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Theorem ndxid 17074
Description: A structure component extractor is defined by its own index. This theorem, together with strfv 17081 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 17089 and the 10 in df-ple 17158, making it easier to change should the need arise.

For example, we can refer to a specific poset with base set 𝐡 and order relation 𝐿 using {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(leβ€˜ndx), 𝐿⟩} rather than {⟨1, 𝐡⟩, ⟨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 ∈ β„•
Assertion
Ref Expression
ndxid 𝐸 = Slot (πΈβ€˜ndx)

Proof of Theorem ndxid
StepHypRef Expression
1 ndxarg.e . . . 4 𝐸 = Slot 𝑁
2 ndxarg.n . . . 4 𝑁 ∈ β„•
31, 2ndxarg 17073 . . 3 (πΈβ€˜ndx) = 𝑁
43eqcomi 2742 . 2 𝑁 = (πΈβ€˜ndx)
5 sloteq 17060 . . 3 (𝑁 = (πΈβ€˜ndx) β†’ Slot 𝑁 = Slot (πΈβ€˜ndx))
61, 5eqtrid 2785 . 2 (𝑁 = (πΈβ€˜ndx) β†’ 𝐸 = Slot (πΈβ€˜ndx))
74, 6ax-mp 5 1 𝐸 = Slot (πΈβ€˜ndx)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   ∈ wcel 2107  β€˜cfv 6497  β„•cn 12158  Slot cslot 17058  ndxcnx 17070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-1cn 11114  ax-addcl 11116
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-nn 12159  df-slot 17059  df-ndx 17071
This theorem is referenced by:  strndxid  17075  setsidvaldOLD  17077  baseid  17091  2strop  17112  2strop1  17116  resslemOLD  17128  plusgid  17165  mulrid  17180  starvid  17189  scaid  17201  vscaid  17206  ipid  17217  tsetid  17239  pleid  17253  ocid  17268  dsid  17272  unifid  17282  homid  17298  ccoid  17300  oppglemOLD  19134  mgplemOLD  19906  opprlemOLD  20060  sralemOLD  20655  zlmlemOLD  20934  znbaslemOLD  20958  opsrbaslemOLD  21467  tnglemOLD  24013  itvid  27423  lngid  27424  ttglemOLD  27862  cchhllemOLD  27878  edgfid  27981  resvlemOLD  32170  hlhilslemOLD  40448  mnringnmulrdOLD  42578
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