Theorem strslfv 12438

Description: Extract a structure component C (such as the base set) from a structure S with a component extractor E (such as the base set extractor df-base 12400). By virtue of ndxslid 12419, this can be done without having to refer to the hard-coded numeric index of E. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Jim Kingdon, 30-Jan-2023.)

Hypotheses

Ref	Expression
strfv.s	|- S Struct X
strslfv.e	|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
strfv.n	|- { <. ( E ` ndx ) , C >. } C_ S

Assertion

Ref	Expression
strslfv	|- ( C e. V -> C = ( E ` S ) )

Proof of Theorem strslfv

Step	Hyp	Ref	Expression
1		strslfv.e	. 2 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
2		strfv.s	. . 3 |- S Struct X
3		structex 12406	. . 3 |- ( S Struct X -> S e. _V )
4	2, 3	mp1i 10	. 2 |- ( C e. V -> S e. _V )
5	2	structfun 12412	. . 3 |- Fun `' `' S
6	5	a1i 9	. 2 |- ( C e. V -> Fun `' `' S )
7		strfv.n	. . 3 |- { <. ( E ` ndx ) , C >. } C_ S
8	1	simpri 112	. . . . 5 |- ( E ` ndx ) e. NN
9		opexg 4206	. . . . 5 |- ( ( ( E ` ndx ) e. NN /\ C e. V ) -> <. ( E ` ndx ) , C >. e. _V )
10	8, 9	mpan 421	. . . 4 |- ( C e. V -> <. ( E ` ndx ) , C >. e. _V )
11		snssg 3709	. . . 4 |- ( <. ( E ` ndx ) , C >. e. _V -> ( <. ( E ` ndx ) , C >. e. S <-> { <. ( E ` ndx ) , C >. } C_ S ) )
12	10, 11	syl 14	. . 3 |- ( C e. V -> ( <. ( E ` ndx ) , C >. e. S <-> { <. ( E ` ndx ) , C >. } C_ S ) )
13	7, 12	mpbiri 167	. 2 |- ( C e. V -> <. ( E ` ndx ) , C >. e. S )
14		id 19	. 2 |- ( C e. V -> C e. V )
15	1, 4, 6, 13, 14	strslfv2d 12436	1 |- ( C e. V -> C = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   _Vcvv 2726   C_ wss 3116   {csn 3576   <.cop 3579   class class class wbr 3982   `'ccnv 4603   Fun wfun 5182   ` cfv 5188   NNcn 8857   Struct cstr 12390   ndxcnx 12391  Slot cslot 12393

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-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fv 5196  df-struct 12396  df-slot 12398

This theorem is referenced by:  strslfv3  12439