Theorem strslfv 13126

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 13087). By virtue of ndxslid 13106, 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 13093	. . 3 |- ( S Struct X -> S e. _V )
4	2, 3	mp1i 10	. 2 |- ( C e. V -> S e. _V )
5	2	structfun 13099	. . 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 113	. . . . 5 |- ( E ` ndx ) e. NN
9		opexg 4320	. . . . 5 |- ( ( ( E ` ndx ) e. NN /\ C e. V ) -> <. ( E ` ndx ) , C >. e. _V )
10	8, 9	mpan 424	. . . 4 |- ( C e. V -> <. ( E ` ndx ) , C >. e. _V )
11		snssg 3807	. . . 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 168	. 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 13124	1 |- ( C e. V -> C = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   _Vcvv 2802   C_ wss 3200   {csn 3669   <.cop 3672   class class class wbr 4088   `'ccnv 4724   Fun wfun 5320   ` cfv 5326   NNcn 9142   Struct cstr 13077   ndxcnx 13078  Slot cslot 13080

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-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fv 5334  df-struct 13083  df-slot 13085

This theorem is referenced by:  cnfldbas  14573  mpocnfldadd  14574  mpocnfldmul  14576  cnfldcj  14578  cnfldtset  14579  cnfldle  14580  cnfldds  14581