Theorem structtocusgr 26342

Description: Any (extensible) structure with a base set can be made a complete simple graph with the set of pairs of elements of the base set regarded as edges. (Contributed by AV, 10-Nov-2021.) (Revised by AV, 17-Nov-2021.)

Hypotheses

Ref	Expression
structtousgr.p	|- P = { x e. ~P ( Base ` S ) | ( # ` x ) = 2 }
structtousgr.s	|- ( ph -> S Struct X )
structtousgr.g	|- G = ( S sSet <. (.ef ` ndx ) , ( _I |` P ) >. )
structtousgr.b	|- ( ph -> ( Base ` ndx ) e. dom S )

Assertion

Ref	Expression
structtocusgr	|- ( ph -> G e. ComplUSGraph )

Distinct variable groups:   x, G   x, P   x, S   ph, x

Allowed substitution hint:   X( x)

Proof of Theorem structtocusgr

Dummy variables e n v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		structtousgr.p	. . 3 |- P = { x e. ~P ( Base ` S ) | ( # ` x ) = 2 }
2		structtousgr.s	. . 3 |- ( ph -> S Struct X )
3		structtousgr.g	. . 3 |- G = ( S sSet <. (.ef ` ndx ) , ( _I |` P ) >. )
4		structtousgr.b	. . 3 |- ( ph -> ( Base ` ndx ) e. dom S )
5	1, 2, 3, 4	structtousgr 26341	. 2 |- ( ph -> G e. USGraph )
6		simpr 477	. . . . 5 |- ( ( ph /\ v e. (Vtx ` G ) ) -> v e. (Vtx ` G ) )
7		eldifi 3732	. . . . . . . 8 |- ( n e. ( (Vtx ` G ) \ { v } ) -> n e. (Vtx ` G ) )
8	6, 7	anim12ci 591	. . . . . . 7 |- ( ( ( ph /\ v e. (Vtx ` G ) ) /\ n e. ( (Vtx ` G ) \ { v } ) ) -> ( n e. (Vtx ` G ) /\ v e. (Vtx ` G ) ) )
9		eldifsni 4320	. . . . . . . 8 |- ( n e. ( (Vtx ` G ) \ { v } ) -> n =/= v )
10	9	adantl 482	. . . . . . 7 |- ( ( ( ph /\ v e. (Vtx ` G ) ) /\ n e. ( (Vtx ` G ) \ { v } ) ) -> n =/= v )
11		fvexd 6203	. . . . . . . . 9 |- ( ( ( ph /\ v e. (Vtx ` G ) ) /\ n e. ( (Vtx ` G ) \ { v } ) ) -> ( Base ` S ) e. _V )
12	3	fveq2i 6194	. . . . . . . . . . . . . 14 |- (Vtx ` G ) = (Vtx ` ( S sSet <. (.ef ` ndx ) , ( _I |` P ) >. ) )
13		eqid 2622	. . . . . . . . . . . . . . 15 |- (.ef ` ndx ) = (.ef ` ndx )
14		fvex 6201	. . . . . . . . . . . . . . . 16 |- ( Base ` S ) e. _V
15	1	cusgrexilem1 26335	. . . . . . . . . . . . . . . 16 |- ( ( Base ` S ) e. _V -> ( _I |` P ) e. _V )
16	14, 15	mp1i 13	. . . . . . . . . . . . . . 15 |- ( ph -> ( _I |` P ) e. _V )
17	13, 2, 4, 16	setsvtx 25927	. . . . . . . . . . . . . 14 |- ( ph -> (Vtx ` ( S sSet <. (.ef ` ndx ) , ( _I |` P ) >. ) ) = ( Base ` S ) )
18	12, 17	syl5eq 2668	. . . . . . . . . . . . 13 |- ( ph -> (Vtx ` G ) = ( Base ` S ) )
19	18	eleq2d 2687	. . . . . . . . . . . 12 |- ( ph -> ( v e. (Vtx ` G ) <-> v e. ( Base ` S ) ) )
20	19	biimpd 219	. . . . . . . . . . 11 |- ( ph -> ( v e. (Vtx ` G ) -> v e. ( Base ` S ) ) )
21	20	imp 445	. . . . . . . . . 10 |- ( ( ph /\ v e. (Vtx ` G ) ) -> v e. ( Base ` S ) )
22	21	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ v e. (Vtx ` G ) ) /\ n e. ( (Vtx ` G ) \ { v } ) ) -> v e. ( Base ` S ) )
23	18	difeq1d 3727	. . . . . . . . . . . . 13 |- ( ph -> ( (Vtx ` G ) \ { v } ) = ( ( Base ` S ) \ { v } ) )
24	23	eleq2d 2687	. . . . . . . . . . . 12 |- ( ph -> ( n e. ( (Vtx ` G ) \ { v } ) <-> n e. ( ( Base ` S ) \ { v } ) ) )
25	24	biimpd 219	. . . . . . . . . . 11 |- ( ph -> ( n e. ( (Vtx ` G ) \ { v } ) -> n e. ( ( Base ` S ) \ { v } ) ) )
26	25	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ v e. (Vtx ` G ) ) -> ( n e. ( (Vtx ` G ) \ { v } ) -> n e. ( ( Base ` S ) \ { v } ) ) )
27	26	imp 445	. . . . . . . . 9 |- ( ( ( ph /\ v e. (Vtx ` G ) ) /\ n e. ( (Vtx ` G ) \ { v } ) ) -> n e. ( ( Base ` S ) \ { v } ) )
28	1	cusgrexilem2 26338	. . . . . . . . 9 |- ( ( ( ( Base ` S ) e. _V /\ v e. ( Base ` S ) ) /\ n e. ( ( Base ` S ) \ { v } ) ) -> E. e e. ran ( _I |` P ) { v , n } C_ e )
29	11, 22, 27, 28	syl21anc 1325	. . . . . . . 8 |- ( ( ( ph /\ v e. (Vtx ` G ) ) /\ n e. ( (Vtx ` G ) \ { v } ) ) -> E. e e. ran ( _I |` P ) { v , n } C_ e )
30		edgval 25941	. . . . . . . . . . 11 |- (Edg ` G ) = ran (iEdg ` G )
31	3	fveq2i 6194	. . . . . . . . . . . . 13 |- (iEdg ` G ) = (iEdg ` ( S sSet <. (.ef ` ndx ) , ( _I |` P ) >. ) )
32	13, 2, 4, 16	setsiedg 25928	. . . . . . . . . . . . 13 |- ( ph -> (iEdg ` ( S sSet <. (.ef ` ndx ) , ( _I |` P ) >. ) ) = ( _I |` P ) )
33	31, 32	syl5eq 2668	. . . . . . . . . . . 12 |- ( ph -> (iEdg ` G ) = ( _I |` P ) )
34	33	rneqd 5353	. . . . . . . . . . 11 |- ( ph -> ran (iEdg ` G ) = ran ( _I |` P ) )
35	30, 34	syl5eq 2668	. . . . . . . . . 10 |- ( ph -> (Edg ` G ) = ran ( _I |` P ) )
36	35	rexeqdv 3145	. . . . . . . . 9 |- ( ph -> ( E. e e. (Edg ` G ) { v , n } C_ e <-> E. e e. ran ( _I |` P ) { v , n } C_ e ) )
37	36	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ v e. (Vtx ` G ) ) /\ n e. ( (Vtx ` G ) \ { v } ) ) -> ( E. e e. (Edg ` G ) { v , n } C_ e <-> E. e e. ran ( _I |` P ) { v , n } C_ e ) )
38	29, 37	mpbird 247	. . . . . . 7 |- ( ( ( ph /\ v e. (Vtx ` G ) ) /\ n e. ( (Vtx ` G ) \ { v } ) ) -> E. e e. (Edg ` G ) { v , n } C_ e )
39	5	elexd 3214	. . . . . . . . 9 |- ( ph -> G e. _V )
40	39	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ v e. (Vtx ` G ) ) /\ n e. ( (Vtx ` G ) \ { v } ) ) -> G e. _V )
41		eqid 2622	. . . . . . . . 9 |- (Vtx ` G ) = (Vtx ` G )
42		eqid 2622	. . . . . . . . 9 |- (Edg ` G ) = (Edg ` G )
43	41, 42	nbgrel 26238	. . . . . . . 8 |- ( G e. _V -> ( n e. ( G NeighbVtx v ) <-> ( ( n e. (Vtx ` G ) /\ v e. (Vtx ` G ) ) /\ n =/= v /\ E. e e. (Edg ` G ) { v , n } C_ e ) ) )
44	40, 43	syl 17	. . . . . . 7 |- ( ( ( ph /\ v e. (Vtx ` G ) ) /\ n e. ( (Vtx ` G ) \ { v } ) ) -> ( n e. ( G NeighbVtx v ) <-> ( ( n e. (Vtx ` G ) /\ v e. (Vtx ` G ) ) /\ n =/= v /\ E. e e. (Edg ` G ) { v , n } C_ e ) ) )
45	8, 10, 38, 44	mpbir3and 1245	. . . . . 6 |- ( ( ( ph /\ v e. (Vtx ` G ) ) /\ n e. ( (Vtx ` G ) \ { v } ) ) -> n e. ( G NeighbVtx v ) )
46	45	ralrimiva 2966	. . . . 5 |- ( ( ph /\ v e. (Vtx ` G ) ) -> A. n e. ( (Vtx ` G ) \ { v } ) n e. ( G NeighbVtx v ) )
47	39	adantr 481	. . . . . 6 |- ( ( ph /\ v e. (Vtx ` G ) ) -> G e. _V )
48	41	uvtxael 26288	. . . . . 6 |- ( G e. _V -> ( v e. (UnivVtx ` G ) <-> ( v e. (Vtx ` G ) /\ A. n e. ( (Vtx ` G ) \ { v } ) n e. ( G NeighbVtx v ) ) ) )
49	47, 48	syl 17	. . . . 5 |- ( ( ph /\ v e. (Vtx ` G ) ) -> ( v e. (UnivVtx ` G ) <-> ( v e. (Vtx ` G ) /\ A. n e. ( (Vtx ` G ) \ { v } ) n e. ( G NeighbVtx v ) ) ) )
50	6, 46, 49	mpbir2and 957	. . . 4 |- ( ( ph /\ v e. (Vtx ` G ) ) -> v e. (UnivVtx ` G ) )
51	50	ralrimiva 2966	. . 3 |- ( ph -> A. v e. (Vtx ` G ) v e. (UnivVtx ` G ) )
52	41	iscplgr 26310	. . . 4 |- ( G e. _V -> ( G e. ComplGraph <-> A. v e. (Vtx ` G ) v e. (UnivVtx ` G ) ) )
53	39, 52	syl 17	. . 3 |- ( ph -> ( G e. ComplGraph <-> A. v e. (Vtx ` G ) v e. (UnivVtx ` G ) ) )
54	51, 53	mpbird 247	. 2 |- ( ph -> G e. ComplGraph )
55		iscusgr 26314	. 2 |- ( G e. ComplUSGraph <-> ( G e. USGraph /\ G e. ComplGraph ) )
56	5, 54, 55	sylanbrc 698	1 |- ( ph -> G e. ComplUSGraph )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ~Pcpw 4158   {csn 4177   {cpr 4179   <.cop 4183   class class class wbr 4653   _I cid 5023   dom cdm 5114   ran crn 5115   |` cres 5116   ` cfv 5888  (class class class)co 6650   2c2 11070   #chash 13117   Struct cstr 15853   ndxcnx 15854   sSet csts 15855   Basecbs 15857  .efcedgf 25867  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   USGraph cusgr 26044   NeighbVtx cnbgr 26224  UnivVtxcuvtxa 26225  ComplGraphccplgr 26226  ComplUSGraphccusgr 26227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-edgf 25868  df-vtx 25876  df-iedg 25877  df-edg 25940  df-usgr 26046  df-nbgr 26228  df-uvtxa 26230  df-cplgr 26231  df-cusgr 26232

This theorem is referenced by:  cffldtocusgr  26343