Theorem ndxid 17158

Description: A structure component extractor is defined by its own index. This theorem, together with strfv 17164 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 17171 and the ;10 in df-ple 17231, 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 17157	. . 3 ⊢ (𝐸‘ndx) = 𝑁
4	3	eqcomi 2748	. 2 ⊢ 𝑁 = (𝐸‘ndx)
5		sloteq 17144	. . 3 ⊢ (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx))
6	1, 5	eqtrid 2786	. 2 ⊢ (𝑁 = (𝐸‘ndx) → 𝐸 = Slot (𝐸‘ndx))
7	4, 6	ax-mp 5	1 ⊢ 𝐸 = Slot (𝐸‘ndx)

Syntax hints:   = wceq 1547   ∈ wcel 2119  ‘cfv 6485  ℕcn 12165  Slot cslot 17142  ndxcnx 17154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-1cn 11087  ax-addcl 11089

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-nn 12166  df-slot 17143  df-ndx 17155

This theorem is referenced by:  strndxid  17159  baseid  17173  2strop  17190  plusgid  17238  mulridx  17249  starvid  17257  scaid  17269  vscaid  17274  ipid  17285  tsetid  17307  pleid  17321  ocid  17336  dsid  17340  unifid  17350  homid  17366  ccoid  17368  itvid  28525  lngid  28526  edgfid  29077  eufid  33375