Theorem ndxslid 13237

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 13257. (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 13236	. 2 |- E = Slot ( E ` ndx )
4	1, 2	ndxarg 13235	. . 3 |- ( E ` ndx ) = N
5	4, 2	eqeltri 2305	. 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 1398   e. wcel 2203   ` cfv 5352   NNcn 9237   ndxcnx 13209  Slot cslot 13211

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fv 5360  df-inn 9238  df-ndx 13215  df-slot 13216

This theorem is referenced by:  base0  13262  baseslid  13270  plusgslid  13325  2stropg  13334  2strop1g  13337  mulrslid  13345  starvslid  13354  scaslid  13366  vscaslid  13376  ipslid  13384  tsetslid  13401  pleslid  13415  dsslid  13430  homslid  13448  ccoslid  13451  prdsbaslemss  13487  zlmlemg  14776  znbaslemnn  14787  iedgvalg  16012  iedgex  16014  edgfiedgval2dom  16030  setsiedg  16047  iedgval0  16049  edgvalg  16054  edgstruct  16059