Theorem ndxid 11820

Description: A structure component extractor is defined by its own index. This theorem, together with strslfv 11840 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 11802, making it easier to change should the need arise.

(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

Step	Hyp	Ref	Expression
1		ndxarg.1	. . . 4 ⊢ 𝐸 = Slot 𝑁
2		ndxarg.2	. . . 4 ⊢ 𝑁 ∈ ℕ
3	1, 2	ndxarg 11819	. . 3 ⊢ (𝐸‘ndx) = 𝑁
4	3	eqcomi 2117	. 2 ⊢ 𝑁 = (𝐸‘ndx)
5		sloteq 11801	. . 3 ⊢ (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx))
6	1, 5	syl5eq 2157	. 2 ⊢ (𝑁 = (𝐸‘ndx) → 𝐸 = Slot (𝐸‘ndx))
7	4, 6	ax-mp 7	1 ⊢ 𝐸 = Slot (𝐸‘ndx)

Syntax hints:   = wceq 1312   ∈ wcel 1461  ‘cfv 5079  ℕcn 8624  ndxcnx 11793  Slot cslot 11795

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-cnex 7630  ax-resscn 7631  ax-1re 7633  ax-addrcl 7636

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-iota 5044  df-fun 5081  df-fv 5087  df-inn 8625  df-ndx 11799  df-slot 11800

This theorem is referenced by:  ndxslid  11821  strndxid  11824  baseid  11849  plusgid  11890  mulrid  11907  starvid  11916  scaid  11924  vscaid  11927  ipid  11935  tsetid  11945  pleid  11952  dsid  11955