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Theorem ndxid 15659
Description: A structure component extractor is defined by its own index. This theorem, together with strfv 15678 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 15643 and the 10 in df-ple 15731, 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.)

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

Proof of Theorem ndxid
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ndxarg.1 . 2 𝐸 = Slot 𝑁
2 df-slot 15642 . . 3 Slot (𝐸‘ndx) = (𝑥 ∈ V ↦ (𝑥‘(𝐸‘ndx)))
3 df-slot 15642 . . . 4 Slot 𝑁 = (𝑥 ∈ V ↦ (𝑥𝑁))
4 ndxarg.2 . . . . . . 7 𝑁 ∈ ℕ
51, 4ndxarg 15658 . . . . . 6 (𝐸‘ndx) = 𝑁
65fveq2i 6088 . . . . 5 (𝑥‘(𝐸‘ndx)) = (𝑥𝑁)
76mpteq2i 4660 . . . 4 (𝑥 ∈ V ↦ (𝑥‘(𝐸‘ndx))) = (𝑥 ∈ V ↦ (𝑥𝑁))
83, 7eqtr4i 2631 . . 3 Slot 𝑁 = (𝑥 ∈ V ↦ (𝑥‘(𝐸‘ndx)))
92, 8eqtr4i 2631 . 2 Slot (𝐸‘ndx) = Slot 𝑁
101, 9eqtr4i 2631 1 𝐸 = Slot (𝐸‘ndx)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wcel 1976  Vcvv 3169  cmpt 4634  cfv 5787  cn 10864  ndxcnx 15635  Slot cslot 15637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-i2m1 9857  ax-1ne0 9858  ax-rrecex 9861  ax-cnre 9862
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-ov 6527  df-om 6932  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-nn 10865  df-ndx 15641  df-slot 15642
This theorem is referenced by:  strndxid  15660  setsidvald  15664  baseid  15690  resslem  15703  plusgid  15747  2strop  15754  2strop1  15757  mulrid  15765  starvid  15771  scaid  15780  vscaid  15782  ipid  15789  tsetid  15807  pleid  15815  pleidOLD  15816  ocid  15827  dsid  15829  unifid  15831  homid  15841  ccoid  15843  oppglem  17546  mgplem  18260  opprlem  18394  sralem  18941  opsrbaslem  19241  opsrbaslemOLD  19242  zlmlem  19626  znbaslem  19647  znbaslemOLD  19648  tnglem  22189  itvid  25055  lngid  25056  ttglem  25471  cchhllem  25482  resvlem  28965  hlhilslem  36048  struct2griedg  40260  uhgrstrrepe  40303
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