Theorem ndxid 17126

Description: A structure component extractor is defined by its own index. This theorem, together with strfv 17133 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 17141 and the ;10 in df-ple 17213, 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 17125	. . 3 ⊢ (𝐸‘ndx) = 𝑁
4	3	eqcomi 2741	. 2 ⊢ 𝑁 = (𝐸‘ndx)
5		sloteq 17112	. . 3 ⊢ (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx))
6	1, 5	eqtrid 2784	. 2 ⊢ (𝑁 = (𝐸‘ndx) → 𝐸 = Slot (𝐸‘ndx))
7	4, 6	ax-mp 5	1 ⊢ 𝐸 = Slot (𝐸‘ndx)

Syntax hints:   = wceq 1541   ∈ wcel 2106  ‘cfv 6540  ℕcn 12208  Slot cslot 17110  ndxcnx 17122

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-1cn 11164  ax-addcl 11166

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-nn 12209  df-slot 17111  df-ndx 17123

This theorem is referenced by:  strndxid  17127  setsidvaldOLD  17129  baseid  17143  2strop  17164  2strop1  17168  resslemOLD  17183  plusgid  17220  mulridx  17235  starvid  17244  scaid  17256  vscaid  17261  ipid  17272  tsetid  17294  pleid  17308  ocid  17323  dsid  17327  unifid  17337  homid  17353  ccoid  17355  oppglemOLD  19209  mgplemOLD  19986  opprlemOLD  20148  sralemOLD  20783  zlmlemOLD  21058  znbaslemOLD  21082  opsrbaslemOLD  21596  tnglemOLD  24141  itvid  27679  lngid  27680  ttglemOLD  28118  cchhllemOLD  28134  edgfid  28237  eufid  32379  resvlemOLD  32434  hlhilslemOLD  40798  mnringnmulrdOLD  42954