Theorem ndxslid 13052

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 13072. (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 13051	. 2 |- E = Slot ( E ` ndx )
4	1, 2	ndxarg 13050	. . 3 |- ( E ` ndx ) = N
5	4, 2	eqeltri 2302	. 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 1395   e. wcel 2200   ` cfv 5317   NNcn 9106   ndxcnx 13024  Slot cslot 13026

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fv 5325  df-inn 9107  df-ndx 13030  df-slot 13031

This theorem is referenced by:  base0  13077  baseslid  13085  plusgslid  13140  2stropg  13149  2strop1g  13152  mulrslid  13160  starvslid  13169  scaslid  13181  vscaslid  13191  ipslid  13199  tsetslid  13216  pleslid  13230  dsslid  13245  homslid  13263  ccoslid  13266  prdsbaslemss  13302  zlmlemg  14586  znbaslemnn  14597  iedgvalg  15812  iedgex  15814  edgfiedgval2dom  15830  setsiedg  15847  iedgval0  15849  edgvalg  15854  edgstruct  15858