Theorem ndxid 15716

Description: A structure component extractor is defined by its own index. This theorem, together with strfv 15735 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 15700 and the ;10 in df-ple 15788, 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.)

Hypotheses

Ref	Expression
ndxarg.1	⊢ 𝐸 = Slot 𝑁
ndxarg.2	⊢ 𝑁 ∈ ℕ

Assertion

Ref	Expression
ndxid	⊢ 𝐸 = Slot (𝐸‘ndx)

Proof of Theorem ndxid

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ndxarg.1	. 2 ⊢ 𝐸 = Slot 𝑁
2		df-slot 15699	. . 3 ⊢ Slot (𝐸‘ndx) = (𝑥 ∈ V ↦ (𝑥‘(𝐸‘ndx)))
3		df-slot 15699	. . . 4 ⊢ Slot 𝑁 = (𝑥 ∈ V ↦ (𝑥‘𝑁))
4		ndxarg.2	. . . . . . 7 ⊢ 𝑁 ∈ ℕ
5	1, 4	ndxarg 15715	. . . . . 6 ⊢ (𝐸‘ndx) = 𝑁
6	5	fveq2i 6106	. . . . 5 ⊢ (𝑥‘(𝐸‘ndx)) = (𝑥‘𝑁)
7	6	mpteq2i 4669	. . . 4 ⊢ (𝑥 ∈ V ↦ (𝑥‘(𝐸‘ndx))) = (𝑥 ∈ V ↦ (𝑥‘𝑁))
8	3, 7	eqtr4i 2635	. . 3 ⊢ Slot 𝑁 = (𝑥 ∈ V ↦ (𝑥‘(𝐸‘ndx)))
9	2, 8	eqtr4i 2635	. 2 ⊢ Slot (𝐸‘ndx) = Slot 𝑁
10	1, 9	eqtr4i 2635	1 ⊢ 𝐸 = Slot (𝐸‘ndx)

Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ↦ cmpt 4643  ‘cfv 5804  ℕcn 10897  ndxcnx 15692  Slot cslot 15694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rrecex 9887  ax-cnre 9888

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-nn 10898  df-ndx 15698  df-slot 15699

This theorem is referenced by:  strndxid  15717  setsidvald  15721  baseid  15747  resslem  15760  plusgid  15804  2strop  15811  2strop1  15814  mulrid  15822  starvid  15828  scaid  15837  vscaid  15839  ipid  15846  tsetid  15864  pleid  15872  pleidOLD  15873  ocid  15884  dsid  15886  unifid  15888  homid  15898  ccoid  15900  oppglem  17603  mgplem  18317  opprlem  18451  sralem  18998  opsrbaslem  19298  opsrbaslemOLD  19299  zlmlem  19684  znbaslem  19705  znbaslemOLD  19706  tnglem  22254  itvid  25141  lngid  25142  ttglem  25556  cchhllem  25567  struct2griedg  25705  uhgrstrrepe  25745  resvlem  29162  hlhilslem  36248