Theorem isubgr3stgr 48876

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 488	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → 𝐺 ∈ USGraph)
2		simpr 490	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
3		simpl 488	. . . . 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 48869	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ∃𝑓(𝑓:𝐶–1-1-onto→𝑊 ∧ (𝑓‘𝑋) = 0))
11	1, 2, 3, 10	syl2an3an 1449	. . . 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 48732	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉
13	6, 12	eqsstri 3980	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 ⊆ 𝑉
14	13	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ 𝑉 → 𝐶 ⊆ 𝑉)
15	14	anim2i 629	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (𝐺 ∈ USGraph ∧ 𝐶 ⊆ 𝑉))
16	15	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → (𝐺 ∈ USGraph ∧ 𝐶 ⊆ 𝑉))
17	4	isubgrvtx 48768	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USGraph ∧ 𝐶 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝐶)) = 𝐶)
18	16, 17	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → (Vtx‘(𝐺 ISubGr 𝐶)) = 𝐶)
19	18	eqcomd 2768	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → 𝐶 = (Vtx‘(𝐺 ISubGr 𝐶)))
20	19	f1oeq2d 6817	. . . . . . . . . 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 232	. . . . . . . . 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 497	. . . . . . . 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 412	. . . . . . 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 6897	. . . . . . . . 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 7226	. . . . . . . 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 2762	. . . . . . . . 9 ⊢ (Edg‘(𝐺 ISubGr 𝐶)) = (Edg‘(𝐺 ISubGr 𝐶))
28		eqid 2762	. . . . . . . . 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 48875	. . . . . . . 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 6809	. . . . . . . . 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 6881	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖)) → (𝑔‘𝑒) = ((𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖))‘𝑒))
32	31	eqeq2d 2773	. . . . . . . . . 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 3187	. . . . . . . . 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 644	. . . . . . . 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 3555	. . . . . . 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 521	. . . . . 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 418	. . . . 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 1950	. . . 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 16	. . 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 48773	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝐶 ⊆ 𝑉) → (𝐺 ISubGr 𝐶) ∈ USGraph)
41	15, 40	syl 18	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (𝐺 ISubGr 𝐶) ∈ USGraph)
42		usgruspgr 29624	. . . . . 6 ⊢ ((𝐺 ISubGr 𝐶) ∈ USGraph → (𝐺 ISubGr 𝐶) ∈ USPGraph)
43		uspgrushgr 29621	. . . . . 6 ⊢ ((𝐺 ISubGr 𝐶) ∈ USPGraph → (𝐺 ISubGr 𝐶) ∈ USHGraph)
44	41, 42, 43	3syl 19	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (𝐺 ISubGr 𝐶) ∈ USHGraph)
45		stgrusgra 48860	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (StarGr‘𝑁) ∈ USGraph)
46		usgruspgr 29624	. . . . . . 7 ⊢ ((StarGr‘𝑁) ∈ USGraph → (StarGr‘𝑁) ∈ USPGraph)
47		uspgrushgr 29621	. . . . . . 7 ⊢ ((StarGr‘𝑁) ∈ USPGraph → (StarGr‘𝑁) ∈ USHGraph)
48	45, 46, 47	3syl 19	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (StarGr‘𝑁) ∈ USHGraph)
49	7, 48	ax-mp 5	. . . . 5 ⊢ (StarGr‘𝑁) ∈ USHGraph
50		eqid 2762	. . . . . 6 ⊢ (Vtx‘(𝐺 ISubGr 𝐶)) = (Vtx‘(𝐺 ISubGr 𝐶))
51	8	fveq2i 6885	. . . . . . 7 ⊢ (Vtx‘𝑆) = (Vtx‘(StarGr‘𝑁))
52	9, 51	eqtri 2785	. . . . . 6 ⊢ 𝑊 = (Vtx‘(StarGr‘𝑁))
53		eqid 2762	. . . . . 6 ⊢ (Edg‘(StarGr‘𝑁)) = (Edg‘(StarGr‘𝑁))
54	50, 52, 27, 53	gricushgr 48818	. . . . 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 598	. . . 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 486	. . 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 260	. 2 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → (𝐺 ISubGr 𝐶) ≃𝑔𝑟 (StarGr‘𝑁))
58	57	ex 418	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸) → (𝐺 ISubGr 𝐶) ≃𝑔𝑟 (StarGr‘𝑁)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ∉ wnel 3063  ∀wral 3078  Vcvv 3453   ⊆ wss 3902  {cpr 4589   class class class wbr 5107   ↦ cmpt 5190   “ cima 5662  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7416  0cc0 11125  ℕ₀cn0 12529  ♯chash 14394  Vtxcvtx 29437  Edgcedg 29488  USHGraphcushgr 29498  USPGraphcuspgr 29592  USGraphcusgr 29593   NeighbVtx cnbgr 29776   ClNeighbVtx cclnbgr 48719   ISubGr cisubgr 48761   ≃_𝑔𝑟 cgric 48777  StarGrcstgr 48852

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-oadd 8462  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9909  df-card 9947  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-nn 12259  df-2 12328  df-3 12329  df-4 12330  df-5 12331  df-6 12332  df-7 12333  df-8 12334  df-9 12335  df-n0 12530  df-xnn0 12603  df-z 12617  df-dec 12738  df-uz 12889  df-fz 13562  df-hash 14395  df-struct 17241  df-slot 17276  df-ndx 17288  df-base 17304  df-edgf 29430  df-vtx 29439  df-iedg 29440  df-edg 29489  df-uhgr 29499  df-ushgr 29500  df-upgr 29523  df-umgr 29524  df-uspgr 29594  df-usgr 29595  df-subgr 29712  df-nbgr 29777  df-clnbgr 48720  df-isubgr 48762  df-grim 48779  df-gric 48782  df-stgr 48853

This theorem is used by:  gpg5gricstgr3  48991