Theorem strslfv 12501

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 12462). By virtue of ndxslid 12481, 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 12468	. . 3 |- ( S Struct X -> S e. _V )
4	2, 3	mp1i 10	. 2 |- ( C e. V -> S e. _V )
5	2	structfun 12474	. . 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 4228	. . . . 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 3726	. . . 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 12499	1 |- ( C e. V -> C = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   _Vcvv 2737   C_ wss 3129   {csn 3592   <.cop 3595   class class class wbr 4003   `'ccnv 4625   Fun wfun 5210   ` cfv 5216   NNcn 8917   Struct cstr 12452   ndxcnx 12453  Slot cslot 12455

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-iota 5178  df-fun 5218  df-fv 5224  df-struct 12458  df-slot 12460

This theorem is referenced by:  strslfv3  12502  cnfldbas  13350  cnfldadd  13351  cnfldmul  13352  cnfldcj  13353