Theorem cplgrop 29522

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 2737	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2737	. . . . . 6 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	iscplgredg 29502	. . . . 5 ⊢ (𝐺 ∈ ComplGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒))
4		edgval 29134	. . . . . . 7 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
5	4	a1i 11	. . . . . 6 ⊢ (𝐺 ∈ ComplGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
6		simpl 482	. . . . . . . . . . . 12 ⊢ (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → (Vtx‘𝑔) = (Vtx‘𝐺))
7	6	adantl 481	. . . . . . . . . . 11 ⊢ (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → (Vtx‘𝑔) = (Vtx‘𝐺))
8	6	difeq1d 4079	. . . . . . . . . . . . 13 ⊢ (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → ((Vtx‘𝑔) ∖ {𝑣}) = ((Vtx‘𝐺) ∖ {𝑣}))
9	8	adantl 481	. . . . . . . . . . . 12 ⊢ (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → ((Vtx‘𝑔) ∖ {𝑣}) = ((Vtx‘𝐺) ∖ {𝑣}))
10		edgval 29134	. . . . . . . . . . . . . . . 16 ⊢ (Edg‘𝑔) = ran (iEdg‘𝑔)
11		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → (iEdg‘𝑔) = (iEdg‘𝐺))
12	11	rneqd 5895	. . . . . . . . . . . . . . . 16 ⊢ (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
13	10, 12	eqtrid 2784	. . . . . . . . . . . . . . 15 ⊢ (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → (Edg‘𝑔) = ran (iEdg‘𝐺))
14	13	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → (Edg‘𝑔) = ran (iEdg‘𝐺))
15		simpl 482	. . . . . . . . . . . . . 14 ⊢ (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → (Edg‘𝐺) = ran (iEdg‘𝐺))
16	14, 15	eqtr4d 2775	. . . . . . . . . . . . 13 ⊢ (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → (Edg‘𝑔) = (Edg‘𝐺))
17	16	rexeqdv 3299	. . . . . . . . . . . 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 3318	. . . . . . . . . . 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 3318	. . . . . . . . . 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 477	. . . . . . . . 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 2737	. . . . . . . . . . 11 ⊢ (Vtx‘𝑔) = (Vtx‘𝑔)
22		eqid 2737	. . . . . . . . . . 11 ⊢ (Edg‘𝑔) = (Edg‘𝑔)
23	21, 22	iscplgredg 29502	. . . . . . . . . 10 ⊢ (𝑔 ∈ V → (𝑔 ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘𝑔)∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒))
24	23	elv 3447	. . . . . . . . 9 ⊢ (𝑔 ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘𝑔)∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒)
25	20, 24	sylibr 234	. . . . . . . 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 413	. . . . . . 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 415	. . . . . 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 240	. . . 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 1929	. 2 ⊢ (𝐺 ∈ ComplGraph → ∀𝑔(((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → 𝑔 ∈ ComplGraph))
32		fvexd 6857	. 2 ⊢ (𝐺 ∈ ComplGraph → (Vtx‘𝐺) ∈ V)
33		fvexd 6857	. 2 ⊢ (𝐺 ∈ ComplGraph → (iEdg‘𝐺) ∈ V)
34	31, 32, 33	gropeld 29118	1 ⊢ (𝐺 ∈ ComplGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplGraph)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ∖ cdif 3900   ⊆ wss 3903  {csn 4582  {cpr 4584  ⟨cop 4588  ran crn 5633  ‘cfv 6500  Vtxcvtx 29081  iEdgciedg 29082  Edgcedg 29132  ComplGraphccplgr 29494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-vtx 29083  df-iedg 29084  df-edg 29133  df-nbgr 29418  df-uvtx 29471  df-cplgr 29496

This theorem is referenced by:  cusgrop  29523