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Theorem ndxslid 12043
 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 12062. (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 12042 . 2 𝐸 = Slot (𝐸‘ndx)
41, 2ndxarg 12041 . . 3 (𝐸‘ndx) = 𝑁
54, 2eqeltri 2213 . 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 1332   ∈ wcel 1481  ‘cfv 5132  ℕcn 8764  ndxcnx 12015  Slot cslot 12017 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-cnex 7755  ax-resscn 7756  ax-1re 7758  ax-addrcl 7761 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-sbc 2915  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-int 3781  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-iota 5097  df-fun 5134  df-fv 5140  df-inn 8765  df-ndx 12021  df-slot 12022 This theorem is referenced by:  base0  12067  baseslid  12074  plusgslid  12113  2stropg  12120  2strop1g  12123  mulrslid  12130  starvslid  12139  scaslid  12147  vscaslid  12150  ipslid  12158  tsetslid  12168  pleslid  12175  dsslid  12178
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