Theorem isubgr3stgr 47969

Description: If a vertex of a simple graph has exactly 𝑁 (different) neighbors, and none of these neighbors are connected by an edge, then the (closed) neighborhood of this vertex induces a subgraph which is isomorphic to an 𝑁-star. (Contributed by AV, 29-Sep-2025.)

Hypotheses

Ref	Expression
isubgr3stgr.v	⊢ 𝑉 = (Vtx‘𝐺)
isubgr3stgr.u	⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)
isubgr3stgr.c	⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)
isubgr3stgr.n	⊢ 𝑁 ∈ ℕ0
isubgr3stgr.s	⊢ 𝑆 = (StarGr‘𝑁)
isubgr3stgr.w	⊢ 𝑊 = (Vtx‘𝑆)
isubgr3stgr.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
isubgr3stgr	⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸) → (𝐺 ISubGr 𝐶) ≃𝑔𝑟 (StarGr‘𝑁)))

Distinct variable groups:   𝑦,𝐶   𝑥,𝐸,𝑦   𝑦,𝐺   𝑥,𝑁,𝑦   𝑥,𝑈,𝑦   𝑦,𝑉   𝑦,𝑊   𝑦,𝑋

Allowed substitution hints:   𝐶(𝑥)   𝑆(𝑥,𝑦)   𝐺(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)

Proof of Theorem isubgr3stgr

Dummy variables 𝑓 𝑔 𝑒 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → 𝐺 ∈ USGraph)
2		simpr 484	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
3		simpl 482	. . . . 5 ⊢ (((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸) → (♯‘𝑈) = 𝑁)
4		isubgr3stgr.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
5		isubgr3stgr.u	. . . . . 6 ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)
6		isubgr3stgr.c	. . . . . 6 ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)
7		isubgr3stgr.n	. . . . . 6 ⊢ 𝑁 ∈ ℕ0
8		isubgr3stgr.s	. . . . . 6 ⊢ 𝑆 = (StarGr‘𝑁)
9		isubgr3stgr.w	. . . . . 6 ⊢ 𝑊 = (Vtx‘𝑆)
10	4, 5, 6, 7, 8, 9	isubgr3stgrlem3 47962	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ∃𝑓(𝑓:𝐶–1-1-onto→𝑊 ∧ (𝑓‘𝑋) = 0))
11	1, 2, 3, 10	syl2an3an 1424	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → ∃𝑓(𝑓:𝐶–1-1-onto→𝑊 ∧ (𝑓‘𝑋) = 0))
12	4	clnbgrssvtx 47825	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉
13	6, 12	eqsstri 3982	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 ⊆ 𝑉
14	13	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ 𝑉 → 𝐶 ⊆ 𝑉)
15	14	anim2i 617	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (𝐺 ∈ USGraph ∧ 𝐶 ⊆ 𝑉))
16	15	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → (𝐺 ∈ USGraph ∧ 𝐶 ⊆ 𝑉))
17	4	isubgrvtx 47861	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USGraph ∧ 𝐶 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝐶)) = 𝐶)
18	16, 17	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → (Vtx‘(𝐺 ISubGr 𝐶)) = 𝐶)
19	18	eqcomd 2735	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → 𝐶 = (Vtx‘(𝐺 ISubGr 𝐶)))
20	19	f1oeq2d 6760	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → (𝑓:𝐶–1-1-onto→𝑊 ↔ 𝑓:(Vtx‘(𝐺 ISubGr 𝐶))–1-1-onto→𝑊))
21	20	biimpd 229	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → (𝑓:𝐶–1-1-onto→𝑊 → 𝑓:(Vtx‘(𝐺 ISubGr 𝐶))–1-1-onto→𝑊))
22	21	adantrd 491	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → ((𝑓:𝐶–1-1-onto→𝑊 ∧ (𝑓‘𝑋) = 0) → 𝑓:(Vtx‘(𝐺 ISubGr 𝐶))–1-1-onto→𝑊))
23	22	imp 406	. . . . . . 7 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝑓:𝐶–1-1-onto→𝑊 ∧ (𝑓‘𝑋) = 0)) → 𝑓:(Vtx‘(𝐺 ISubGr 𝐶))–1-1-onto→𝑊)
24		fvexd 6837	. . . . . . . . 9 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝑓:𝐶–1-1-onto→𝑊 ∧ (𝑓‘𝑋) = 0)) → (Edg‘(𝐺 ISubGr 𝐶)) ∈ V)
25	24	mptexd 7160	. . . . . . . 8 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝑓:𝐶–1-1-onto→𝑊 ∧ (𝑓‘𝑋) = 0)) → (𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖)) ∈ V)
26		isubgr3stgr.e	. . . . . . . . 9 ⊢ 𝐸 = (Edg‘𝐺)
27		eqid 2729	. . . . . . . . 9 ⊢ (Edg‘(𝐺 ISubGr 𝐶)) = (Edg‘(𝐺 ISubGr 𝐶))
28		eqid 2729	. . . . . . . . 9 ⊢ (𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖)) = (𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖))
29	4, 5, 6, 7, 8, 9, 26, 27, 28	isubgr3stgrlem9 47968	. . . . . . . 8 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝑓:𝐶–1-1-onto→𝑊 ∧ (𝑓‘𝑋) = 0)) → ((𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖)):(Edg‘(𝐺 ISubGr 𝐶))–1-1-onto→(Edg‘(StarGr‘𝑁)) ∧ ∀𝑒 ∈ (Edg‘(𝐺 ISubGr 𝐶))(𝑓 “ 𝑒) = ((𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖))‘𝑒)))
30		f1oeq1 6752	. . . . . . . . 9 ⊢ (𝑔 = (𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖)) → (𝑔:(Edg‘(𝐺 ISubGr 𝐶))–1-1-onto→(Edg‘(StarGr‘𝑁)) ↔ (𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖)):(Edg‘(𝐺 ISubGr 𝐶))–1-1-onto→(Edg‘(StarGr‘𝑁))))
31		fveq1 6821	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖)) → (𝑔‘𝑒) = ((𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖))‘𝑒))
32	31	eqeq2d 2740	. . . . . . . . . 10 ⊢ (𝑔 = (𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖)) → ((𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ (𝑓 “ 𝑒) = ((𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖))‘𝑒)))
33	32	ralbidv 3152	. . . . . . . . 9 ⊢ (𝑔 = (𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖)) → (∀𝑒 ∈ (Edg‘(𝐺 ISubGr 𝐶))(𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ ∀𝑒 ∈ (Edg‘(𝐺 ISubGr 𝐶))(𝑓 “ 𝑒) = ((𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖))‘𝑒)))
34	30, 33	anbi12d 632	. . . . . . . 8 ⊢ (𝑔 = (𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖)) → ((𝑔:(Edg‘(𝐺 ISubGr 𝐶))–1-1-onto→(Edg‘(StarGr‘𝑁)) ∧ ∀𝑒 ∈ (Edg‘(𝐺 ISubGr 𝐶))(𝑓 “ 𝑒) = (𝑔‘𝑒)) ↔ ((𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖)):(Edg‘(𝐺 ISubGr 𝐶))–1-1-onto→(Edg‘(StarGr‘𝑁)) ∧ ∀𝑒 ∈ (Edg‘(𝐺 ISubGr 𝐶))(𝑓 “ 𝑒) = ((𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖))‘𝑒))))
35	25, 29, 34	spcedv 3553	. . . . . . 7 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝑓:𝐶–1-1-onto→𝑊 ∧ (𝑓‘𝑋) = 0)) → ∃𝑔(𝑔:(Edg‘(𝐺 ISubGr 𝐶))–1-1-onto→(Edg‘(StarGr‘𝑁)) ∧ ∀𝑒 ∈ (Edg‘(𝐺 ISubGr 𝐶))(𝑓 “ 𝑒) = (𝑔‘𝑒)))
36	23, 35	jca 511	. . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝑓:𝐶–1-1-onto→𝑊 ∧ (𝑓‘𝑋) = 0)) → (𝑓:(Vtx‘(𝐺 ISubGr 𝐶))–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:(Edg‘(𝐺 ISubGr 𝐶))–1-1-onto→(Edg‘(StarGr‘𝑁)) ∧ ∀𝑒 ∈ (Edg‘(𝐺 ISubGr 𝐶))(𝑓 “ 𝑒) = (𝑔‘𝑒))))
37	36	ex 412	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → ((𝑓:𝐶–1-1-onto→𝑊 ∧ (𝑓‘𝑋) = 0) → (𝑓:(Vtx‘(𝐺 ISubGr 𝐶))–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:(Edg‘(𝐺 ISubGr 𝐶))–1-1-onto→(Edg‘(StarGr‘𝑁)) ∧ ∀𝑒 ∈ (Edg‘(𝐺 ISubGr 𝐶))(𝑓 “ 𝑒) = (𝑔‘𝑒)))))
38	37	eximdv 1917	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → (∃𝑓(𝑓:𝐶–1-1-onto→𝑊 ∧ (𝑓‘𝑋) = 0) → ∃𝑓(𝑓:(Vtx‘(𝐺 ISubGr 𝐶))–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:(Edg‘(𝐺 ISubGr 𝐶))–1-1-onto→(Edg‘(StarGr‘𝑁)) ∧ ∀𝑒 ∈ (Edg‘(𝐺 ISubGr 𝐶))(𝑓 “ 𝑒) = (𝑔‘𝑒)))))
39	11, 38	mpd 15	. . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → ∃𝑓(𝑓:(Vtx‘(𝐺 ISubGr 𝐶))–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:(Edg‘(𝐺 ISubGr 𝐶))–1-1-onto→(Edg‘(StarGr‘𝑁)) ∧ ∀𝑒 ∈ (Edg‘(𝐺 ISubGr 𝐶))(𝑓 “ 𝑒) = (𝑔‘𝑒))))
40	4	isubgrusgr 47866	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝐶 ⊆ 𝑉) → (𝐺 ISubGr 𝐶) ∈ USGraph)
41	15, 40	syl 17	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (𝐺 ISubGr 𝐶) ∈ USGraph)
42		usgruspgr 29125	. . . . . 6 ⊢ ((𝐺 ISubGr 𝐶) ∈ USGraph → (𝐺 ISubGr 𝐶) ∈ USPGraph)
43		uspgrushgr 29122	. . . . . 6 ⊢ ((𝐺 ISubGr 𝐶) ∈ USPGraph → (𝐺 ISubGr 𝐶) ∈ USHGraph)
44	41, 42, 43	3syl 18	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (𝐺 ISubGr 𝐶) ∈ USHGraph)
45		stgrusgra 47953	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (StarGr‘𝑁) ∈ USGraph)
46		usgruspgr 29125	. . . . . . 7 ⊢ ((StarGr‘𝑁) ∈ USGraph → (StarGr‘𝑁) ∈ USPGraph)
47		uspgrushgr 29122	. . . . . . 7 ⊢ ((StarGr‘𝑁) ∈ USPGraph → (StarGr‘𝑁) ∈ USHGraph)
48	45, 46, 47	3syl 18	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (StarGr‘𝑁) ∈ USHGraph)
49	7, 48	ax-mp 5	. . . . 5 ⊢ (StarGr‘𝑁) ∈ USHGraph
50		eqid 2729	. . . . . 6 ⊢ (Vtx‘(𝐺 ISubGr 𝐶)) = (Vtx‘(𝐺 ISubGr 𝐶))
51	8	fveq2i 6825	. . . . . . 7 ⊢ (Vtx‘𝑆) = (Vtx‘(StarGr‘𝑁))
52	9, 51	eqtri 2752	. . . . . 6 ⊢ 𝑊 = (Vtx‘(StarGr‘𝑁))
53		eqid 2729	. . . . . 6 ⊢ (Edg‘(StarGr‘𝑁)) = (Edg‘(StarGr‘𝑁))
54	50, 52, 27, 53	gricushgr 47911	. . . . 5 ⊢ (((𝐺 ISubGr 𝐶) ∈ USHGraph ∧ (StarGr‘𝑁) ∈ USHGraph) → ((𝐺 ISubGr 𝐶) ≃𝑔𝑟 (StarGr‘𝑁) ↔ ∃𝑓(𝑓:(Vtx‘(𝐺 ISubGr 𝐶))–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:(Edg‘(𝐺 ISubGr 𝐶))–1-1-onto→(Edg‘(StarGr‘𝑁)) ∧ ∀𝑒 ∈ (Edg‘(𝐺 ISubGr 𝐶))(𝑓 “ 𝑒) = (𝑔‘𝑒)))))
55	44, 49, 54	sylancl 586	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → ((𝐺 ISubGr 𝐶) ≃𝑔𝑟 (StarGr‘𝑁) ↔ ∃𝑓(𝑓:(Vtx‘(𝐺 ISubGr 𝐶))–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:(Edg‘(𝐺 ISubGr 𝐶))–1-1-onto→(Edg‘(StarGr‘𝑁)) ∧ ∀𝑒 ∈ (Edg‘(𝐺 ISubGr 𝐶))(𝑓 “ 𝑒) = (𝑔‘𝑒)))))
56	55	adantr 480	. . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → ((𝐺 ISubGr 𝐶) ≃𝑔𝑟 (StarGr‘𝑁) ↔ ∃𝑓(𝑓:(Vtx‘(𝐺 ISubGr 𝐶))–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:(Edg‘(𝐺 ISubGr 𝐶))–1-1-onto→(Edg‘(StarGr‘𝑁)) ∧ ∀𝑒 ∈ (Edg‘(𝐺 ISubGr 𝐶))(𝑓 “ 𝑒) = (𝑔‘𝑒)))))
57	39, 56	mpbird 257	. 2 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → (𝐺 ISubGr 𝐶) ≃𝑔𝑟 (StarGr‘𝑁))
58	57	ex 412	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸) → (𝐺 ISubGr 𝐶) ≃𝑔𝑟 (StarGr‘𝑁)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ∉ wnel 3029  ∀wral 3044  Vcvv 3436   ⊆ wss 3903  {cpr 4579   class class class wbr 5092   ↦ cmpt 5173   “ cima 5622  –1-1-onto→wf1o 6481  ‘cfv 6482  (class class class)co 7349  0cc0 11009  ℕ₀cn0 12384  ♯chash 14237  Vtxcvtx 28941  Edgcedg 28992  USHGraphcushgr 29002  USPGraphcuspgr 29093  USGraphcusgr 29094   NeighbVtx cnbgr 29277   ClNeighbVtx cclnbgr 47812   ISubGr cisubgr 47854   ≃_𝑔𝑟 cgric 47870  StarGrcstgr 47945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-xnn0 12458  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-hash 14238  df-struct 17058  df-slot 17093  df-ndx 17105  df-base 17121  df-edgf 28934  df-vtx 28943  df-iedg 28944  df-edg 28993  df-uhgr 29003  df-ushgr 29004  df-upgr 29027  df-umgr 29028  df-uspgr 29095  df-usgr 29096  df-subgr 29213  df-nbgr 29278  df-clnbgr 47813  df-isubgr 47855  df-grim 47872  df-gric 47875  df-stgr 47946

This theorem is referenced by:  gpg5gricstgr3  48084