Theorem ndxid 12355

Description: A structure component extractor is defined by its own index. This theorem, together with strslfv 12375 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 12337, 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 12354	. . 3 ⊢ (𝐸‘ndx) = 𝑁
4	3	eqcomi 2168	. 2 ⊢ 𝑁 = (𝐸‘ndx)
5		sloteq 12336	. . 3 ⊢ (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx))
6	1, 5	syl5eq 2209	. 2 ⊢ (𝑁 = (𝐸‘ndx) → 𝐸 = Slot (𝐸‘ndx))
7	4, 6	ax-mp 5	1 ⊢ 𝐸 = Slot (𝐸‘ndx)

Syntax hints:   = wceq 1342   ∈ wcel 2135  ‘cfv 5182  ℕcn 8848  ndxcnx 12328  Slot cslot 12330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-cnex 7835  ax-resscn 7836  ax-1re 7838  ax-addrcl 7841

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-sbc 2947  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-iota 5147  df-fun 5184  df-fv 5190  df-inn 8849  df-ndx 12334  df-slot 12335

This theorem is referenced by:  ndxslid  12356  strndxid  12359  baseid  12384  plusgid  12425  mulrid  12442  starvid  12451  scaid  12459  vscaid  12462  ipid  12470  tsetid  12480  pleid  12487  dsid  12490