Theorem strslfv3 13073

Description: Variant on strslfv 13072 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) (Revised by Jim Kingdon, 30-Jan-2023.)

Hypotheses

Ref	Expression
strfv3.u	|- ( ph -> U = S )
strslfv3.s	|- ( ph -> S Struct X )
strslfv3.e	|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
strslfv3.n	|- ( ph -> { <. ( E ` ndx ) , C >. } C_ S )
strfv3.c	|- ( ph -> C e. V )
strfv3.a	|- A = ( E ` U )

Assertion

Ref	Expression
strslfv3	|- ( ph -> A = C )

Proof of Theorem strslfv3

Step	Hyp	Ref	Expression
1		strfv3.a	. 2 |- A = ( E ` U )
2		strslfv3.e	. . 3 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
3		strfv3.u	. . . 4 |- ( ph -> U = S )
4		strslfv3.s	. . . . 5 |- ( ph -> S Struct X )
5		structex 13039	. . . . 5 |- ( S Struct X -> S e. _V )
6	4, 5	syl 14	. . . 4 |- ( ph -> S e. _V )
7	3, 6	eqeltrd 2306	. . 3 |- ( ph -> U e. _V )
8		structfung 13044	. . . . 5 |- ( S Struct X -> Fun `' `' S )
9	4, 8	syl 14	. . . 4 |- ( ph -> Fun `' `' S )
10	3	cnveqd 4897	. . . . . 6 |- ( ph -> `' U = `' S )
11	10	cnveqd 4897	. . . . 5 |- ( ph -> `' `' U = `' `' S )
12	11	funeqd 5339	. . . 4 |- ( ph -> ( Fun `' `' U <-> Fun `' `' S ) )
13	9, 12	mpbird 167	. . 3 |- ( ph -> Fun `' `' U )
14		strslfv3.n	. . . . 5 |- ( ph -> { <. ( E ` ndx ) , C >. } C_ S )
15	2	simpri 113	. . . . . . 7 |- ( E ` ndx ) e. NN
16		strfv3.c	. . . . . . 7 |- ( ph -> C e. V )
17		opexg 4313	. . . . . . 7 |- ( ( ( E ` ndx ) e. NN /\ C e. V ) -> <. ( E ` ndx ) , C >. e. _V )
18	15, 16, 17	sylancr 414	. . . . . 6 |- ( ph -> <. ( E ` ndx ) , C >. e. _V )
19		snssg 3801	. . . . . 6 |- ( <. ( E ` ndx ) , C >. e. _V -> ( <. ( E ` ndx ) , C >. e. S <-> { <. ( E ` ndx ) , C >. } C_ S ) )
20	18, 19	syl 14	. . . . 5 |- ( ph -> ( <. ( E ` ndx ) , C >. e. S <-> { <. ( E ` ndx ) , C >. } C_ S ) )
21	14, 20	mpbird 167	. . . 4 |- ( ph -> <. ( E ` ndx ) , C >. e. S )
22	21, 3	eleqtrrd 2309	. . 3 |- ( ph -> <. ( E ` ndx ) , C >. e. U )
23	2, 7, 13, 22, 16	strslfv2d 13070	. 2 |- ( ph -> C = ( E ` U ) )
24	1, 23	eqtr4id 2281	1 |- ( ph -> A = C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2799   C_ wss 3197   {csn 3666   <.cop 3669   class class class wbr 4082   `'ccnv 4717   Fun wfun 5311   ` cfv 5317   NNcn 9106   Struct cstr 13023   ndxcnx 13024  Slot cslot 13026

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fv 5325  df-struct 13029  df-slot 13031

This theorem is referenced by:  prdsbaslemss  13302