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Theorem vdgn1frgrv2 41463
Description: Any vertex in a friendship graph does not have degree 1, see remark 2 in [MertziosUnger] p. 153 (after Proposition 1): "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 4-Apr-2021.)
Hypothesis
Ref Expression
vdn1frgrv2.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
vdgn1frgrv2 ((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) → (1 < (#‘𝑉) → ((VtxDeg‘𝐺)‘𝑁) ≠ 1))

Proof of Theorem vdgn1frgrv2
Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrusgr 41429 . . . . . 6 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
21anim1i 589 . . . . 5 ((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) → (𝐺 ∈ USGraph ∧ 𝑁𝑉))
32adantr 479 . . . 4 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → (𝐺 ∈ USGraph ∧ 𝑁𝑉))
4 vdn1frgrv2.v . . . . 5 𝑉 = (Vtx‘𝐺)
5 eqid 2604 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
6 eqid 2604 . . . . 5 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
7 eqid 2604 . . . . 5 (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
84, 5, 6, 7vtxdusgrval 40699 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ((VtxDeg‘𝐺)‘𝑁) = (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
93, 8syl 17 . . 3 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((VtxDeg‘𝐺)‘𝑁) = (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
10 eqid 2604 . . . . . . 7 (Edg‘𝐺) = (Edg‘𝐺)
114, 103cyclfrgrrn2 41454 . . . . . 6 ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))))
1211adantlr 746 . . . . 5 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))))
13 preq1 4206 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑁 → {𝑎, 𝑏} = {𝑁, 𝑏})
1413eleq1d 2666 . . . . . . . . . . . . . . 15 (𝑎 = 𝑁 → ({𝑎, 𝑏} ∈ (Edg‘𝐺) ↔ {𝑁, 𝑏} ∈ (Edg‘𝐺)))
15 preq2 4207 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑁 → {𝑐, 𝑎} = {𝑐, 𝑁})
1615eleq1d 2666 . . . . . . . . . . . . . . 15 (𝑎 = 𝑁 → ({𝑐, 𝑎} ∈ (Edg‘𝐺) ↔ {𝑐, 𝑁} ∈ (Edg‘𝐺)))
1714, 163anbi13d 1392 . . . . . . . . . . . . . 14 (𝑎 = 𝑁 → (({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺)) ↔ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))))
1817anbi2d 735 . . . . . . . . . . . . 13 (𝑎 = 𝑁 → ((𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) ↔ (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)))))
19182rexbidv 3033 . . . . . . . . . . . 12 (𝑎 = 𝑁 → (∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) ↔ ∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)))))
2019rspcva 3274 . . . . . . . . . . 11 ((𝑁𝑉 ∧ ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺)))) → ∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))))
211adantl 480 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁𝑉) ∧ 𝐺 ∈ FriendGraph ) → 𝐺 ∈ USGraph )
22 simplll 793 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁𝑉) ∧ 𝐺 ∈ FriendGraph ) → 𝑏𝑐)
23 3simpb 1051 . . . . . . . . . . . . . . . . . 18 (({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)) → ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)))
2423ad3antlr 762 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁𝑉) ∧ 𝐺 ∈ FriendGraph ) → ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)))
255, 10usgr2edg1 40436 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ USGraph ∧ 𝑏𝑐) ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
2621, 22, 24, 25syl21anc 1316 . . . . . . . . . . . . . . . 16 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁𝑉) ∧ 𝐺 ∈ FriendGraph ) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
2726a1d 25 . . . . . . . . . . . . . . 15 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁𝑉) ∧ 𝐺 ∈ FriendGraph ) → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
2827ex 448 . . . . . . . . . . . . . 14 (((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁𝑉) → (𝐺 ∈ FriendGraph → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))))
2928ex 448 . . . . . . . . . . . . 13 ((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) → (𝑁𝑉 → (𝐺 ∈ FriendGraph → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
3029a1i 11 . . . . . . . . . . . 12 ((𝑏𝑉𝑐𝑉) → ((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) → (𝑁𝑉 → (𝐺 ∈ FriendGraph → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))))))
3130rexlimivv 3012 . . . . . . . . . . 11 (∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) → (𝑁𝑉 → (𝐺 ∈ FriendGraph → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
3220, 31syl 17 . . . . . . . . . 10 ((𝑁𝑉 ∧ ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺)))) → (𝑁𝑉 → (𝐺 ∈ FriendGraph → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
3332ex 448 . . . . . . . . 9 (𝑁𝑉 → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → (𝑁𝑉 → (𝐺 ∈ FriendGraph → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))))))
3433pm2.43a 51 . . . . . . . 8 (𝑁𝑉 → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → (𝐺 ∈ FriendGraph → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
3534com24 92 . . . . . . 7 (𝑁𝑉 → (1 < (#‘𝑉) → (𝐺 ∈ FriendGraph → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
3635com3r 84 . . . . . 6 (𝐺 ∈ FriendGraph → (𝑁𝑉 → (1 < (#‘𝑉) → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
3736imp31 446 . . . . 5 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
3812, 37mpd 15 . . . 4 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
39 fvex 6093 . . . . . . . . 9 (iEdg‘𝐺) ∈ V
4039dmex 6963 . . . . . . . 8 dom (iEdg‘𝐺) ∈ V
4140a1i 11 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → dom (iEdg‘𝐺) ∈ V)
42 rabexg 4729 . . . . . . 7 (dom (iEdg‘𝐺) ∈ V → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V)
43 hash1snb 13015 . . . . . . 7 ({𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V → ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = 1 ↔ ∃𝑖{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑖}))
4441, 42, 433syl 18 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = 1 ↔ ∃𝑖{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑖}))
45 reusn 4200 . . . . . 6 (∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ∃𝑖{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑖})
4644, 45syl6bbr 276 . . . . 5 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = 1 ↔ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
4746necon3abid 2812 . . . 4 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≠ 1 ↔ ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
4838, 47mpbird 245 . . 3 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≠ 1)
499, 48eqnetrd 2843 . 2 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((VtxDeg‘𝐺)‘𝑁) ≠ 1)
5049ex 448 1 ((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) → (1 < (#‘𝑉) → ((VtxDeg‘𝐺)‘𝑁) ≠ 1))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1975  wne 2774  wral 2890  wrex 2891  ∃!wreu 2892  {crab 2894  Vcvv 3167  {csn 4119  {cpr 4121   class class class wbr 4572  dom cdm 5023  cfv 5785  1c1 9788   < clt 9925  #chash 12929  Vtxcvtx 40226  iEdgciedg 40227  Edgcedga 40348   USGraph cusgr 40376  VtxDegcvtxdg 40678   FriendGraph cfrgr 41425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-cnex 9843  ax-resscn 9844  ax-1cn 9845  ax-icn 9846  ax-addcl 9847  ax-addrcl 9848  ax-mulcl 9849  ax-mulrcl 9850  ax-mulcom 9851  ax-addass 9852  ax-mulass 9853  ax-distr 9854  ax-i2m1 9855  ax-1ne0 9856  ax-1rid 9857  ax-rnegex 9858  ax-rrecex 9859  ax-cnre 9860  ax-pre-lttri 9861  ax-pre-lttrn 9862  ax-pre-ltadd 9863  ax-pre-mulgt0 9864
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-nel 2777  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-om 6930  df-1st 7031  df-2nd 7032  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-1o 7419  df-oadd 7423  df-er 7601  df-en 7814  df-dom 7815  df-sdom 7816  df-fin 7817  df-card 8620  df-cda 8845  df-pnf 9927  df-mnf 9928  df-xr 9929  df-ltxr 9930  df-le 9931  df-sub 10114  df-neg 10115  df-nn 10863  df-2 10921  df-n0 11135  df-z 11206  df-uz 11515  df-xadd 11774  df-fz 12148  df-hash 12930  df-xnn0 40195  df-uhgr 40277  df-upgr 40305  df-umgr 40306  df-edga 40349  df-usgr 40378  df-vtxdg 40679  df-frgr 41426
This theorem is referenced by:  vdgn1frgrv3  41464  vdgfrgrgt2  41465
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