Theorem ndxid 13425

Description: A structure component extractor is defined by its own index. This theorem, together with strslfv 13446 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 13407, 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 13424	. . 3 |- ( E ` ndx ) = N
4	3	eqcomi 2242	. 2 |- N = ( E ` ndx )
5		sloteq 13406	. . 3 |- ( N = ( E ` ndx ) -> Slot N = Slot ( E ` ndx ) )
6	1, 5	eqtrid 2283	. 2 |- ( N = ( E ` ndx ) -> E = Slot ( E ` ndx ) )
7	4, 6	ax-mp 5	1 |- E = Slot ( E ` ndx )

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209   ` cfv 5377   NNcn 9306   ndxcnx 13398  Slot cslot 13400

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fv 5385  df-inn 9307  df-ndx 13404  df-slot 13405

This theorem is used by:  ndxslid  13426  strndxid  13429  baseid  13455  plusgid  13513  mulridx  13534  starvid  13543  scaid  13555  vscaid  13561  ipid  13573  tsetid  13590  pleid  13604  ocid  13615  dsid  13619  unifid  13630  homid  13637  ccoid  13640  edgfid  16345