Theorem ndxid 12775

Description: A structure component extractor is defined by its own index. This theorem, together with strslfv 12796 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 12757, 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 12774	. . 3 ⊢ (𝐸‘ndx) = 𝑁
4	3	eqcomi 2208	. 2 ⊢ 𝑁 = (𝐸‘ndx)
5		sloteq 12756	. . 3 ⊢ (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx))
6	1, 5	eqtrid 2249	. 2 ⊢ (𝑁 = (𝐸‘ndx) → 𝐸 = Slot (𝐸‘ndx))
7	4, 6	ax-mp 5	1 ⊢ 𝐸 = Slot (𝐸‘ndx)

Syntax hints:   = wceq 1372   ∈ wcel 2175  ‘cfv 5268  ℕcn 9018  ndxcnx 12748  Slot cslot 12750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-cnex 7998  ax-resscn 7999  ax-1re 8001  ax-addrcl 8004

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-iota 5229  df-fun 5270  df-fv 5276  df-inn 9019  df-ndx 12754  df-slot 12755

This theorem is referenced by:  ndxslid  12776  strndxid  12779  baseid  12805  plusgid  12861  mulridx  12881  starvid  12890  scaid  12902  vscaid  12908  ipid  12920  tsetid  12937  pleid  12951  ocid  12962  dsid  12966  unifid  12977  homid  12984  ccoid  12987