Theorem ndxid 12889

Description: A structure component extractor is defined by its own index. This theorem, together with strslfv 12910 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 12871, 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 12888	. . 3 |- ( E ` ndx ) = N
4	3	eqcomi 2209	. 2 |- N = ( E ` ndx )
5		sloteq 12870	. . 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 5272   NNcn 9038   ndxcnx 12862  Slot cslot 12864

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 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-cnex 8018  ax-resscn 8019  ax-1re 8021  ax-addrcl 8024

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 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-iota 5233  df-fun 5274  df-fv 5280  df-inn 9039  df-ndx 12868  df-slot 12869

This theorem is referenced by:  ndxslid  12890  strndxid  12893  baseid  12919  plusgid  12975  mulridx  12996  starvid  13005  scaid  13017  vscaid  13023  ipid  13035  tsetid  13052  pleid  13066  ocid  13077  dsid  13081  unifid  13092  homid  13099  ccoid  13102  edgfid  15638