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Theorem ndxid 13260
Description: A structure component extractor is defined by its own index. This theorem, together with strfv 13271 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the  1 in df-base 13244 and the  10 in df-ple 13319, making it easier to change should the need arise. For example, we can refer to a specific poset with base set  B and order relation  L using  { <. ( Base `  ndx ) ,  B >. ,  <. ( le `  ndx ) ,  L >. } rather than  { <. 1 ,  B >. ,  <. 10 ,  L >. }. The latter, while shorter to state, requires revision if we later change  10 to some other number, and it may also be harder to remember. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.)
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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ndxarg.1 . 2  |-  E  = Slot 
N
2 df-slot 13243 . . 3  |- Slot  ( E `
 ndx )  =  ( x  e.  _V  |->  ( x `  ( E `  ndx ) ) )
3 df-slot 13243 . . . 4  |- Slot  N  =  ( x  e.  _V  |->  ( x `  N
) )
4 ndxarg.2 . . . . . . 7  |-  N  e.  NN
51, 4ndxarg 13259 . . . . . 6  |-  ( E `
 ndx )  =  N
65fveq2i 5608 . . . . 5  |-  ( x `
 ( E `  ndx ) )  =  ( x `  N )
76mpteq2i 4182 . . . 4  |-  ( x  e.  _V  |->  ( x `
 ( E `  ndx ) ) )  =  ( x  e.  _V  |->  ( x `  N
) )
83, 7eqtr4i 2381 . . 3  |- Slot  N  =  ( x  e.  _V  |->  ( x `  ( E `  ndx ) ) )
92, 8eqtr4i 2381 . 2  |- Slot  ( E `
 ndx )  = Slot 
N
101, 9eqtr4i 2381 1  |-  E  = Slot  ( E `  ndx )
Colors of variables: wff set class
Syntax hints:    = wceq 1642    e. wcel 1710   _Vcvv 2864    e. cmpt 4156   ` cfv 5334   NNcn 9833   ndxcnx 13236  Slot cslot 13238
This theorem is referenced by:  baseid  13281  resslem  13292  plusgid  13334  2strop  13337  mulrid  13345  starvid  13351  scaid  13360  vscaid  13362  ipid  13377  tsetid  13385  pleid  13392  ocid  13399  dsid  13401  unifid  13403  homid  13413  ccoid  13415  oppchomfval  13710  oppccofval  13712  oppcbas  13714  reschom  13800  rescco  13802  fuccofval  13926  fuchom  13928  setchomfval  14004  setccofval  14007  catchomfval  14023  catccofval  14025  oppglem  14916  mgplem  15423  opprlem  15503  sralem  16023  opsrbaslem  16312  zlmlem  16571  znbaslem  16592  tnglem  18252  tuslem  23565  hlhilslem  32183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-i2m1 8892  ax-1ne0 8893  ax-rrecex 8896  ax-cnre 8897
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-recs 6472  df-rdg 6507  df-nn 9834  df-ndx 13242  df-slot 13243
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