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Theorem ndxslid 11684
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 11703. (Contributed by Jim Kingdon, 29-Jan-2023.)
Hypotheses
Ref Expression
ndxarg.1  |-  E  = Slot 
N
ndxarg.2  |-  N  e.  NN
Assertion
Ref Expression
ndxslid  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )

Proof of Theorem ndxslid
StepHypRef Expression
1 ndxarg.1 . . 3  |-  E  = Slot 
N
2 ndxarg.2 . . 3  |-  N  e.  NN
31, 2ndxid 11683 . 2  |-  E  = Slot  ( E `  ndx )
41, 2ndxarg 11682 . . 3  |-  ( E `
 ndx )  =  N
54, 2eqeltri 2167 . 2  |-  ( E `
 ndx )  e.  NN
63, 5pm3.2i 267 1  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1296    e. wcel 1445   ` cfv 5049   NNcn 8520   ndxcnx 11656  Slot cslot 11658
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 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-cnex 7533  ax-resscn 7534  ax-1re 7536  ax-addrcl 7539
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-iota 5014  df-fun 5051  df-fv 5057  df-inn 8521  df-ndx 11662  df-slot 11663
This theorem is referenced by:  base0  11708  baseslid  11715  plusgslid  11754  2stropg  11761  2strop1g  11764  mulrslid  11771  starvslid  11780  scaslid  11788  vscaslid  11791  ipslid  11799  tsetslid  11809  pleslid  11816  dsslid  11819
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