Theorem ndxid 12727

Description: A structure component extractor is defined by its own index. This theorem, together with strslfv 12748 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 12709, 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 12726	. . 3 |- ( E ` ndx ) = N
4	3	eqcomi 2200	. 2 |- N = ( E ` ndx )
5		sloteq 12708	. . 3 |- ( N = ( E ` ndx ) -> Slot N = Slot ( E ` ndx ) )
6	1, 5	eqtrid 2241	. 2 |- ( N = ( E ` ndx ) -> E = Slot ( E ` ndx ) )
7	4, 6	ax-mp 5	1 |- E = Slot ( E ` ndx )

Syntax hints:   = wceq 1364   e. wcel 2167   ` cfv 5259   NNcn 9007   ndxcnx 12700  Slot cslot 12702

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fv 5267  df-inn 9008  df-ndx 12706  df-slot 12707

This theorem is referenced by:  ndxslid  12728  strndxid  12731  baseid  12757  plusgid  12813  mulridx  12833  starvid  12842  scaid  12854  vscaid  12860  ipid  12872  tsetid  12889  pleid  12903  ocid  12914  dsid  12918  unifid  12929  homid  12936  ccoid  12939