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Theorem ndxslid 11766
Description: A structure component extractor is defined by its own index. That the index is a natural number will also be needed in quite a few contexts so it is included in the conclusion of this theorem which can be used as a hypothesis of theorems like strslfv 11785. (Contributed by Jim Kingdon, 29-Jan-2023.)
Hypotheses
Ref Expression
ndxarg.1 𝐸 = Slot 𝑁
ndxarg.2 𝑁 ∈ ℕ
Assertion
Ref Expression
ndxslid (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)

Proof of Theorem ndxslid
StepHypRef Expression
1 ndxarg.1 . . 3 𝐸 = Slot 𝑁
2 ndxarg.2 . . 3 𝑁 ∈ ℕ
31, 2ndxid 11765 . 2 𝐸 = Slot (𝐸‘ndx)
41, 2ndxarg 11764 . . 3 (𝐸‘ndx) = 𝑁
54, 2eqeltri 2172 . 2 (𝐸‘ndx) ∈ ℕ
63, 5pm3.2i 268 1 (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1299  wcel 1448  cfv 5059  cn 8578  ndxcnx 11738  Slot cslot 11740
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 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-cnex 7586  ax-resscn 7587  ax-1re 7589  ax-addrcl 7592
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-iota 5024  df-fun 5061  df-fv 5067  df-inn 8579  df-ndx 11744  df-slot 11745
This theorem is referenced by:  base0  11790  baseslid  11797  plusgslid  11836  2stropg  11843  2strop1g  11846  mulrslid  11853  starvslid  11862  scaslid  11870  vscaslid  11873  ipslid  11881  tsetslid  11891  pleslid  11898  dsslid  11901
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