Theorem ndxid 17129

Description: A structure component extractor is defined by its own index. This theorem, together with strfv 17136 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 17144 and the ;10 in df-ple 17216, 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 17128	. . 3 ⊢ (𝐸‘ndx) = 𝑁
4	3	eqcomi 2733	. 2 ⊢ 𝑁 = (𝐸‘ndx)
5		sloteq 17115	. . 3 ⊢ (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx))
6	1, 5	eqtrid 2776	. 2 ⊢ (𝑁 = (𝐸‘ndx) → 𝐸 = Slot (𝐸‘ndx))
7	4, 6	ax-mp 5	1 ⊢ 𝐸 = Slot (𝐸‘ndx)

Syntax hints:   = wceq 1533   ∈ wcel 2098  ‘cfv 6533  ℕcn 12209  Slot cslot 17113  ndxcnx 17125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-1cn 11164  ax-addcl 11166

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-om 7849  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-nn 12210  df-slot 17114  df-ndx 17126

This theorem is referenced by:  strndxid  17130  setsidvaldOLD  17132  baseid  17146  2strop  17167  2strop1  17171  resslemOLD  17186  plusgid  17223  mulridx  17238  starvid  17247  scaid  17259  vscaid  17264  ipid  17275  tsetid  17297  pleid  17311  ocid  17326  dsid  17330  unifid  17340  homid  17356  ccoid  17358  oppglemOLD  19257  mgplemOLD  20034  opprlemOLD  20232  sralemOLD  21015  zlmlemOLD  21372  znbaslemOLD  21398  opsrbaslemOLD  21915  tnglemOLD  24472  itvid  28159  lngid  28160  ttglemOLD  28598  cchhllemOLD  28614  edgfid  28717  eufid  32857  resvlemOLD  32912  hlhilslemOLD  41300  mnringnmulrdOLD  43458