Theorem cplgrop 30051

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 2763	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2763	. . . . . 6 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	iscplgredg 30031	. . . . 5 ⊢ (𝐺 ∈ ComplGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒))
4		edgval 29660	. . . . . . 7 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
5	4	a1i 11	. . . . . 6 ⊢ (𝐺 ∈ ComplGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
6		simpl 488	. . . . . . . . . . . 12 ⊢ (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → (Vtx‘𝑔) = (Vtx‘𝐺))
7	6	adantl 487	. . . . . . . . . . 11 ⊢ (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → (Vtx‘𝑔) = (Vtx‘𝐺))
8	6	difeq1d 4065	. . . . . . . . . . . . 13 ⊢ (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → ((Vtx‘𝑔) ∖ {𝑣}) = ((Vtx‘𝐺) ∖ {𝑣}))
9	8	adantl 487	. . . . . . . . . . . 12 ⊢ (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → ((Vtx‘𝑔) ∖ {𝑣}) = ((Vtx‘𝐺) ∖ {𝑣}))
10		edgval 29660	. . . . . . . . . . . . . . . 16 ⊢ (Edg‘𝑔) = ran (iEdg‘𝑔)
11		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → (iEdg‘𝑔) = (iEdg‘𝐺))
12	11	rneqd 5928	. . . . . . . . . . . . . . . 16 ⊢ (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
13	10, 12	eqtrid 2810	. . . . . . . . . . . . . . 15 ⊢ (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → (Edg‘𝑔) = ran (iEdg‘𝐺))
14	13	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → (Edg‘𝑔) = ran (iEdg‘𝐺))
15		simpl 488	. . . . . . . . . . . . . 14 ⊢ (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → (Edg‘𝐺) = ran (iEdg‘𝐺))
16	14, 15	eqtr4d 2801	. . . . . . . . . . . . 13 ⊢ (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → (Edg‘𝑔) = (Edg‘𝐺))
17	16	rexeqdv 3324	. . . . . . . . . . . 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 3338	. . . . . . . . . . 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 3338	. . . . . . . . . 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 483	. . . . . . . . 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 2763	. . . . . . . . . . 11 ⊢ (Vtx‘𝑔) = (Vtx‘𝑔)
22		eqid 2763	. . . . . . . . . . 11 ⊢ (Edg‘𝑔) = (Edg‘𝑔)
23	21, 22	iscplgredg 30031	. . . . . . . . . 10 ⊢ (𝑔 ∈ V → (𝑔 ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘𝑔)∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒))
24	23	elv 3459	. . . . . . . . 9 ⊢ (𝑔 ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘𝑔)∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒)
25	20, 24	sylibr 237	. . . . . . . 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 419	. . . . . . 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 421	. . . . . 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 32	. . . . 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 243	. . . 4 ⊢ (𝐺 ∈ ComplGraph → (𝐺 ∈ ComplGraph → (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → 𝑔 ∈ ComplGraph)))
30	29	pm2.43i 53	. . 3 ⊢ (𝐺 ∈ ComplGraph → (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → 𝑔 ∈ ComplGraph))
31	30	alrimiv 1962	. 2 ⊢ (𝐺 ∈ ComplGraph → ∀𝑔(((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → 𝑔 ∈ ComplGraph))
32		fvexd 6926	. 2 ⊢ (𝐺 ∈ ComplGraph → (Vtx‘𝐺) ∈ V)
33		fvexd 6926	. 2 ⊢ (𝐺 ∈ ComplGraph → (iEdg‘𝐺) ∈ V)
34	31, 32, 33	gropeld 29644	1 ⊢ (𝐺 ∈ ComplGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplGraph)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048  {csn 4588  {cpr 4590  ⟨cop 4838  ran crn 5657  ‘cfv 6561  Vtxcvtx 29607  iEdgciedg 29608  Edgcedg 29658  ComplGraphccplgr 30023

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-vtx 29609  df-iedg 29610  df-edg 29659  df-nbgr 29947  df-uvtx 30000  df-cplgr 30025

This theorem is used by:  cusgrop  30052