Theorem strslfv 12877

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 12838). By virtue of ndxslid 12857, 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 12844	. . 3 |- ( S Struct X -> S e. _V )
4	2, 3	mp1i 10	. 2 |- ( C e. V -> S e. _V )
5	2	structfun 12850	. . 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 4272	. . . . 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 3767	. . . 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 12875	1 |- ( C e. V -> C = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   _Vcvv 2772   C_ wss 3166   {csn 3633   <.cop 3636   class class class wbr 4044   `'ccnv 4674   Fun wfun 5265   ` cfv 5271   NNcn 9036   Struct cstr 12828   ndxcnx 12829  Slot cslot 12831

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fv 5279  df-struct 12834  df-slot 12836

This theorem is referenced by:  cnfldbas  14322  mpocnfldadd  14323  mpocnfldmul  14325  cnfldcj  14327  cnfldtset  14328  cnfldle  14329  cnfldds  14330