Theorem strndxid 13324

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 13320	. . 3 |- E = Slot ( E ` ndx )
4		strndxid.s	. . 3 |- ( ph -> S e. V )
5	1, 2	ndxarg 13319	. . . . 5 |- ( E ` ndx ) = N
6	5, 2	eqeltri 2307	. . . 4 |- ( E ` ndx ) e. NN
7	6	a1i 9	. . 3 |- ( ph -> ( E ` ndx ) e. NN )
8	3, 4, 7	strnfvnd 13316	. 2 |- ( ph -> ( E ` S ) = ( S ` ( E ` ndx ) ) )
9	8	eqcomd 2240	1 |- ( ph -> ( S ` ( E ` ndx ) ) = ( E ` S ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   ` cfv 5357   NNcn 9254   ndxcnx 13293  Slot cslot 13295

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fv 5365  df-inn 9255  df-ndx 13299  df-slot 13300

This theorem is referenced by:  imasbas  13571  imasplusg  13572  imasmulr  13573