Theorem isubgr3stgrlem8 49071

Description: Lemma 8 for isubgr3stgr 49073. (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 5208	. . 3 ⊢ (𝑖 ∈ 𝐼 ↦ (𝐹 “ 𝑖)) = (𝑘 ∈ 𝐼 ↦ (𝐹 “ 𝑘))
4	1, 3	eqtri 2783	. 2 ⊢ 𝐻 = (𝑘 ∈ 𝐼 ↦ (𝐹 “ 𝑘))
5		f1of 6824	. . . . 5 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → 𝐹:𝐶⟶𝑊)
6	5	ad2antrl 741	. . . 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 49068	. . . 4 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑘 ∈ 𝐼) → (𝐻‘𝑘) = (𝐹 “ 𝑘))
16	6, 15	sylan 592	. . 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 49069	. . . 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 7084	. . 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 2861	. 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 49070	. . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁))) → (◡𝐹 “ 𝑗) ∈ 𝐼)
21	20	ad4ant134 1193	. 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 6832	. . . . . 6 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → 𝐹:𝐶–onto→𝑊)
23	22	ad2antrl 741	. . . . 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 49057	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (StarGr‘𝑁) ∈ USGraph)
25	10, 24	mp1i 14	. . . . . 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 490	. . . . . 6 ⊢ ((𝑘 ∈ 𝐼 ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁))) → 𝑗 ∈ (Edg‘(StarGr‘𝑁)))
27	11	fveq2i 6888	. . . . . . . . 9 ⊢ (Vtx‘𝑆) = (Vtx‘(StarGr‘𝑁))
28	12, 27	eqtri 2783	. . . . . . . 8 ⊢ 𝑊 = (Vtx‘(StarGr‘𝑁))
29		eqid 2760	. . . . . . . 8 ⊢ (Edg‘(StarGr‘𝑁)) = (Edg‘(StarGr‘𝑁))
30	28, 29	edgssv2 29779	. . . . . . 7 ⊢ (((StarGr‘𝑁) ∈ USGraph ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁))) → (𝑗 ⊆ 𝑊 ∧ (♯‘𝑗) = 2))
31	30	simpld 500	. . . . . 6 ⊢ (((StarGr‘𝑁) ∈ USGraph ∧ 𝑗 ∈ (Edg‘(StarGr‘𝑁))) → 𝑗 ⊆ 𝑊)
32	25, 26, 31	syl2an 608	. . . . 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 6842	. . . . 5 ⊢ ((𝐹:𝐶–onto→𝑊 ∧ 𝑗 ⊆ 𝑊) → (𝐹 “ (◡𝐹 “ 𝑗)) = 𝑗)
34	23, 32, 33	syl2an2r 698	. . . 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 2766	. . . 4 ⊢ (𝑘 = (◡𝐹 “ 𝑗) → (𝐹 “ (◡𝐹 “ 𝑗)) = (𝐹 “ 𝑘))
37	34, 36	sylan9req 2816	. . 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 6823	. . . . . . 7 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → 𝐹:𝐶–1-1→𝑊)
40	39	ad2antrl 741	. . . . . 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 29767	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)
42	41	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → 𝐺 ∈ UHGraph)
43	7	clnbgrssvtx 48929	. . . . . . . . . . 11 ⊢ (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉
44	9, 43	eqsstri 3954	. . . . . . . . . 10 ⊢ 𝐶 ⊆ 𝑉
45	44	a1i 11	. . . . . . . . 9 ⊢ ((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) → 𝐶 ⊆ 𝑉)
46		eqid 2760	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
47		eqid 2760	. . . . . . . . . 10 ⊢ (𝐺 ISubGr 𝐶) = (𝐺 ISubGr 𝐶)
48	7, 46, 47, 14	isubgredg 48964	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐶 ⊆ 𝑉) → (𝑘 ∈ 𝐼 ↔ (𝑘 ∈ (Edg‘𝐺) ∧ 𝑘 ⊆ 𝐶)))
49	42, 45, 48	syl2an 608	. . . . . . . 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 490	. . . . . . . . 9 ⊢ ((𝑘 ∈ (Edg‘𝐺) ∧ 𝑘 ⊆ 𝐶) → 𝑘 ⊆ 𝐶)
51	50	a1d 26	. . . . . . . 8 ⊢ ((𝑘 ∈ (Edg‘𝐺) ∧ 𝑘 ⊆ 𝐶) → (𝑗 ∈ (Edg‘(StarGr‘𝑁)) → 𝑘 ⊆ 𝐶))
52	49, 51	biimtrdi 256	. . . . . . 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 424	. . . . . 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 6841	. . . . . 6 ⊢ ((𝐹:𝐶–1-1→𝑊 ∧ 𝑘 ⊆ 𝐶) → (◡𝐹 “ (𝐹 “ 𝑘)) = 𝑘)
55	40, 53, 54	syl2an2r 698	. . . . 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 2817	. . . 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 2766	. . 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 813	. 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 7675	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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ∉ wnel 3061  ∀wral 3076   ⊆ wss 3895  {cpr 4585   ↦ cmpt 5185  ^◡ccnv 5649   “ cima 5653  ⟶wf 6534  –1-1→wf1 6535  –onto→wfo 6536  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420  0cc0 11200  2c2 12397  ℕ₀cn0 12606  ♯chash 14474  Vtxcvtx 29574  Edgcedg 29625  UHGraphcuhgr 29634  USGraphcusgr 29730   NeighbVtx cnbgr 29913   ClNeighbVtx cclnbgr 48916   ISubGr cisubgr 48958  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-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-upgr 29660  df-umgr 29661  df-uspgr 29731  df-usgr 29732  df-nbgr 29914  df-clnbgr 48917  df-isubgr 48959  df-stgr 49050

This theorem is used by:  isubgr3stgrlem9  49072