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Theorem ndxid 12469
Description: A structure component extractor is defined by its own index. This theorem, together with strslfv 12489 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the  1 in df-base 12451, 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  e.  NN
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  e.  NN
31, 2ndxarg 12468 . . 3  |-  ( E `
 ndx )  =  N
43eqcomi 2181 . 2  |-  N  =  ( E `  ndx )
5 sloteq 12450 . . 3  |-  ( N  =  ( E `  ndx )  -> Slot  N  = Slot  ( E `  ndx ) )
61, 5eqtrid 2222 . 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 1353    e. wcel 2148   ` cfv 5212   NNcn 8908   ndxcnx 12442  Slot cslot 12444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fv 5220  df-inn 8909  df-ndx 12448  df-slot 12449
This theorem is referenced by:  ndxslid  12470  strndxid  12473  baseid  12498  plusgid  12550  mulrid  12568  starvid  12577  scaid  12585  vscaid  12591  ipid  12599  tsetid  12609  pleid  12623  dsid  12626
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