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Theorem cplgrop 27227
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.
StepHypRef Expression
1 eqid 2798 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2798 . . . . . 6 (Edg‘𝐺) = (Edg‘𝐺)
31, 2iscplgredg 27207 . . . . 5 (𝐺 ∈ ComplGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘𝐺)∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒))
4 edgval 26842 . . . . . . 7 (Edg‘𝐺) = ran (iEdg‘𝐺)
54a1i 11 . . . . . 6 (𝐺 ∈ ComplGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
6 simpl 486 . . . . . . . . . . . 12 (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → (Vtx‘𝑔) = (Vtx‘𝐺))
76adantl 485 . . . . . . . . . . 11 (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → (Vtx‘𝑔) = (Vtx‘𝐺))
86difeq1d 4049 . . . . . . . . . . . . 13 (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → ((Vtx‘𝑔) ∖ {𝑣}) = ((Vtx‘𝐺) ∖ {𝑣}))
98adantl 485 . . . . . . . . . . . 12 (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → ((Vtx‘𝑔) ∖ {𝑣}) = ((Vtx‘𝐺) ∖ {𝑣}))
10 edgval 26842 . . . . . . . . . . . . . . . 16 (Edg‘𝑔) = ran (iEdg‘𝑔)
11 simpr 488 . . . . . . . . . . . . . . . . 17 (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → (iEdg‘𝑔) = (iEdg‘𝐺))
1211rneqd 5772 . . . . . . . . . . . . . . . 16 (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
1310, 12syl5eq 2845 . . . . . . . . . . . . . . 15 (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → (Edg‘𝑔) = ran (iEdg‘𝐺))
1413adantl 485 . . . . . . . . . . . . . 14 (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → (Edg‘𝑔) = ran (iEdg‘𝐺))
15 simpl 486 . . . . . . . . . . . . . 14 (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → (Edg‘𝐺) = ran (iEdg‘𝐺))
1614, 15eqtr4d 2836 . . . . . . . . . . . . 13 (((Edg‘𝐺) = ran (iEdg‘𝐺) ∧ ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺))) → (Edg‘𝑔) = (Edg‘𝐺))
1716rexeqdv 3365 . . . . . . . . . . . 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))
189, 17raleqbidv 3354 . . . . . . . . . . 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))
197, 18raleqbidv 3354 . . . . . . . . . 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))
2019biimpar 481 . . . . . . . . 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 2798 . . . . . . . . . . 11 (Vtx‘𝑔) = (Vtx‘𝑔)
22 eqid 2798 . . . . . . . . . . 11 (Edg‘𝑔) = (Edg‘𝑔)
2321, 22iscplgredg 27207 . . . . . . . . . 10 (𝑔 ∈ V → (𝑔 ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘𝑔)∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒))
2423elv 3446 . . . . . . . . 9 (𝑔 ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘𝑔)∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒)
2520, 24sylibr 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)
2625expcom 417 . . . . . . 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))
2726expd 419 . . . . . 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)))
285, 27syl5com 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)))
293, 28sylbid 243 . . . 4 (𝐺 ∈ ComplGraph → (𝐺 ∈ ComplGraph → (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → 𝑔 ∈ ComplGraph)))
3029pm2.43i 52 . . 3 (𝐺 ∈ ComplGraph → (((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → 𝑔 ∈ ComplGraph))
3130alrimiv 1928 . 2 (𝐺 ∈ ComplGraph → ∀𝑔(((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (iEdg‘𝑔) = (iEdg‘𝐺)) → 𝑔 ∈ ComplGraph))
32 fvexd 6660 . 2 (𝐺 ∈ ComplGraph → (Vtx‘𝐺) ∈ V)
33 fvexd 6660 . 2 (𝐺 ∈ ComplGraph → (iEdg‘𝐺) ∈ V)
3431, 32, 33gropeld 26826 1 (𝐺 ∈ ComplGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  wral 3106  wrex 3107  Vcvv 3441  cdif 3878  wss 3881  {csn 4525  {cpr 4527  cop 4531  ran crn 5520  cfv 6324  Vtxcvtx 26789  iEdgciedg 26790  Edgcedg 26840  ComplGraphccplgr 27199
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-vtx 26791  df-iedg 26792  df-edg 26841  df-nbgr 27123  df-uvtx 27176  df-cplgr 27201
This theorem is referenced by:  cusgrop  27228
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