Theorem ndxid 11765

Description: A structure component extractor is defined by its own index. This theorem, together with strslfv 11785 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 11747, 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 11764	. . 3 |- ( E ` ndx ) = N
4	3	eqcomi 2104	. 2 |- N = ( E ` ndx )
5		sloteq 11746	. . 3 |- ( N = ( E ` ndx ) -> Slot N = Slot ( E ` ndx ) )
6	1, 5	syl5eq 2144	. 2 |- ( N = ( E ` ndx ) -> E = Slot ( E ` ndx ) )
7	4, 6	ax-mp 7	1 |- E = Slot ( E ` ndx )

Syntax hints:   = wceq 1299   e. wcel 1448   ` cfv 5059   NNcn 8578   ndxcnx 11738  Slot cslot 11740

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-cnex 7586  ax-resscn 7587  ax-1re 7589  ax-addrcl 7592

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-iota 5024  df-fun 5061  df-fv 5067  df-inn 8579  df-ndx 11744  df-slot 11745

This theorem is referenced by:  ndxslid  11766  strndxid  11769  baseid  11794  plusgid  11835  mulrid  11852  starvid  11861  scaid  11869  vscaid  11872  ipid  11880  tsetid  11890  pleid  11897  dsid  11900