Theorem cplgrop 27804

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

Assertion

Ref	Expression
cplgrop	⊢ (𝐺 ∈ ComplGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ 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 2738	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2738	. . . . . 6 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	iscplgredg 27784	. . . . 5 ⊢ (𝐺 ∈ ComplGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒))
4		edgval 27419	. . . . . . 7 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
5	4	a1i 11	. . . . . 6 ⊢ (𝐺 ∈ ComplGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
6		simpl 483	. . . . . . . . . . . 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 4056	. . . . . . . . . . . . 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 27419	. . . . . . . . . . . . . . . 16 ⊢ (Edg‘𝑔) = ran (iEdg‘𝑔)
11		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → (iEdg‘𝑔) = (iEdg‘𝐺))
12	11	rneqd 5847	. . . . . . . . . . . . . . . 16 ⊢ (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
13	10, 12	eqtrid 2790	. . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 14 ⊢ (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → (Edg‘𝐺) = ran (iEdg‘𝐺))
16	14, 15	eqtr4d 2781	. . . . . . . . . . . . 13 ⊢ (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → (Edg‘𝑔) = (Edg‘𝐺))
17	16	rexeqdv 3349	. . . . . . . . . . . 12 ⊢ (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → (∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒))
18	9, 17	raleqbidv 3336	. . . . . . . . . . 11 ⊢ (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → (∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒 ↔ ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒))
19	7, 18	raleqbidv 3336	. . . . . . . . . 10 ⊢ (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → (∀𝑣 ∈ (Vtx‘𝑔)∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒 ↔ ∀𝑣 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒))
20	19	biimpar 478	. . . . . . . . 9 ⊢ ((((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) ∧ ∀𝑣 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒) → ∀𝑣 ∈ (Vtx‘𝑔)∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒)
21		eqid 2738	. . . . . . . . . . 11 ⊢ (Vtx‘𝑔) = (Vtx‘𝑔)
22		eqid 2738	. . . . . . . . . . 11 ⊢ (Edg‘𝑔) = (Edg‘𝑔)
23	21, 22	iscplgredg 27784	. . . . . . . . . 10 ⊢ (𝑔 ∈ V → (𝑔 ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘𝑔)∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒))
24	23	elv 3438	. . . . . . . . 9 ⊢ (𝑔 ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘𝑔)∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒)
25	20, 24	sylibr 233	. . . . . . . 8 ⊢ ((((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) ∧ ∀𝑣 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒) → 𝑔 ∈ ComplGraph)
26	25	expcom 414	. . . . . . 7 ⊢ (∀𝑣 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒 → (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → 𝑔 ∈ ComplGraph))
27	26	expd 416	. . . . . 6 ⊢ (∀𝑣 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒 → ((Edg‘𝐺) = ran (iEdg‘𝐺) → (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → 𝑔 ∈ ComplGraph)))
28	5, 27	syl5com 31	. . . . 5 ⊢ (𝐺 ∈ ComplGraph → (∀𝑣 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒 → (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → 𝑔 ∈ ComplGraph)))
29	3, 28	sylbid 239	. . . 4 ⊢ (𝐺 ∈ ComplGraph → (𝐺 ∈ ComplGraph → (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → 𝑔 ∈ ComplGraph)))
30	29	pm2.43i 52	. . 3 ⊢ (𝐺 ∈ ComplGraph → (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → 𝑔 ∈ ComplGraph))
31	30	alrimiv 1930	. 2 ⊢ (𝐺 ∈ ComplGraph → ∀𝑔(((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → 𝑔 ∈ ComplGraph))
32		fvexd 6789	. 2 ⊢ (𝐺 ∈ ComplGraph → (Vtx‘𝐺) ∈ V)
33		fvexd 6789	. 2 ⊢ (𝐺 ∈ ComplGraph → (iEdg‘𝐺) ∈ V)
34	31, 32, 33	gropeld 27403	1 ⊢ (𝐺 ∈ ComplGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplGraph)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∖ cdif 3884   ⊆ wss 3887  {csn 4561  {cpr 4563  ⟨cop 4567  ran crn 5590  ‘cfv 6433  Vtxcvtx 27366  iEdgciedg 27367  Edgcedg 27417  ComplGraphccplgr 27776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-vtx 27368  df-iedg 27369  df-edg 27418  df-nbgr 27700  df-uvtx 27753  df-cplgr 27778

This theorem is referenced by:  cusgrop  27805