Theorem ndxid 12856

Description: A structure component extractor is defined by its own index. This theorem, together with strslfv 12877 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 12838, 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

Step	Hyp	Ref	Expression
1		ndxarg.1	. . . 4 |- E = Slot N
2		ndxarg.2	. . . 4 |- N e. NN
3	1, 2	ndxarg 12855	. . 3 |- ( E ` ndx ) = N
4	3	eqcomi 2209	. 2 |- N = ( E ` ndx )
5		sloteq 12837	. . 3 |- ( N = ( E ` ndx ) -> Slot N = Slot ( E ` ndx ) )
6	1, 5	eqtrid 2250	. 2 |- ( N = ( E ` ndx ) -> E = Slot ( E ` ndx ) )
7	4, 6	ax-mp 5	1 |- E = Slot ( E ` ndx )

Syntax hints:   = wceq 1373   e. wcel 2176   ` cfv 5271   NNcn 9036   ndxcnx 12829  Slot cslot 12831

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fv 5279  df-inn 9037  df-ndx 12835  df-slot 12836

This theorem is referenced by:  ndxslid  12857  strndxid  12860  baseid  12886  plusgid  12942  mulridx  12963  starvid  12972  scaid  12984  vscaid  12990  ipid  13002  tsetid  13019  pleid  13033  ocid  13044  dsid  13048  unifid  13059  homid  13066  ccoid  13069  edgfid  15605