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Theorem ndxslid 12428
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 12447. (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 12427 . 2  |-  E  = Slot  ( E `  ndx )
41, 2ndxarg 12426 . . 3  |-  ( E `
 ndx )  =  N
54, 2eqeltri 2243 . 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 1348    e. wcel 2141   ` cfv 5196   NNcn 8865   ndxcnx 12400  Slot cslot 12402
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-cnex 7852  ax-resscn 7853  ax-1re 7855  ax-addrcl 7858
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-iota 5158  df-fun 5198  df-fv 5204  df-inn 8866  df-ndx 12406  df-slot 12407
This theorem is referenced by:  base0  12452  baseslid  12459  plusgslid  12500  2stropg  12507  2strop1g  12510  mulrslid  12517  starvslid  12526  scaslid  12534  vscaslid  12537  ipslid  12545  tsetslid  12555  pleslid  12562  dsslid  12565
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