Theorem isubgr3stgr 49073

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 49066	. . . . 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 48929	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉
13	6, 12	eqsstri 3954	. . . . . . . . . . . . . . . 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 48965	. . . . . . . . . . . . 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 2766	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → 𝐶 = (Vtx‘(𝐺 ISubGr 𝐶)))
20	19	f1oeq2d 6820	. . . . . . . . . 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 6900	. . . . . . . . 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 7230	. . . . . . . 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 2760	. . . . . . . . 9 ⊢ (Edg‘(𝐺 ISubGr 𝐶)) = (Edg‘(𝐺 ISubGr 𝐶))
28		eqid 2760	. . . . . . . . 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 49072	. . . . . . . 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 6812	. . . . . . . . 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 6884	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖)) → (𝑔‘𝑒) = ((𝑖 ∈ (Edg‘(𝐺 ISubGr 𝐶)) ↦ (𝑓 “ 𝑖))‘𝑒))
32	31	eqeq2d 2771	. . . . . . . . . 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 3185	. . . . . . . . 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 3552	. . . . . . 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 1949	. . . 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 48970	. . . . . . 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 29761	. . . . . 6 ⊢ ((𝐺 ISubGr 𝐶) ∈ USGraph → (𝐺 ISubGr 𝐶) ∈ USPGraph)
43		uspgrushgr 29758	. . . . . 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 49057	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (StarGr‘𝑁) ∈ USGraph)
46		usgruspgr 29761	. . . . . . 7 ⊢ ((StarGr‘𝑁) ∈ USGraph → (StarGr‘𝑁) ∈ USPGraph)
47		uspgrushgr 29758	. . . . . . 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 2760	. . . . . 6 ⊢ (Vtx‘(𝐺 ISubGr 𝐶)) = (Vtx‘(𝐺 ISubGr 𝐶))
51	8	fveq2i 6888	. . . . . . 7 ⊢ (Vtx‘𝑆) = (Vtx‘(StarGr‘𝑁))
52	9, 51	eqtri 2783	. . . . . 6 ⊢ 𝑊 = (Vtx‘(StarGr‘𝑁))
53		eqid 2760	. . . . . 6 ⊢ (Edg‘(StarGr‘𝑁)) = (Edg‘(StarGr‘𝑁))
54	50, 52, 27, 53	gricushgr 49015	. . . . 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 1569  ∃wex 1811   ∈ wcel 2144   ∉ wnel 3061  ∀wral 3076  Vcvv 3450   ⊆ wss 3895  {cpr 4585   class class class wbr 5102   ↦ cmpt 5185   “ cima 5653  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420  0cc0 11200  ℕ₀cn0 12606  ♯chash 14474  Vtxcvtx 29574  Edgcedg 29625  USHGraphcushgr 29635  USPGraphcuspgr 29729  USGraphcusgr 29730   NeighbVtx cnbgr 29913   ClNeighbVtx cclnbgr 48916   ISubGr cisubgr 48958   ≃_𝑔𝑟 cgric 48974  StarGrcstgr 49049

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-dju 9982  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-xnn0 12680  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-hash 14475  df-struct 17325  df-slot 17360  df-ndx 17372  df-base 17388  df-edgf 29567  df-vtx 29576  df-iedg 29577  df-edg 29626  df-uhgr 29636  df-ushgr 29637  df-upgr 29660  df-umgr 29661  df-uspgr 29731  df-usgr 29732  df-subgr 29849  df-nbgr 29914  df-clnbgr 48917  df-isubgr 48959  df-grim 48976  df-gric 48979  df-stgr 49050

This theorem is used by:  gpg5gricstgr3  49188