Theorem isubgr3stgrlem8 47976

Description: Lemma 8 for isubgr3stgr 47978. (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 6030	. . . 4 ⊢ (𝑖 = 𝑘 → (𝐹 “ 𝑖) = (𝐹 “ 𝑘))
3	2	cbvmptv 5214	. . 3 ⊢ (𝑖 ∈ 𝐼 ↦ (𝐹 “ 𝑖)) = (𝑘 ∈ 𝐼 ↦ (𝐹 “ 𝑘))
4	1, 3	eqtri 2753	. 2 ⊢ 𝐻 = (𝑘 ∈ 𝐼 ↦ (𝐹 “ 𝑘))
5		f1of 6803	. . . . 5 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → 𝐹:𝐶⟶𝑊)
6	5	ad2antrl 728	. . . 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 47973	. . . 4 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑘 ∈ 𝐼) → (𝐻‘𝑘) = (𝐹 “ 𝑘))
16	6, 15	sylan 580	. . 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 47974	. . . 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 7059	. . 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 2830	. 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 47975	. . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁))) → (◡𝐹 “ 𝑗) ∈ 𝐼)
21	20	ad4ant134 1175	. 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 6810	. . . . . 6 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → 𝐹:𝐶–onto→𝑊)
23	22	ad2antrl 728	. . . . 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 47962	. . . . . . 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 484	. . . . . 6 ⊢ ((𝑘 ∈ 𝐼 ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁))) → 𝑗 ∈ (Edg‘(StarGr‘𝑁)))
27	11	fveq2i 6864	. . . . . . . . 9 ⊢ (Vtx‘𝑆) = (Vtx‘(StarGr‘𝑁))
28	12, 27	eqtri 2753	. . . . . . . 8 ⊢ 𝑊 = (Vtx‘(StarGr‘𝑁))
29		eqid 2730	. . . . . . . 8 ⊢ (Edg‘(StarGr‘𝑁)) = (Edg‘(StarGr‘𝑁))
30	28, 29	edgssv2 29132	. . . . . . 7 ⊢ (((StarGr‘𝑁) ∈ USGraph ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁))) → (𝑗 ⊆ 𝑊 ∧ (♯‘𝑗) = 2))
31	30	simpld 494	. . . . . 6 ⊢ (((StarGr‘𝑁) ∈ USGraph ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁))) → 𝑗 ⊆ 𝑊)
32	25, 26, 31	syl2an 596	. . . . 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 6820	. . . . 5 ⊢ ((𝐹:𝐶–onto→𝑊 ∧ 𝑗 ⊆ 𝑊) → (𝐹 “ (◡𝐹 “ 𝑗)) = 𝑗)
34	23, 32, 33	syl2an2r 685	. . . 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 6030	. . . . 5 ⊢ (𝑘 = (◡𝐹 “ 𝑗) → (𝐹 “ 𝑘) = (𝐹 “ (◡𝐹 “ 𝑗)))
36	35	eqcomd 2736	. . . 4 ⊢ (𝑘 = (◡𝐹 “ 𝑗) → (𝐹 “ (◡𝐹 “ 𝑗)) = (𝐹 “ 𝑘))
37	34, 36	sylan9req 2786	. . 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 6030	. . . . 5 ⊢ (𝑗 = (𝐹 “ 𝑘) → (◡𝐹 “ 𝑗) = (◡𝐹 “ (𝐹 “ 𝑘)))
39		f1of1 6802	. . . . . . 7 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → 𝐹:𝐶–1-1→𝑊)
40	39	ad2antrl 728	. . . . . 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 29120	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)
42	41	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → 𝐺 ∈ UHGraph)
43	7	clnbgrssvtx 47836	. . . . . . . . . . 11 ⊢ (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉
44	9, 43	eqsstri 3996	. . . . . . . . . 10 ⊢ 𝐶 ⊆ 𝑉
45	44	a1i 11	. . . . . . . . 9 ⊢ ((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) → 𝐶 ⊆ 𝑉)
46		eqid 2730	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
47		eqid 2730	. . . . . . . . . 10 ⊢ (𝐺 ISubGr 𝐶) = (𝐺 ISubGr 𝐶)
48	7, 46, 47, 14	isubgredg 47870	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐶 ⊆ 𝑉) → (𝑘 ∈ 𝐼 ↔ (𝑘 ∈ (Edg‘𝐺) ∧ 𝑘 ⊆ 𝐶)))
49	42, 45, 48	syl2an 596	. . . . . . . 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 484	. . . . . . . . 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 253	. . . . . . 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 418	. . . . . 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 6819	. . . . . 6 ⊢ ((𝐹:𝐶–1-1→𝑊 ∧ 𝑘 ⊆ 𝐶) → (◡𝐹 “ (𝐹 “ 𝑘)) = 𝑘)
55	40, 53, 54	syl2an2r 685	. . . . 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 2787	. . . 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 2736	. . 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 800	. 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 7646	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∉ wnel 3030  ∀wral 3045   ⊆ wss 3917  {cpr 4594   ↦ cmpt 5191  ^◡ccnv 5640   “ cima 5644  ⟶wf 6510  –1-1→wf1 6511  –onto→wfo 6512  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390  0cc0 11075  2c2 12248  ℕ₀cn0 12449  ♯chash 14302  Vtxcvtx 28930  Edgcedg 28981  UHGraphcuhgr 28990  USGraphcusgr 29083   NeighbVtx cnbgr 29266   ClNeighbVtx cclnbgr 47823   ISubGr cisubgr 47864  StarGrcstgr 47954

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  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 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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-xnn0 12523  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-hash 14303  df-struct 17124  df-slot 17159  df-ndx 17171  df-base 17187  df-edgf 28923  df-vtx 28932  df-iedg 28933  df-edg 28982  df-uhgr 28992  df-upgr 29016  df-umgr 29017  df-uspgr 29084  df-usgr 29085  df-nbgr 29267  df-clnbgr 47824  df-isubgr 47865  df-stgr 47955

This theorem is referenced by:  isubgr3stgrlem9  47977