Theorem isubgr3stgr 48535

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 485	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → 𝐺 ∈ USGraph)
2		simpr 487	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
3		simpl 485	. . . . 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 48528	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ∃𝑓(𝑓:𝐶–1-1-onto→𝑊 ∧ (𝑓‘𝑋) = 0))
11	1, 2, 3, 10	syl2an3an 1433	. . . 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 48391	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉
13	6, 12	eqsstri 3973	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 ⊆ 𝑉
14	13	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ 𝑉 → 𝐶 ⊆ 𝑉)
15	14	anim2i 625	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (𝐺 ∈ USGraph ∧ 𝐶 ⊆ 𝑉))
16	15	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → (𝐺 ∈ USGraph ∧ 𝐶 ⊆ 𝑉))
17	4	isubgrvtx 48427	. . . . . . . . . . . . 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 2758	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → 𝐶 = (Vtx‘(𝐺 ISubGr 𝐶)))
20	19	f1oeq2d 6787	. . . . . . . . . 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 231	. . . . . . . . 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 494	. . . . . . . 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 409	. . . . . . 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 6867	. . . . . . . . 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 7193	. . . . . . . 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 2752	. . . . . . . . 9 ⊢ (Edg‘(𝐺 ISubGr 𝐶)) = (Edg‘(𝐺 ISubGr 𝐶))
28		eqid 2752	. . . . . . . . 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 48534	. . . . . . . 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 6779	. . . . . . . . 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 6851	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖)) → (𝑔‘𝑒) = ((𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖))‘𝑒))
32	31	eqeq2d 2763	. . . . . . . . . 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 3175	. . . . . . . . 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 640	. . . . . . . 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 3548	. . . . . . 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 518	. . . . . 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 415	. . . . 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 1927	. . . 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 48432	. . . . . . 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 29316	. . . . . 6 ⊢ ((𝐺 ISubGr 𝐶) ∈ USGraph → (𝐺 ISubGr 𝐶) ∈ USPGraph)
43		uspgrushgr 29313	. . . . . 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 48519	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (StarGr‘𝑁) ∈ USGraph)
46		usgruspgr 29316	. . . . . . 7 ⊢ ((StarGr‘𝑁) ∈ USGraph → (StarGr‘𝑁) ∈ USPGraph)
47		uspgrushgr 29313	. . . . . . 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 2752	. . . . . 6 ⊢ (Vtx‘(𝐺 ISubGr 𝐶)) = (Vtx‘(𝐺 ISubGr 𝐶))
51	8	fveq2i 6855	. . . . . . 7 ⊢ (Vtx‘𝑆) = (Vtx‘(StarGr‘𝑁))
52	9, 51	eqtri 2775	. . . . . 6 ⊢ 𝑊 = (Vtx‘(StarGr‘𝑁))
53		eqid 2752	. . . . . 6 ⊢ (Edg‘(StarGr‘𝑁)) = (Edg‘(StarGr‘𝑁))
54	50, 52, 27, 53	gricushgr 48477	. . . . 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 594	. . . 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 483	. . 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 259	. 2 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → (𝐺 ISubGr 𝐶) ≃𝑔𝑟 (StarGr‘𝑁))
58	57	ex 415	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 208   ∧ wa 398   = wceq 1550  ∃wex 1789   ∈ wcel 2132   ∉ wnel 3051  ∀wral 3066  Vcvv 3444   ⊆ wss 3895  {cpr 4574   class class class wbr 5090   ↦ cmpt 5171   “ cima 5639  –1-1-onto→wf1o 6505  ‘cfv 6506  (class class class)co 7381  0cc0 11059  ℕ₀cn0 12467  ♯chash 14329  Vtxcvtx 29132  Edgcedg 29183  USHGraphcushgr 29193  USPGraphcuspgr 29284  USGraphcusgr 29285   NeighbVtx cnbgr 29468   ClNeighbVtx cclnbgr 48378   ISubGr cisubgr 48420   ≃_𝑔𝑟 cgric 48436  StarGrcstgr 48511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-nel 3052  df-ral 3067  df-rex 3077  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-int 4896  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-1st 7955  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-2o 8422  df-oadd 8425  df-er 8662  df-map 8794  df-en 8913  df-dom 8914  df-sdom 8915  df-fin 8916  df-dju 9845  df-card 9883  df-pnf 11204  df-mnf 11205  df-xr 11206  df-ltxr 11207  df-le 11208  df-sub 11402  df-neg 11403  df-nn 12197  df-2 12266  df-3 12267  df-4 12268  df-5 12269  df-6 12270  df-7 12271  df-8 12272  df-9 12273  df-n0 12468  df-xnn0 12541  df-z 12555  df-dec 12675  df-uz 12826  df-fz 13499  df-hash 14330  df-struct 17155  df-slot 17190  df-ndx 17202  df-base 17218  df-edgf 29125  df-vtx 29134  df-iedg 29135  df-edg 29184  df-uhgr 29194  df-ushgr 29195  df-upgr 29218  df-umgr 29219  df-uspgr 29286  df-usgr 29287  df-subgr 29404  df-nbgr 29469  df-clnbgr 48379  df-isubgr 48421  df-grim 48438  df-gric 48441  df-stgr 48512

This theorem is referenced by:  gpg5gricstgr3  48650