Theorem ndxslid 12643

Description: A structure component extractor is defined by its own index. That the index is a natural number will also be needed in quite a few contexts so it is included in the conclusion of this theorem which can be used as a hypothesis of theorems like strslfv 12663. (Contributed by Jim Kingdon, 29-Jan-2023.)

Hypotheses

Ref	Expression
ndxarg.1	|- E = Slot N
ndxarg.2	|- N e. NN

Assertion

Ref	Expression
ndxslid	|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )

Proof of Theorem ndxslid

Step	Hyp	Ref	Expression
1		ndxarg.1	. . 3 |- E = Slot N
2		ndxarg.2	. . 3 |- N e. NN
3	1, 2	ndxid 12642	. 2 |- E = Slot ( E ` ndx )
4	1, 2	ndxarg 12641	. . 3 |- ( E ` ndx ) = N
5	4, 2	eqeltri 2266	. 2 |- ( E ` ndx ) e. NN
6	3, 5	pm3.2i 272	1 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2164   ` cfv 5254   NNcn 8982   ndxcnx 12615  Slot cslot 12617

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fv 5262  df-inn 8983  df-ndx 12621  df-slot 12622

This theorem is referenced by:  base0  12668  baseslid  12675  plusgslid  12730  2stropg  12738  2strop1g  12741  mulrslid  12749  starvslid  12758  scaslid  12770  vscaslid  12780  ipslid  12788  tsetslid  12805  pleslid  12819  dsslid  12830  homslid  12847  ccoslid  12849  zlmlemg  14116  znbaslemnn  14127