Theorem ndxslid 12646

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 12666. (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 12645	. 2 |- E = Slot ( E ` ndx )
4	1, 2	ndxarg 12644	. . 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 5255   NNcn 8984   ndxcnx 12618  Slot cslot 12620

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 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

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 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fv 5263  df-inn 8985  df-ndx 12624  df-slot 12625

This theorem is referenced by:  base0  12671  baseslid  12678  plusgslid  12733  2stropg  12741  2strop1g  12744  mulrslid  12752  starvslid  12761  scaslid  12773  vscaslid  12783  ipslid  12791  tsetslid  12808  pleslid  12822  dsslid  12833  homslid  12850  ccoslid  12852  zlmlemg  14127  znbaslemnn  14138