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Theorem ndxslid 11998
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 12017. (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 11997 . 2  |-  E  = Slot  ( E `  ndx )
41, 2ndxarg 11996 . . 3  |-  ( E `
 ndx )  =  N
54, 2eqeltri 2212 . 2  |-  ( E `
 ndx )  e.  NN
63, 5pm3.2i 270 1  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1331    e. wcel 1480   ` cfv 5123   NNcn 8732   ndxcnx 11970  Slot cslot 11972
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-cnex 7723  ax-resscn 7724  ax-1re 7726  ax-addrcl 7729
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fv 5131  df-inn 8733  df-ndx 11976  df-slot 11977
This theorem is referenced by:  base0  12022  baseslid  12029  plusgslid  12068  2stropg  12075  2strop1g  12078  mulrslid  12085  starvslid  12094  scaslid  12102  vscaslid  12105  ipslid  12113  tsetslid  12123  pleslid  12130  dsslid  12133
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