Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ndxid Structured version   Visualization version   GIF version

Theorem ndxid 15864
 Description: A structure component extractor is defined by its own index. This theorem, together with strfv 15888 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 15844 and the ;10 in df-ple 15942, 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
StepHypRef Expression
1 ndxarg.1 . . . 4 𝐸 = Slot 𝑁
2 ndxarg.2 . . . 4 𝑁 ∈ ℕ
31, 2ndxarg 15863 . . 3 (𝐸‘ndx) = 𝑁
43eqcomi 2629 . 2 𝑁 = (𝐸‘ndx)
5 sloteq 15843 . . 3 (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx))
61, 5syl5eq 2666 . 2 (𝑁 = (𝐸‘ndx) → 𝐸 = Slot (𝐸‘ndx))
74, 6ax-mp 5 1 𝐸 = Slot (𝐸‘ndx)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1481   ∈ wcel 1988  ‘cfv 5876  ℕcn 11005  ndxcnx 15835  Slot cslot 15837 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-i2m1 9989  ax-1ne0 9990  ax-rrecex 9993  ax-cnre 9994 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-nn 11006  df-ndx 15841  df-slot 15842 This theorem is referenced by:  strndxid  15866  setsidvald  15870  baseid  15900  resslem  15914  plusgid  15958  2strop  15966  2strop1  15969  mulrid  15978  starvid  15986  scaid  15995  vscaid  15997  ipid  16004  tsetid  16022  pleid  16030  pleidOLD  16031  ocid  16042  dsid  16044  unifid  16046  homid  16056  ccoid  16058  oppglem  17761  mgplem  18475  opprlem  18609  sralem  19158  opsrbaslem  19458  opsrbaslemOLD  19459  zlmlem  19846  znbaslem  19867  znbaslemOLD  19868  tnglem  22425  itvid  25322  lngid  25323  ttglem  25737  cchhllem  25748  edgfid  25850  resvlem  29805  hlhilslem  37049
 Copyright terms: Public domain W3C validator