isubgr3stgrlem7 - Mathbox for Alexander van der Vekens

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Theorem isubgr3stgrlem7 48071

Description: Lemma 7 for isubgr3stgr 48074. (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
isubgr3stgrlem7	⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ 𝐽 ∈ (Edg‘(StarGr‘𝑁))) → (^◡𝐹 “ 𝐽) ∈ 𝐼)

Distinct variable groups: 𝐶,𝑖 𝑖,𝐹 𝑖,𝐼 𝑖,𝑊 𝑖,𝐸 𝑖,𝐺 𝑖,𝑁 𝑈,𝑖 𝑖,𝑉 𝑖,𝑋 Allowed substitution hints: 𝑆(𝑖) 𝐻(𝑖) 𝐽(𝑖)

Proof of Theorem isubgr3stgrlem7 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isubgr3stgr.n	. . . 4 ⊢ 𝑁 ∈ ℕ₀
2		stgredgel 48056	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (𝐽 ∈ (Edg‘(StarGr‘𝑁)) ↔ (𝐽 ⊆ (0...𝑁) ∧ ∃𝑦 ∈ (1...𝑁)𝐽 = {0, 𝑦})))
3	1, 2	mp1i 13	. . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → (𝐽 ∈ (Edg‘(StarGr‘𝑁)) ↔ (𝐽 ⊆ (0...𝑁) ∧ ∃𝑦 ∈ (1...𝑁)𝐽 = {0, 𝑦})))
4		c0ex 11106	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
5	4	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → 0 ∈ V)
6		prssg 4768	. . . . . . . . . . . 12 ⊢ ((0 ∈ V ∧ 𝑦 ∈ (1...𝑁)) → ((0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁)) ↔ {0, 𝑦} ⊆ (0...𝑁)))
7	5, 6	sylan 580	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ((0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁)) ↔ {0, 𝑦} ⊆ (0...𝑁)))
8		f1ocnv 6775	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → ^◡𝐹:𝑊–1-1-onto→𝐶)
9		f1ofn 6764	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡𝐹:𝑊–1-1-onto→𝐶 → ^◡𝐹 Fn 𝑊)
10		isubgr3stgr.w	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑊 = (Vtx‘𝑆)
11		isubgr3stgr.s	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑆 = (StarGr‘𝑁)
12	11	fveq2i 6825	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Vtx‘𝑆) = (Vtx‘(StarGr‘𝑁))
13		stgrvtx 48053	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → (Vtx‘(StarGr‘𝑁)) = (0...𝑁))
14	1, 13	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Vtx‘(StarGr‘𝑁)) = (0...𝑁)
15	10, 12, 14	3eqtri 2758	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑊 = (0...𝑁)
16	15	fneq2i 6579	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡𝐹 Fn 𝑊 ↔ ^◡𝐹 Fn (0...𝑁))
17	9, 16	sylib 218	. . . . . . . . . . . . . . . . 17 ⊢ (^◡𝐹:𝑊–1-1-onto→𝐶 → ^◡𝐹 Fn (0...𝑁))
18	8, 17	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → ^◡𝐹 Fn (0...𝑁))
19	18	ad2antrl 728	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → ^◡𝐹 Fn (0...𝑁))
20	19	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ^◡𝐹 Fn (0...𝑁))
21	20	anim1i 615	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁))) → (^◡𝐹 Fn (0...𝑁) ∧ (0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁))))
22		3anass 1094	. . . . . . . . . . . . 13 ⊢ ((^◡𝐹 Fn (0...𝑁) ∧ 0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁)) ↔ (^◡𝐹 Fn (0...𝑁) ∧ (0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁))))
23	21, 22	sylibr 234	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁))) → (^◡𝐹 Fn (0...𝑁) ∧ 0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁)))
24	23	ex 412	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ((0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁)) → (^◡𝐹 Fn (0...𝑁) ∧ 0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁))))
25	7, 24	sylbird 260	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ({0, 𝑦} ⊆ (0...𝑁) → (^◡𝐹 Fn (0...𝑁) ∧ 0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁))))
26	25	imp 406	. . . . . . . . 9 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ {0, 𝑦} ⊆ (0...𝑁)) → (^◡𝐹 Fn (0...𝑁) ∧ 0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁)))
27		fnimapr 6905	. . . . . . . . 9 ⊢ ((^◡𝐹 Fn (0...𝑁) ∧ 0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁)) → (^◡𝐹 “ {0, 𝑦}) = {(^◡𝐹‘0), (^◡𝐹‘𝑦)})
28	26, 27	syl 17	. . . . . . . 8 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ {0, 𝑦} ⊆ (0...𝑁)) → (^◡𝐹 “ {0, 𝑦}) = {(^◡𝐹‘0), (^◡𝐹‘𝑦)})
29		isubgr3stgr.v	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑉 = (Vtx‘𝐺)
30	29	clnbgrvtxel 47928	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (𝐺 ClNeighbVtx 𝑋))
31	30	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝐺 ClNeighbVtx 𝑋))
32		isubgr3stgr.c	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)
33	31, 32	eleqtrrdi 2842	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐶)
34		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) → 𝐹:𝐶–1-1-onto→𝑊)
35	33, 34	anim12ci 614	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → (𝐹:𝐶–1-1-onto→𝑊 ∧ 𝑋 ∈ 𝐶))
36		simprr 772	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → (𝐹‘𝑋) = 0)
37	35, 36	jca 511	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → ((𝐹:𝐶–1-1-onto→𝑊 ∧ 𝑋 ∈ 𝐶) ∧ (𝐹‘𝑋) = 0))
38	37	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ((𝐹:𝐶–1-1-onto→𝑊 ∧ 𝑋 ∈ 𝐶) ∧ (𝐹‘𝑋) = 0))
39		f1ocnvfv 7212	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝐶–1-1-onto→𝑊 ∧ 𝑋 ∈ 𝐶) → ((𝐹‘𝑋) = 0 → (^◡𝐹‘0) = 𝑋))
40	39	imp 406	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ 𝑋 ∈ 𝐶) ∧ (𝐹‘𝑋) = 0) → (^◡𝐹‘0) = 𝑋)
41	38, 40	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (^◡𝐹‘0) = 𝑋)
42	30, 32	eleqtrrdi 2842	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝐶)
43	42	ad3antlr 731	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → 𝑋 ∈ 𝐶)
44		f1of 6763	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡𝐹:𝑊–1-1-onto→𝐶 → ^◡𝐹:𝑊⟶𝐶)
45	8, 44	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → ^◡𝐹:𝑊⟶𝐶)
46	45	ad2antrl 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → ^◡𝐹:𝑊⟶𝐶)
47	46	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ^◡𝐹:𝑊⟶𝐶)
48		fz1ssfz0 13523	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑁) ⊆ (0...𝑁)
49	48	sseli 3925	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ (0...𝑁))
50	49, 15	eleqtrrdi 2842	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ 𝑊)
51	50	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → 𝑦 ∈ 𝑊)
52	47, 51	ffvelcdmd 7018	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (^◡𝐹‘𝑦) ∈ 𝐶)
53	43, 52	jca 511	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (𝑋 ∈ 𝐶 ∧ (^◡𝐹‘𝑦) ∈ 𝐶))
54	32	eleq2i 2823	. . . . . . . . . . . . . . . . 17 ⊢ ((^◡𝐹‘𝑦) ∈ 𝐶 ↔ (^◡𝐹‘𝑦) ∈ (𝐺 ClNeighbVtx 𝑋))
55		usgrupgr 29163	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)
56	55	anim1i 615	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉))
57	56	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉))
58	29	clnbgrssvtx 47930	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉
59	32, 58	eqsstri 3976	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐶 ⊆ 𝑉
60	59, 52	sselid 3927	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (^◡𝐹‘𝑦) ∈ 𝑉)
61		df-3an 1088	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉 ∧ (^◡𝐹‘𝑦) ∈ 𝑉) ↔ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉) ∧ (^◡𝐹‘𝑦) ∈ 𝑉))
62	57, 60, 61	sylanbrc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉 ∧ (^◡𝐹‘𝑦) ∈ 𝑉))
63	62	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (^◡𝐹‘0) = 𝑋) ∧ 𝑋 ∈ 𝐶) → (𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉 ∧ (^◡𝐹‘𝑦) ∈ 𝑉))
64		isubgr3stgr.e	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐸 = (Edg‘𝐺)
65	29, 64	clnbupgrel 47933	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉 ∧ (^◡𝐹‘𝑦) ∈ 𝑉) → ((^◡𝐹‘𝑦) ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((^◡𝐹‘𝑦) = 𝑋 ∨ {(^◡𝐹‘𝑦), 𝑋} ∈ 𝐸)))
66	63, 65	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (^◡𝐹‘0) = 𝑋) ∧ 𝑋 ∈ 𝐶) → ((^◡𝐹‘𝑦) ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((^◡𝐹‘𝑦) = 𝑋 ∨ {(^◡𝐹‘𝑦), 𝑋} ∈ 𝐸)))
67	54, 66	bitrid 283	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (^◡𝐹‘0) = 𝑋) ∧ 𝑋 ∈ 𝐶) → ((^◡𝐹‘𝑦) ∈ 𝐶 ↔ ((^◡𝐹‘𝑦) = 𝑋 ∨ {(^◡𝐹‘𝑦), 𝑋} ∈ 𝐸)))
68		eqeq2 2743	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((^◡𝐹‘0) = 𝑋 → ((^◡𝐹‘𝑦) = (^◡𝐹‘0) ↔ (^◡𝐹‘𝑦) = 𝑋))
69	68	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (^◡𝐹‘0) = 𝑋) → ((^◡𝐹‘𝑦) = (^◡𝐹‘0) ↔ (^◡𝐹‘𝑦) = 𝑋))
70		f1of1 6762	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (^◡𝐹:𝑊–1-1-onto→𝐶 → ^◡𝐹:𝑊–1-1→𝐶)
71	8, 70	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → ^◡𝐹:𝑊–1-1→𝐶)
72	71	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → ^◡𝐹:𝑊–1-1→𝐶)
73		0elfz 13524	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ₀ → 0 ∈ (0...𝑁))
74	1, 73	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ (0...𝑁)
75	74, 15	eleqtrri 2830	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ 𝑊
76	50, 75	jctir 520	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (1...𝑁) → (𝑦 ∈ 𝑊 ∧ 0 ∈ 𝑊))
77		f1veqaeq 7190	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((^◡𝐹:𝑊–1-1→𝐶 ∧ (𝑦 ∈ 𝑊 ∧ 0 ∈ 𝑊)) → ((^◡𝐹‘𝑦) = (^◡𝐹‘0) → 𝑦 = 0))
78	72, 76, 77	syl2an 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ((^◡𝐹‘𝑦) = (^◡𝐹‘0) → 𝑦 = 0))
79		elfznn 13453	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
80		nnne0 12159	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0)
81		eqneqall 2939	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 0 → (𝑦 ≠ 0 → {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸))
82	80, 81	syl5com 31	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℕ → (𝑦 = 0 → {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸))
83	79, 82	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (1...𝑁) → (𝑦 = 0 → {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸))
84	83	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (𝑦 = 0 → {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸))
85	78, 84	syld 47	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ((^◡𝐹‘𝑦) = (^◡𝐹‘0) → {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸))
86	85	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (^◡𝐹‘0) = 𝑋) → ((^◡𝐹‘𝑦) = (^◡𝐹‘0) → {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸))
87	69, 86	sylbird 260	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (^◡𝐹‘0) = 𝑋) → ((^◡𝐹‘𝑦) = 𝑋 → {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸))
88	87	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (^◡𝐹‘0) = 𝑋) ∧ 𝑋 ∈ 𝐶) → ((^◡𝐹‘𝑦) = 𝑋 → {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸))
89		prcom 4682	. . . . . . . . . . . . . . . . . . . 20 ⊢ {(^◡𝐹‘𝑦), 𝑋} = {𝑋, (^◡𝐹‘𝑦)}
90	89	eleq1i 2822	. . . . . . . . . . . . . . . . . . 19 ⊢ ({(^◡𝐹‘𝑦), 𝑋} ∈ 𝐸 ↔ {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸)
91	90	biimpi 216	. . . . . . . . . . . . . . . . . 18 ⊢ ({(^◡𝐹‘𝑦), 𝑋} ∈ 𝐸 → {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸)
92	91	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (^◡𝐹‘0) = 𝑋) ∧ 𝑋 ∈ 𝐶) → ({(^◡𝐹‘𝑦), 𝑋} ∈ 𝐸 → {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸))
93	88, 92	jaod 859	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (^◡𝐹‘0) = 𝑋) ∧ 𝑋 ∈ 𝐶) → (((^◡𝐹‘𝑦) = 𝑋 ∨ {(^◡𝐹‘𝑦), 𝑋} ∈ 𝐸) → {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸))
94	67, 93	sylbid 240	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (^◡𝐹‘0) = 𝑋) ∧ 𝑋 ∈ 𝐶) → ((^◡𝐹‘𝑦) ∈ 𝐶 → {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸))
95	94	impr 454	. . . . . . . . . . . . . 14 ⊢ ((((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (^◡𝐹‘0) = 𝑋) ∧ (𝑋 ∈ 𝐶 ∧ (^◡𝐹‘𝑦) ∈ 𝐶)) → {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸)
96		prssi 4770	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝐶 ∧ (^◡𝐹‘𝑦) ∈ 𝐶) → {𝑋, (^◡𝐹‘𝑦)} ⊆ 𝐶)
97	96	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (^◡𝐹‘0) = 𝑋) ∧ (𝑋 ∈ 𝐶 ∧ (^◡𝐹‘𝑦) ∈ 𝐶)) → {𝑋, (^◡𝐹‘𝑦)} ⊆ 𝐶)
98	95, 97	jca 511	. . . . . . . . . . . . 13 ⊢ ((((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (^◡𝐹‘0) = 𝑋) ∧ (𝑋 ∈ 𝐶 ∧ (^◡𝐹‘𝑦) ∈ 𝐶)) → ({𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸 ∧ {𝑋, (^◡𝐹‘𝑦)} ⊆ 𝐶))
99	53, 98	mpidan 689	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (^◡𝐹‘0) = 𝑋) → ({𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸 ∧ {𝑋, (^◡𝐹‘𝑦)} ⊆ 𝐶))
100		preq1 4683	. . . . . . . . . . . . . . 15 ⊢ ((^◡𝐹‘0) = 𝑋 → {(^◡𝐹‘0), (^◡𝐹‘𝑦)} = {𝑋, (^◡𝐹‘𝑦)})
101	100	eleq1d 2816	. . . . . . . . . . . . . 14 ⊢ ((^◡𝐹‘0) = 𝑋 → ({(^◡𝐹‘0), (^◡𝐹‘𝑦)} ∈ 𝐸 ↔ {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸))
102	100	sseq1d 3961	. . . . . . . . . . . . . 14 ⊢ ((^◡𝐹‘0) = 𝑋 → ({(^◡𝐹‘0), (^◡𝐹‘𝑦)} ⊆ 𝐶 ↔ {𝑋, (^◡𝐹‘𝑦)} ⊆ 𝐶))
103	101, 102	anbi12d 632	. . . . . . . . . . . . 13 ⊢ ((^◡𝐹‘0) = 𝑋 → (({(^◡𝐹‘0), (^◡𝐹‘𝑦)} ∈ 𝐸 ∧ {(^◡𝐹‘0), (^◡𝐹‘𝑦)} ⊆ 𝐶) ↔ ({𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸 ∧ {𝑋, (^◡𝐹‘𝑦)} ⊆ 𝐶)))
104	103	adantl 481	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (^◡𝐹‘0) = 𝑋) → (({(^◡𝐹‘0), (^◡𝐹‘𝑦)} ∈ 𝐸 ∧ {(^◡𝐹‘0), (^◡𝐹‘𝑦)} ⊆ 𝐶) ↔ ({𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸 ∧ {𝑋, (^◡𝐹‘𝑦)} ⊆ 𝐶)))
105	99, 104	mpbird 257	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (^◡𝐹‘0) = 𝑋) → ({(^◡𝐹‘0), (^◡𝐹‘𝑦)} ∈ 𝐸 ∧ {(^◡𝐹‘0), (^◡𝐹‘𝑦)} ⊆ 𝐶))
106	41, 105	mpdan 687	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ({(^◡𝐹‘0), (^◡𝐹‘𝑦)} ∈ 𝐸 ∧ {(^◡𝐹‘0), (^◡𝐹‘𝑦)} ⊆ 𝐶))
107	106	adantr 480	. . . . . . . . 9 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ {0, 𝑦} ⊆ (0...𝑁)) → ({(^◡𝐹‘0), (^◡𝐹‘𝑦)} ∈ 𝐸 ∧ {(^◡𝐹‘0), (^◡𝐹‘𝑦)} ⊆ 𝐶))
108		usgruhgr 29164	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)
109	108	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → 𝐺 ∈ UHGraph)
110	59	a1i 11	. . . . . . . . . 10 ⊢ ({0, 𝑦} ⊆ (0...𝑁) → 𝐶 ⊆ 𝑉)
111		eqid 2731	. . . . . . . . . . 11 ⊢ (𝐺 ISubGr 𝐶) = (𝐺 ISubGr 𝐶)
112		isubgr3stgr.i	. . . . . . . . . . 11 ⊢ 𝐼 = (Edg‘(𝐺 ISubGr 𝐶))
113	29, 64, 111, 112	isubgredg 47965	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐶 ⊆ 𝑉) → ({(^◡𝐹‘0), (^◡𝐹‘𝑦)} ∈ 𝐼 ↔ ({(^◡𝐹‘0), (^◡𝐹‘𝑦)} ∈ 𝐸 ∧ {(^◡𝐹‘0), (^◡𝐹‘𝑦)} ⊆ 𝐶)))
114	109, 110, 113	syl2an 596	. . . . . . . . 9 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ {0, 𝑦} ⊆ (0...𝑁)) → ({(^◡𝐹‘0), (^◡𝐹‘𝑦)} ∈ 𝐼 ↔ ({(^◡𝐹‘0), (^◡𝐹‘𝑦)} ∈ 𝐸 ∧ {(^◡𝐹‘0), (^◡𝐹‘𝑦)} ⊆ 𝐶)))
115	107, 114	mpbird 257	. . . . . . . 8 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ {0, 𝑦} ⊆ (0...𝑁)) → {(^◡𝐹‘0), (^◡𝐹‘𝑦)} ∈ 𝐼)
116	28, 115	eqeltrd 2831	. . . . . . 7 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ {0, 𝑦} ⊆ (0...𝑁)) → (^◡𝐹 “ {0, 𝑦}) ∈ 𝐼)
117	116	ex 412	. . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ({0, 𝑦} ⊆ (0...𝑁) → (^◡𝐹 “ {0, 𝑦}) ∈ 𝐼))
118		sseq1 3955	. . . . . . 7 ⊢ (𝐽 = {0, 𝑦} → (𝐽 ⊆ (0...𝑁) ↔ {0, 𝑦} ⊆ (0...𝑁)))
119		imaeq2 6004	. . . . . . . 8 ⊢ (𝐽 = {0, 𝑦} → (^◡𝐹 “ 𝐽) = (^◡𝐹 “ {0, 𝑦}))
120	119	eleq1d 2816	. . . . . . 7 ⊢ (𝐽 = {0, 𝑦} → ((^◡𝐹 “ 𝐽) ∈ 𝐼 ↔ (^◡𝐹 “ {0, 𝑦}) ∈ 𝐼))
121	118, 120	imbi12d 344	. . . . . 6 ⊢ (𝐽 = {0, 𝑦} → ((𝐽 ⊆ (0...𝑁) → (^◡𝐹 “ 𝐽) ∈ 𝐼) ↔ ({0, 𝑦} ⊆ (0...𝑁) → (^◡𝐹 “ {0, 𝑦}) ∈ 𝐼)))
122	117, 121	syl5ibrcom 247	. . . . 5 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (𝐽 = {0, 𝑦} → (𝐽 ⊆ (0...𝑁) → (^◡𝐹 “ 𝐽) ∈ 𝐼)))
123	122	rexlimdva 3133	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → (∃𝑦 ∈ (1...𝑁)𝐽 = {0, 𝑦} → (𝐽 ⊆ (0...𝑁) → (^◡𝐹 “ 𝐽) ∈ 𝐼)))
124	123	impcomd 411	. . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → ((𝐽 ⊆ (0...𝑁) ∧ ∃𝑦 ∈ (1...𝑁)𝐽 = {0, 𝑦}) → (^◡𝐹 “ 𝐽) ∈ 𝐼))
125	3, 124	sylbid 240	. 2 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → (𝐽 ∈ (Edg‘(StarGr‘𝑁)) → (^◡𝐹 “ 𝐽) ∈ 𝐼))
126	125	3impia 1117	1 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ 𝐽 ∈ (Edg‘(StarGr‘𝑁))) → (^◡𝐹 “ 𝐽) ∈ 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∃wrex 3056 Vcvv 3436 ⊆ wss 3897 {cpr 4575 ↦ cmpt 5170 ^◡ccnv 5613 “ cima 5617 Fn wfn 6476 ⟶wf 6477 –1-1→wf1 6478 –1-1-onto→wf1o 6480 ‘cfv 6481 (class class class)co 7346 0cc0 11006 1c1 11007 ℕcn 12125 ℕ₀cn0 12381 ...cfz 13407 Vtxcvtx 28974 Edgcedg 29025 UHGraphcuhgr 29034 UPGraphcupgr 29058 USGraphcusgr 29127 NeighbVtx cnbgr 29310 ClNeighbVtx cclnbgr 47917 ISubGr cisubgr 47959 StarGrcstgr 48050

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-dju 9794 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-xnn0 12455 df-z 12469 df-dec 12589 df-uz 12733 df-fz 13408 df-hash 14238 df-struct 17058 df-slot 17093 df-ndx 17105 df-base 17121 df-edgf 28967 df-vtx 28976 df-iedg 28977 df-edg 29026 df-uhgr 29036 df-upgr 29060 df-uspgr 29128 df-usgr 29129 df-nbgr 29311 df-clnbgr 47918 df-isubgr 47960 df-stgr 48051

This theorem is referenced by: isubgr3stgrlem8 48072

W3C validator