Theorem ndxslid 13355

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 13375. (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 13354	. 2 |- E = Slot ( E ` ndx )
4	1, 2	ndxarg 13353	. . 3 |- ( E ` ndx ) = N
5	4, 2	eqeltri 2311	. 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 1402   e. wcel 2209   ` cfv 5372   NNcn 9283   ndxcnx 13327  Slot cslot 13329

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 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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fv 5380  df-inn 9284  df-ndx 13333  df-slot 13334

This theorem is referenced by:  base0  13380  baseslid  13388  plusgslid  13443  2stropg  13452  2strop1g  13455  mulrslid  13463  starvslid  13472  scaslid  13484  vscaslid  13494  ipslid  13502  tsetslid  13519  pleslid  13533  dsslid  13548  homslid  13566  ccoslid  13569  prdsbaslemss  14151  zlmlemg  14935  znbaslemnn  14946  iedgvalg  16172  iedgex  16174  edgfiedgval2dom  16190  setsiedg  16207  iedgval0  16209  edgvalg  16214  edgstruct  16219