ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ndxid GIF version

Theorem ndxid 12020
Description: A structure component extractor is defined by its own index. This theorem, together with strslfv 12040 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 12002, 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
StepHypRef Expression
1 ndxarg.1 . . . 4 𝐸 = Slot 𝑁
2 ndxarg.2 . . . 4 𝑁 ∈ ℕ
31, 2ndxarg 12019 . . 3 (𝐸‘ndx) = 𝑁
43eqcomi 2144 . 2 𝑁 = (𝐸‘ndx)
5 sloteq 12001 . . 3 (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx))
61, 5syl5eq 2185 . 2 (𝑁 = (𝐸‘ndx) → 𝐸 = Slot (𝐸‘ndx))
74, 6ax-mp 5 1 𝐸 = Slot (𝐸‘ndx)
Colors of variables: wff set class
Syntax hints:   = wceq 1332  wcel 1481  cfv 5130  cn 8743  ndxcnx 11993  Slot cslot 11995
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 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-cnex 7734  ax-resscn 7735  ax-1re 7737  ax-addrcl 7740
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 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-iota 5095  df-fun 5132  df-fv 5138  df-inn 8744  df-ndx 11999  df-slot 12000
This theorem is referenced by:  ndxslid  12021  strndxid  12024  baseid  12049  plusgid  12090  mulrid  12107  starvid  12116  scaid  12124  vscaid  12127  ipid  12135  tsetid  12145  pleid  12152  dsid  12155
  Copyright terms: Public domain W3C validator