Theorem strndxid 13060

Description: The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.)

Hypotheses

Ref	Expression
strndxid.s	|- ( ph -> S e. V )
strndxid.e	|- E = Slot N
strndxid.n	|- N e. NN

Assertion

Ref	Expression
strndxid	|- ( ph -> ( S ` ( E ` ndx ) ) = ( E ` S ) )

Proof of Theorem strndxid

Step	Hyp	Ref	Expression
1		strndxid.e	. . . 4 |- E = Slot N
2		strndxid.n	. . . 4 |- N e. NN
3	1, 2	ndxid 13056	. . 3 |- E = Slot ( E ` ndx )
4		strndxid.s	. . 3 |- ( ph -> S e. V )
5	1, 2	ndxarg 13055	. . . . 5 |- ( E ` ndx ) = N
6	5, 2	eqeltri 2302	. . . 4 |- ( E ` ndx ) e. NN
7	6	a1i 9	. . 3 |- ( ph -> ( E ` ndx ) e. NN )
8	3, 4, 7	strnfvnd 13052	. 2 |- ( ph -> ( E ` S ) = ( S ` ( E ` ndx ) ) )
9	8	eqcomd 2235	1 |- ( ph -> ( S ` ( E ` ndx ) ) = ( E ` S ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   ` cfv 5318   NNcn 9110   ndxcnx 13029  Slot cslot 13031

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 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096

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 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fv 5326  df-inn 9111  df-ndx 13035  df-slot 13036

This theorem is referenced by:  imasbas  13340  imasplusg  13341  imasmulr  13342