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Theorem ndxid 13521
Description: A structure component extractor is defined by its own index. This theorem, together with strfv 13532 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the  1 in df-base 13505 and the  10 in df-ple 13580, 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 13504 . . 3  |- Slot  ( E `
 ndx )  =  ( x  e.  _V  |->  ( x `  ( E `  ndx ) ) )
3 df-slot 13504 . . . 4  |- Slot  N  =  ( x  e.  _V  |->  ( x `  N
) )
4 ndxarg.2 . . . . . . 7  |-  N  e.  NN
51, 4ndxarg 13520 . . . . . 6  |-  ( E `
 ndx )  =  N
65fveq2i 5760 . . . . 5  |-  ( x `
 ( E `  ndx ) )  =  ( x `  N )
76mpteq2i 4317 . . . 4  |-  ( x  e.  _V  |->  ( x `
 ( E `  ndx ) ) )  =  ( x  e.  _V  |->  ( x `  N
) )
83, 7eqtr4i 2465 . . 3  |- Slot  N  =  ( x  e.  _V  |->  ( x `  ( E `  ndx ) ) )
92, 8eqtr4i 2465 . 2  |- Slot  ( E `
 ndx )  = Slot 
N
101, 9eqtr4i 2465 1  |-  E  = Slot  ( E `  ndx )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1727   _Vcvv 2962    e. cmpt 4291   ` cfv 5483   NNcn 10031   ndxcnx 13497  Slot cslot 13499
This theorem is referenced by:  baseid  13542  resslem  13553  plusgid  13595  2strop  13598  mulrid  13606  starvid  13612  scaid  13621  vscaid  13623  ipid  13638  tsetid  13646  pleid  13653  ocid  13660  dsid  13662  unifid  13664  homid  13674  ccoid  13676  oppglem  15177  mgplem  15684  opprlem  15764  sralem  16280  opsrbaslem  16569  zlmlem  16829  znbaslem  16850  tnglem  18712  hlhilslem  32837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-i2m1 9089  ax-1ne0 9090  ax-rrecex 9093  ax-cnre 9094
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-recs 6662  df-rdg 6697  df-nn 10032  df-ndx 13503  df-slot 13504
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