrid  16619  starvid  16627  scaid  16636  vscaid  16638  ipid  16645  tsetid  16663  pleid  16670  ocid  16677  dsid  16679  unifid  16681  homid  16691  ccoid  16693  oppglem  18481  mgplem  19247  opprlem  19381  sralem  19952  opsrbaslem  20261  zlmlem  20667  znbaslem  20688  tnglem  23252  itvid  26231  lngid  26232  ttglem  26665  cchhllem  26676  edgfid  26779  resvlem  30908  hlhilslem  39078