Theorem ndxslid 11984

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 12003. (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 11983	. 2 |- E = Slot ( E ` ndx )
4	1, 2	ndxarg 11982	. . 3 |- ( E ` ndx ) = N
5	4, 2	eqeltri 2212	. 2 |- ( E ` ndx ) e. NN
6	3, 5	pm3.2i 270	1 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )

Syntax hints:   /\ wa 103   = wceq 1331   e. wcel 1480   ` cfv 5123   NNcn 8720   ndxcnx 11956  Slot cslot 11958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-cnex 7711  ax-resscn 7712  ax-1re 7714  ax-addrcl 7717

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fv 5131  df-inn 8721  df-ndx 11962  df-slot 11963

This theorem is referenced by:  base0  12008  baseslid  12015  plusgslid  12054  2stropg  12061  2strop1g  12064  mulrslid  12071  starvslid  12080  scaslid  12088  vscaslid  12091  ipslid  12099  tsetslid  12109  pleslid  12116  dsslid  12119