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Theorem ndxslid 11911
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 11930. (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 11910 . 2 𝐸 = Slot (𝐸‘ndx)
41, 2ndxarg 11909 . . 3 (𝐸‘ndx) = 𝑁
54, 2eqeltri 2190 . 2 (𝐸‘ndx) ∈ ℕ
63, 5pm3.2i 270 1 (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1316  wcel 1465  cfv 5093  cn 8688  ndxcnx 11883  Slot cslot 11885
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-cnex 7679  ax-resscn 7680  ax-1re 7682  ax-addrcl 7685
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-iota 5058  df-fun 5095  df-fv 5101  df-inn 8689  df-ndx 11889  df-slot 11890
This theorem is referenced by:  base0  11935  baseslid  11942  plusgslid  11981  2stropg  11988  2strop1g  11991  mulrslid  11998  starvslid  12007  scaslid  12015  vscaslid  12018  ipslid  12026  tsetslid  12036  pleslid  12043  dsslid  12046
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