Theorem cplgrop 26333

Description: A complete graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.)

Assertion

Ref	Expression
cplgrop	|- ( G e. ComplGraph -> <. (Vtx ` G ) , (iEdg ` G ) >. e. ComplGraph )

Proof of Theorem cplgrop

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2622	. . . . . 6 |- (Edg ` G ) = (Edg ` G )
3	1, 2	iscplgredg 26313	. . . . 5 |- ( G e. ComplGraph -> ( G e. ComplGraph <-> A. v e. (Vtx ` G ) A. n e. ( (Vtx ` G ) \ { v } ) E. e e. (Edg ` G ) { v , n } C_ e ) )
4		edgval 25941	. . . . . . 7 |- (Edg ` G ) = ran (iEdg ` G )
5	4	a1i 11	. . . . . 6 |- ( G e. ComplGraph -> (Edg ` G ) = ran (iEdg ` G ) )
6		simpl 473	. . . . . . . . . . . 12 |- ( ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) -> (Vtx ` g ) = (Vtx ` G ) )
7	6	adantl 482	. . . . . . . . . . 11 |- ( ( (Edg ` G ) = ran (iEdg ` G ) /\ ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) ) -> (Vtx ` g ) = (Vtx ` G ) )
8	6	difeq1d 3727	. . . . . . . . . . . . 13 |- ( ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) -> ( (Vtx ` g ) \ { v } ) = ( (Vtx ` G ) \ { v } ) )
9	8	adantl 482	. . . . . . . . . . . 12 |- ( ( (Edg ` G ) = ran (iEdg ` G ) /\ ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) ) -> ( (Vtx ` g ) \ { v } ) = ( (Vtx ` G ) \ { v } ) )
10		edgval 25941	. . . . . . . . . . . . . . . 16 |- (Edg ` g ) = ran (iEdg ` g )
11		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) -> (iEdg ` g ) = (iEdg ` G ) )
12	11	rneqd 5353	. . . . . . . . . . . . . . . 16 |- ( ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) -> ran (iEdg ` g ) = ran (iEdg ` G ) )
13	10, 12	syl5eq 2668	. . . . . . . . . . . . . . 15 |- ( ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) -> (Edg ` g ) = ran (iEdg ` G ) )
14	13	adantl 482	. . . . . . . . . . . . . 14 |- ( ( (Edg ` G ) = ran (iEdg ` G ) /\ ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) ) -> (Edg ` g ) = ran (iEdg ` G ) )
15		simpl 473	. . . . . . . . . . . . . 14 |- ( ( (Edg ` G ) = ran (iEdg ` G ) /\ ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) ) -> (Edg ` G ) = ran (iEdg ` G ) )
16	14, 15	eqtr4d 2659	. . . . . . . . . . . . 13 |- ( ( (Edg ` G ) = ran (iEdg ` G ) /\ ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) ) -> (Edg ` g ) = (Edg ` G ) )
17	16	rexeqdv 3145	. . . . . . . . . . . 12 |- ( ( (Edg ` G ) = ran (iEdg ` G ) /\ ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) ) -> ( E. e e. (Edg ` g ) { v , n } C_ e <-> E. e e. (Edg ` G ) { v , n } C_ e ) )
18	9, 17	raleqbidv 3152	. . . . . . . . . . 11 |- ( ( (Edg ` G ) = ran (iEdg ` G ) /\ ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) ) -> ( A. n e. ( (Vtx ` g ) \ { v } ) E. e e. (Edg ` g ) { v , n } C_ e <-> A. n e. ( (Vtx ` G ) \ { v } ) E. e e. (Edg ` G ) { v , n } C_ e ) )
19	7, 18	raleqbidv 3152	. . . . . . . . . 10 |- ( ( (Edg ` G ) = ran (iEdg ` G ) /\ ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) ) -> ( A. v e. (Vtx ` g ) A. n e. ( (Vtx ` g ) \ { v } ) E. e e. (Edg ` g ) { v , n } C_ e <-> A. v e. (Vtx ` G ) A. n e. ( (Vtx ` G ) \ { v } ) E. e e. (Edg ` G ) { v , n } C_ e ) )
20	19	biimpar 502	. . . . . . . . 9 |- ( ( ( (Edg ` G ) = ran (iEdg ` G ) /\ ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) ) /\ A. v e. (Vtx ` G ) A. n e. ( (Vtx ` G ) \ { v } ) E. e e. (Edg ` G ) { v , n } C_ e ) -> A. v e. (Vtx ` g ) A. n e. ( (Vtx ` g ) \ { v } ) E. e e. (Edg ` g ) { v , n } C_ e )
21		vex 3203	. . . . . . . . . 10 |- g e. _V
22		eqid 2622	. . . . . . . . . . 11 |- (Vtx ` g ) = (Vtx ` g )
23		eqid 2622	. . . . . . . . . . 11 |- (Edg ` g ) = (Edg ` g )
24	22, 23	iscplgredg 26313	. . . . . . . . . 10 |- ( g e. _V -> ( g e. ComplGraph <-> A. v e. (Vtx ` g ) A. n e. ( (Vtx ` g ) \ { v } ) E. e e. (Edg ` g ) { v , n } C_ e ) )
25	21, 24	ax-mp 5	. . . . . . . . 9 |- ( g e. ComplGraph <-> A. v e. (Vtx ` g ) A. n e. ( (Vtx ` g ) \ { v } ) E. e e. (Edg ` g ) { v , n } C_ e )
26	20, 25	sylibr 224	. . . . . . . 8 |- ( ( ( (Edg ` G ) = ran (iEdg ` G ) /\ ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) ) /\ A. v e. (Vtx ` G ) A. n e. ( (Vtx ` G ) \ { v } ) E. e e. (Edg ` G ) { v , n } C_ e ) -> g e. ComplGraph )
27	26	expcom 451	. . . . . . 7 |- ( A. v e. (Vtx ` G ) A. n e. ( (Vtx ` G ) \ { v } ) E. e e. (Edg ` G ) { v , n } C_ e -> ( ( (Edg ` G ) = ran (iEdg ` G ) /\ ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) ) -> g e. ComplGraph ) )
28	27	expd 452	. . . . . 6 |- ( A. v e. (Vtx ` G ) A. n e. ( (Vtx ` G ) \ { v } ) E. e e. (Edg ` G ) { v , n } C_ e -> ( (Edg ` G ) = ran (iEdg ` G ) -> ( ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) -> g e. ComplGraph ) ) )
29	5, 28	syl5com 31	. . . . 5 |- ( G e. ComplGraph -> ( A. v e. (Vtx ` G ) A. n e. ( (Vtx ` G ) \ { v } ) E. e e. (Edg ` G ) { v , n } C_ e -> ( ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) -> g e. ComplGraph ) ) )
30	3, 29	sylbid 230	. . . 4 |- ( G e. ComplGraph -> ( G e. ComplGraph -> ( ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) -> g e. ComplGraph ) ) )
31	30	pm2.43i 52	. . 3 |- ( G e. ComplGraph -> ( ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) -> g e. ComplGraph ) )
32	31	alrimiv 1855	. 2 |- ( G e. ComplGraph -> A. g ( ( (Vtx ` g ) = (Vtx ` G ) /\ (iEdg ` g ) = (iEdg ` G ) ) -> g e. ComplGraph ) )
33		fvexd 6203	. 2 |- ( G e. ComplGraph -> (Vtx ` G ) e. _V )
34		fvexd 6203	. 2 |- ( G e. ComplGraph -> (iEdg ` G ) e. _V )
35	32, 33, 34	gropeld 25925	1 |- ( G e. ComplGraph -> <. (Vtx ` G ) , (iEdg ` G ) >. e. ComplGraph )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   <.cop 4183   ran crn 5115   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939  ComplGraphccplgr 26226

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-vtx 25876  df-iedg 25877  df-edg 25940  df-nbgr 26228  df-uvtxa 26230  df-cplgr 26231

This theorem is referenced by:  cusgrop  26334