Theorem strslfv 12663

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 12624). By virtue of ndxslid 12643, 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 12630	. . 3 |- ( S Struct X -> S e. _V )
4	2, 3	mp1i 10	. 2 |- ( C e. V -> S e. _V )
5	2	structfun 12636	. . 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 4257	. . . . 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 3752	. . . 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 12661	1 |- ( C e. V -> C = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   C_ wss 3153   {csn 3618   <.cop 3621   class class class wbr 4029   `'ccnv 4658   Fun wfun 5248   ` cfv 5254   NNcn 8982   Struct cstr 12614   ndxcnx 12615  Slot cslot 12617

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fv 5262  df-struct 12620  df-slot 12622

This theorem is referenced by:  strslfv3  12664  cnfldbas  14051  cnfldadd  14052  cnfldmul  14054  cnfldcj  14056