Theorem ndxid 17115

Description: A structure component extractor is defined by its own index. This theorem, together with strfv 17121 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 17128 and the ;10 in df-ple 17188, 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 17114	. . 3 ⊢ (𝐸‘ndx) = 𝑁
4	3	eqcomi 2742	. 2 ⊢ 𝑁 = (𝐸‘ndx)
5		sloteq 17101	. . 3 ⊢ (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx))
6	1, 5	eqtrid 2780	. 2 ⊢ (𝑁 = (𝐸‘ndx) → 𝐸 = Slot (𝐸‘ndx))
7	4, 6	ax-mp 5	1 ⊢ 𝐸 = Slot (𝐸‘ndx)

Syntax hints:   = wceq 1541   ∈ wcel 2113  ‘cfv 6489  ℕcn 12136  Slot cslot 17099  ndxcnx 17111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-1cn 11075  ax-addcl 11077

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-om 7806  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-nn 12137  df-slot 17100  df-ndx 17112

This theorem is referenced by:  strndxid  17116  baseid  17130  2strop  17147  plusgid  17195  mulridx  17206  starvid  17214  scaid  17226  vscaid  17231  ipid  17242  tsetid  17264  pleid  17278  ocid  17293  dsid  17297  unifid  17307  homid  17323  ccoid  17325  itvid  28437  lngid  28438  edgfid  28989  eufid  33301