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Theorem ndxid 16090
 Description: A structure component extractor is defined by its own index. This theorem, together with strfv 16114 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 16070 and the ;10 in df-ple 16169, 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 16089 . . 3 (𝐸‘ndx) = 𝑁
43eqcomi 2780 . 2 𝑁 = (𝐸‘ndx)
5 sloteq 16069 . . 3 (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx))
61, 5syl5eq 2817 . 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 1631   ∈ wcel 2145  ‘cfv 6031  ℕcn 11222  ndxcnx 16061  Slot cslot 16063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-i2m1 10206  ax-1ne0 10207  ax-rrecex 10210  ax-cnre 10211 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-nn 11223  df-ndx 16067  df-slot 16068 This theorem is referenced by:  strndxid  16092  setsidvald  16096  baseid  16126  resslem  16140  plusgid  16185  2strop  16193  2strop1  16196  mulrid  16205  starvid  16213  scaid  16222  vscaid  16224  ipid  16231  tsetid  16249  pleid  16257  pleidOLD  16258  ocid  16269  dsid  16271  unifid  16273  homid  16283  ccoid  16285  oppglem  17987  mgplem  18702  opprlem  18836  sralem  19392  opsrbaslem  19692  opsrbaslemOLD  19693  zlmlem  20080  znbaslem  20101  znbaslemOLD  20102  tnglem  22664  itvid  25562  lngid  25563  ttglem  25977  cchhllem  25988  edgfid  26090  resvlem  30171  hlhilslem  37748
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