Theorem isubgr3stgrlem8 48600

Description: Lemma 8 for isubgr3stgr 48602. (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)
isubgr3stgr.i	⊢ 𝐼 = (Edg‘(𝐺 ISubGr 𝐶))
isubgr3stgr.h	⊢ 𝐻 = (𝑖 ∈ 𝐼 ↦ (𝐹 “ 𝑖))

Assertion

Ref	Expression
isubgr3stgrlem8	⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → 𝐻:𝐼–1-1-onto→(Edg‘(StarGr‘𝑁)))

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

Allowed substitution hints:   𝐶(𝑥)   𝑆(𝑥,𝑦,𝑖)   𝐹(𝑥)   𝐺(𝑥)   𝐻(𝑥,𝑦,𝑖)   𝐼(𝑥)   𝑁(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)

Proof of Theorem isubgr3stgrlem8

Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isubgr3stgr.h	. . 3 ⊢ 𝐻 = (𝑖 ∈ 𝐼 ↦ (𝐹 “ 𝑖))
2		imaeq2 6047	. . . 4 ⊢ (𝑖 = 𝑘 → (𝐹 “ 𝑖) = (𝐹 “ 𝑘))
3	2	cbvmptv 5206	. . 3 ⊢ (𝑖 ∈ 𝐼 ↦ (𝐹 “ 𝑖)) = (𝑘 ∈ 𝐼 ↦ (𝐹 “ 𝑘))
4	1, 3	eqtri 2787	. 2 ⊢ 𝐻 = (𝑘 ∈ 𝐼 ↦ (𝐹 “ 𝑘))
5		f1of 6808	. . . . 5 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → 𝐹:𝐶⟶𝑊)
6	5	ad2antrl 738	. . . 4 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → 𝐹:𝐶⟶𝑊)
7		isubgr3stgr.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
8		isubgr3stgr.u	. . . . 5 ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)
9		isubgr3stgr.c	. . . . 5 ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)
10		isubgr3stgr.n	. . . . 5 ⊢ 𝑁 ∈ ℕ0
11		isubgr3stgr.s	. . . . 5 ⊢ 𝑆 = (StarGr‘𝑁)
12		isubgr3stgr.w	. . . . 5 ⊢ 𝑊 = (Vtx‘𝑆)
13		isubgr3stgr.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
14		isubgr3stgr.i	. . . . 5 ⊢ 𝐼 = (Edg‘(𝐺 ISubGr 𝐶))
15	7, 8, 9, 10, 11, 12, 13, 14, 1	isubgr3stgrlem5 48597	. . . 4 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑘 ∈ 𝐼) → (𝐻‘𝑘) = (𝐹 “ 𝑘))
16	6, 15	sylan 589	. . 3 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑘 ∈ 𝐼) → (𝐻‘𝑘) = (𝐹 “ 𝑘))
17	7, 8, 9, 10, 11, 12, 13, 14, 1	isubgr3stgrlem6 48598	. . . 4 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → 𝐻:𝐼⟶(Edg‘(StarGr‘𝑁)))
18	17	ffvelcdmda 7067	. . 3 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑘 ∈ 𝐼) → (𝐻‘𝑘) ∈ (Edg‘(StarGr‘𝑁)))
19	16, 18	eqeltrrd 2865	. 2 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑘 ∈ 𝐼) → (𝐹 “ 𝑘) ∈ (Edg‘(StarGr‘𝑁)))
20	7, 8, 9, 10, 11, 12, 13, 14, 1	isubgr3stgrlem7 48599	. . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁))) → (◡𝐹 “ 𝑗) ∈ 𝐼)
21	20	ad4ant134 1189	. 2 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁))) → (◡𝐹 “ 𝑗) ∈ 𝐼)
22		f1ofo 6816	. . . . . 6 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → 𝐹:𝐶–onto→𝑊)
23	22	ad2antrl 738	. . . . 5 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → 𝐹:𝐶–onto→𝑊)
24		stgrusgra 48586	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (StarGr‘𝑁) ∈ USGraph)
25	10, 24	mp1i 13	. . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → (StarGr‘𝑁) ∈ USGraph)
26		simpr 488	. . . . . 6 ⊢ ((𝑘 ∈ 𝐼 ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁))) → 𝑗 ∈ (Edg‘(StarGr‘𝑁)))
27	11	fveq2i 6872	. . . . . . . . 9 ⊢ (Vtx‘𝑆) = (Vtx‘(StarGr‘𝑁))
28	12, 27	eqtri 2787	. . . . . . . 8 ⊢ 𝑊 = (Vtx‘(StarGr‘𝑁))
29		eqid 2764	. . . . . . . 8 ⊢ (Edg‘(StarGr‘𝑁)) = (Edg‘(StarGr‘𝑁))
30	28, 29	edgssv2 29401	. . . . . . 7 ⊢ (((StarGr‘𝑁) ∈ USGraph ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁))) → (𝑗 ⊆ 𝑊 ∧ (♯‘𝑗) = 2))
31	30	simpld 498	. . . . . 6 ⊢ (((StarGr‘𝑁) ∈ USGraph ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁))) → 𝑗 ⊆ 𝑊)
32	25, 26, 31	syl2an 605	. . . . 5 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ (𝑘 ∈ 𝐼 ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁)))) → 𝑗 ⊆ 𝑊)
33		foimacnv 6826	. . . . 5 ⊢ ((𝐹:𝐶–onto→𝑊 ∧ 𝑗 ⊆ 𝑊) → (𝐹 “ (◡𝐹 “ 𝑗)) = 𝑗)
34	23, 32, 33	syl2an2r 695	. . . 4 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ (𝑘 ∈ 𝐼 ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁)))) → (𝐹 “ (◡𝐹 “ 𝑗)) = 𝑗)
35		imaeq2 6047	. . . . 5 ⊢ (𝑘 = (◡𝐹 “ 𝑗) → (𝐹 “ 𝑘) = (𝐹 “ (◡𝐹 “ 𝑗)))
36	35	eqcomd 2770	. . . 4 ⊢ (𝑘 = (◡𝐹 “ 𝑗) → (𝐹 “ (◡𝐹 “ 𝑗)) = (𝐹 “ 𝑘))
37	34, 36	sylan9req 2820	. . 3 ⊢ ((((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ (𝑘 ∈ 𝐼 ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁)))) ∧ 𝑘 = (◡𝐹 “ 𝑗)) → 𝑗 = (𝐹 “ 𝑘))
38		imaeq2 6047	. . . . 5 ⊢ (𝑗 = (𝐹 “ 𝑘) → (◡𝐹 “ 𝑗) = (◡𝐹 “ (𝐹 “ 𝑘)))
39		f1of1 6807	. . . . . . 7 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → 𝐹:𝐶–1-1→𝑊)
40	39	ad2antrl 738	. . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → 𝐹:𝐶–1-1→𝑊)
41		usgruhgr 29389	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)
42	41	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → 𝐺 ∈ UHGraph)
43	7	clnbgrssvtx 48458	. . . . . . . . . . 11 ⊢ (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉
44	9, 43	eqsstri 3984	. . . . . . . . . 10 ⊢ 𝐶 ⊆ 𝑉
45	44	a1i 11	. . . . . . . . 9 ⊢ ((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) → 𝐶 ⊆ 𝑉)
46		eqid 2764	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
47		eqid 2764	. . . . . . . . . 10 ⊢ (𝐺 ISubGr 𝐶) = (𝐺 ISubGr 𝐶)
48	7, 46, 47, 14	isubgredg 48493	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐶 ⊆ 𝑉) → (𝑘 ∈ 𝐼 ↔ (𝑘 ∈ (Edg‘𝐺) ∧ 𝑘 ⊆ 𝐶)))
49	42, 45, 48	syl2an 605	. . . . . . . 8 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → (𝑘 ∈ 𝐼 ↔ (𝑘 ∈ (Edg‘𝐺) ∧ 𝑘 ⊆ 𝐶)))
50		simpr 488	. . . . . . . . 9 ⊢ ((𝑘 ∈ (Edg‘𝐺) ∧ 𝑘 ⊆ 𝐶) → 𝑘 ⊆ 𝐶)
51	50	a1d 25	. . . . . . . 8 ⊢ ((𝑘 ∈ (Edg‘𝐺) ∧ 𝑘 ⊆ 𝐶) → (𝑗 ∈ (Edg‘(StarGr‘𝑁)) → 𝑘 ⊆ 𝐶))
52	49, 51	biimtrdi 255	. . . . . . 7 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → (𝑘 ∈ 𝐼 → (𝑗 ∈ (Edg‘(StarGr‘𝑁)) → 𝑘 ⊆ 𝐶)))
53	52	imp32 422	. . . . . 6 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ (𝑘 ∈ 𝐼 ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁)))) → 𝑘 ⊆ 𝐶)
54		f1imacnv 6825	. . . . . 6 ⊢ ((𝐹:𝐶–1-1→𝑊 ∧ 𝑘 ⊆ 𝐶) → (◡𝐹 “ (𝐹 “ 𝑘)) = 𝑘)
55	40, 53, 54	syl2an2r 695	. . . . 5 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ (𝑘 ∈ 𝐼 ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁)))) → (◡𝐹 “ (𝐹 “ 𝑘)) = 𝑘)
56	38, 55	sylan9eqr 2821	. . . 4 ⊢ ((((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ (𝑘 ∈ 𝐼 ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁)))) ∧ 𝑗 = (𝐹 “ 𝑘)) → (◡𝐹 “ 𝑗) = 𝑘)
57	56	eqcomd 2770	. . 3 ⊢ ((((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ (𝑘 ∈ 𝐼 ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁)))) ∧ 𝑗 = (𝐹 “ 𝑘)) → 𝑘 = (◡𝐹 “ 𝑗))
58	37, 57	impbida 810	. 2 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ (𝑘 ∈ 𝐼 ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁)))) → (𝑘 = (◡𝐹 “ 𝑗) ↔ 𝑗 = (𝐹 “ 𝑘)))
59	4, 19, 21, 58	f1o2d 7652	1 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → 𝐻:𝐼–1-1-onto→(Edg‘(StarGr‘𝑁)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144   ∉ wnel 3063  ∀wral 3078   ⊆ wss 3906  {cpr 4586   ↦ cmpt 5183  ^◡ccnv 5648   “ cima 5652  ⟶wf 6519  –1-1→wf1 6520  –onto→wfo 6521  –1-1-onto→wf1o 6522  ‘cfv 6523  (class class class)co 7398  0cc0 11075  2c2 12274  ℕ₀cn0 12483  ♯chash 14345  Vtxcvtx 29199  Edgcedg 29250  UHGraphcuhgr 29259  USGraphcusgr 29352   NeighbVtx cnbgr 29535   ClNeighbVtx cclnbgr 48445   ISubGr cisubgr 48487  StarGrcstgr 48578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-2o 8440  df-oadd 8443  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-dju 9861  df-card 9899  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-nn 12213  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12484  df-xnn0 12557  df-z 12571  df-dec 12691  df-uz 12842  df-fz 13515  df-hash 14346  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17248  df-edgf 29192  df-vtx 29201  df-iedg 29202  df-edg 29251  df-uhgr 29261  df-upgr 29285  df-umgr 29286  df-uspgr 29353  df-usgr 29354  df-nbgr 29536  df-clnbgr 48446  df-isubgr 48488  df-stgr 48579

This theorem is referenced by:  isubgr3stgrlem9  48601