Theorem isubgr3stgrlem8 47940

Description: Lemma 8 for isubgr3stgr 47942. (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 6074	. . . 4 ⊢ (𝑖 = 𝑘 → (𝐹 “ 𝑖) = (𝐹 “ 𝑘))
3	2	cbvmptv 5255	. . 3 ⊢ (𝑖 ∈ 𝐼 ↦ (𝐹 “ 𝑖)) = (𝑘 ∈ 𝐼 ↦ (𝐹 “ 𝑘))
4	1, 3	eqtri 2765	. 2 ⊢ 𝐻 = (𝑘 ∈ 𝐼 ↦ (𝐹 “ 𝑘))
5		f1of 6848	. . . . 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 47937	. . . 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 47938	. . . 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 7104	. . 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 2842	. 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 47939	. . 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 6855	. . . . . 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 47926	. . . . . . 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 6909	. . . . . . . . 9 ⊢ (Vtx‘𝑆) = (Vtx‘(StarGr‘𝑁))
28	12, 27	eqtri 2765	. . . . . . . 8 ⊢ 𝑊 = (Vtx‘(StarGr‘𝑁))
29		eqid 2737	. . . . . . . 8 ⊢ (Edg‘(StarGr‘𝑁)) = (Edg‘(StarGr‘𝑁))
30	28, 29	edgssv2 29215	. . . . . . 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 6865	. . . . 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 6074	. . . . 5 ⊢ (𝑘 = (◡𝐹 “ 𝑗) → (𝐹 “ 𝑘) = (𝐹 “ (◡𝐹 “ 𝑗)))
36	35	eqcomd 2743	. . . 4 ⊢ (𝑘 = (◡𝐹 “ 𝑗) → (𝐹 “ (◡𝐹 “ 𝑗)) = (𝐹 “ 𝑘))
37	34, 36	sylan9req 2798	. . 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 6074	. . . . 5 ⊢ (𝑗 = (𝐹 “ 𝑘) → (◡𝐹 “ 𝑗) = (◡𝐹 “ (𝐹 “ 𝑘)))
39		f1of1 6847	. . . . . . 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 29203	. . . . . . . . . 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 47818	. . . . . . . . . . 11 ⊢ (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉
44	9, 43	eqsstri 4030	. . . . . . . . . 10 ⊢ 𝐶 ⊆ 𝑉
45	44	a1i 11	. . . . . . . . 9 ⊢ ((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) → 𝐶 ⊆ 𝑉)
46		eqid 2737	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
47		eqid 2737	. . . . . . . . . 10 ⊢ (𝐺 ISubGr 𝐶) = (𝐺 ISubGr 𝐶)
48	7, 46, 47, 14	isubgredg 47852	. . . . . . . . 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 6864	. . . . . 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 2799	. . . 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 2743	. . 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 801	. 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 7687	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 2108   ∉ wnel 3046  ∀wral 3061   ⊆ wss 3951  {cpr 4628   ↦ cmpt 5225  ^◡ccnv 5684   “ cima 5688  ⟶wf 6557  –1-1→wf1 6558  –onto→wfo 6559  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431  0cc0 11155  2c2 12321  ℕ₀cn0 12526  ♯chash 14369  Vtxcvtx 29013  Edgcedg 29064  UHGraphcuhgr 29073  USGraphcusgr 29166   NeighbVtx cnbgr 29349   ClNeighbVtx cclnbgr 47805   ISubGr cisubgr 47846  StarGrcstgr 47918

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-hash 14370  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-edgf 29004  df-vtx 29015  df-iedg 29016  df-edg 29065  df-uhgr 29075  df-upgr 29099  df-umgr 29100  df-uspgr 29167  df-usgr 29168  df-nbgr 29350  df-clnbgr 47806  df-isubgr 47847  df-stgr 47919

This theorem is referenced by:  isubgr3stgrlem9  47941