isubgr3stgrlem7 - Mathbox for Alexander van der Vekens

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Theorem isubgr3stgrlem7 48477

Description: Lemma 7 for isubgr3stgr 48480. (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 48462	. . . 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 11133	. . . . . . . . . . . . 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 4753	. . . . . . . . . . . 12 ⊢ ((0 ∈ V ∧ 𝑦 ∈ (1...𝑁)) → ((0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁)) ↔ {0, 𝑦} ⊆ (0...𝑁)))
7	5, 6	sylan 587	. . . . . . . . . . 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 6783	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → ^◡𝐹:𝑊–1-1-onto→𝐶)
9		f1ofn 6772	. . . . . . . . . . . . . . . . . 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 6834	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Vtx‘𝑆) = (Vtx‘(StarGr‘𝑁))
13		stgrvtx 48459	. . . . . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑊 = (0...𝑁)
16	15	fneq2i 6587	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡𝐹 Fn 𝑊 ↔ ^◡𝐹 Fn (0...𝑁))
17	9, 16	sylib 220	. . . . . . . . . . . . . . . . 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 735	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → ^◡𝐹 Fn (0...𝑁))
20	19	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ^◡𝐹 Fn (0...𝑁))
21	20	anim1i 622	. . . . . . . . . . . . 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 1101	. . . . . . . . . . . . 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 236	. . . . . . . . . . . 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 414	. . . . . . . . . . 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 262	. . . . . . . . . 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 408	. . . . . . . . 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 6914	. . . . . . . . 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 48334	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (𝐺 ClNeighbVtx 𝑋))
31	30	adantl 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝐺 ClNeighbVtx 𝑋))
32		isubgr3stgr.c	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)
33	31, 32	eleqtrrdi 2852	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐶)
34		simpl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) → 𝐹:𝐶–1-1-onto→𝑊)
35	33, 34	anim12ci 621	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → (𝐹:𝐶–1-1-onto→𝑊 ∧ 𝑋 ∈ 𝐶))
36		simprr 779	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → (𝐹‘𝑋) = 0)
37	35, 36	jca 517	. . . . . . . . . . . . 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 482	. . . . . . . . . . . 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 7226	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝐶–1-1-onto→𝑊 ∧ 𝑋 ∈ 𝐶) → ((𝐹‘𝑋) = 0 → (^◡𝐹‘0) = 𝑋))
40	39	imp 408	. . . . . . . . . . . 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 2852	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝐶)
43	42	ad3antlr 738	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → 𝑋 ∈ 𝐶)
44		f1of 6771	. . . . . . . . . . . . . . . . . 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 735	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → ^◡𝐹:𝑊⟶𝐶)
47	46	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ^◡𝐹:𝑊⟶𝐶)
48		fz1ssfz0 13572	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑁) ⊆ (0...𝑁)
49	48	sseli 3913	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ (0...𝑁))
50	49, 15	eleqtrrdi 2852	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ 𝑊)
51	50	adantl 483	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → 𝑦 ∈ 𝑊)
52	47, 51	ffvelcdmd 7030	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (^◡𝐹‘𝑦) ∈ 𝐶)
53	43, 52	jca 517	. . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . . . . 17 ⊢ ((^◡𝐹‘𝑦) ∈ 𝐶 ↔ (^◡𝐹‘𝑦) ∈ (𝐺 ClNeighbVtx 𝑋))
55		usgrupgr 29276	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)
56	55	anim1i 622	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉))
57	56	ad2antrr 733	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉))
58	29	clnbgrssvtx 48336	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉
59	32, 58	eqsstri 3963	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐶 ⊆ 𝑉
60	59, 52	sselid 3915	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (^◡𝐹‘𝑦) ∈ 𝑉)
61		df-3an 1095	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉 ∧ (^◡𝐹‘𝑦) ∈ 𝑉) ↔ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉) ∧ (^◡𝐹‘𝑦) ∈ 𝑉))
62	57, 60, 61	sylanbrc 590	. . . . . . . . . . . . . . . . . . 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 733	. . . . . . . . . . . . . . . . . 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 48339	. . . . . . . . . . . . . . . . . 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 285	. . . . . . . . . . . . . . . 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 2753	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((^◡𝐹‘0) = 𝑋 → ((^◡𝐹‘𝑦) = (^◡𝐹‘0) ↔ (^◡𝐹‘𝑦) = 𝑋))
69	68	adantl 483	. . . . . . . . . . . . . . . . . . 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 6770	. . . . . . . . . . . . . . . . . . . . . . . 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 735	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → ^◡𝐹:𝑊–1-1→𝐶)
73		0elfz 13573	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ₀ → 0 ∈ (0...𝑁))
74	1, 73	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ (0...𝑁)
75	74, 15	eleqtrri 2840	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ 𝑊
76	50, 75	jctir 526	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (1...𝑁) → (𝑦 ∈ 𝑊 ∧ 0 ∈ 𝑊))
77		f1veqaeq 7204	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((^◡𝐹:𝑊–1-1→𝐶 ∧ (𝑦 ∈ 𝑊 ∧ 0 ∈ 𝑊)) → ((^◡𝐹‘𝑦) = (^◡𝐹‘0) → 𝑦 = 0))
78	72, 76, 77	syl2an 603	. . . . . . . . . . . . . . . . . . . . 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 13502	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
80		nnne0 12206	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0)
81		eqneqall 2947	. . . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . 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 262	. . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . 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 4667	. . . . . . . . . . . . . . . . . . . 20 ⊢ {(^◡𝐹‘𝑦), 𝑋} = {𝑋, (^◡𝐹‘𝑦)}
90	89	eleq1i 2832	. . . . . . . . . . . . . . . . . . 19 ⊢ ({(^◡𝐹‘𝑦), 𝑋} ∈ 𝐸 ↔ {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸)
91	90	biimpi 218	. . . . . . . . . . . . . . . . . 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 866	. . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . 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 456	. . . . . . . . . . . . . 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 4755	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝐶 ∧ (^◡𝐹‘𝑦) ∈ 𝐶) → {𝑋, (^◡𝐹‘𝑦)} ⊆ 𝐶)
97	96	adantl 483	. . . . . . . . . . . . . 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 517	. . . . . . . . . . . . 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 696	. . . . . . . . . . . 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 4668	. . . . . . . . . . . . . . 15 ⊢ ((^◡𝐹‘0) = 𝑋 → {(^◡𝐹‘0), (^◡𝐹‘𝑦)} = {𝑋, (^◡𝐹‘𝑦)})
101	100	eleq1d 2826	. . . . . . . . . . . . . 14 ⊢ ((^◡𝐹‘0) = 𝑋 → ({(^◡𝐹‘0), (^◡𝐹‘𝑦)} ∈ 𝐸 ↔ {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸))
102	100	sseq1d 3948	. . . . . . . . . . . . . 14 ⊢ ((^◡𝐹‘0) = 𝑋 → ({(^◡𝐹‘0), (^◡𝐹‘𝑦)} ⊆ 𝐶 ↔ {𝑋, (^◡𝐹‘𝑦)} ⊆ 𝐶))
103	101, 102	anbi12d 639	. . . . . . . . . . . . 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 483	. . . . . . . . . . . 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 259	. . . . . . . . . . 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 694	. . . . . . . . . 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 482	. . . . . . . . 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 29277	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)
109	108	ad3antrrr 737	. . . . . . . . . 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 2741	. . . . . . . . . . 11 ⊢ (𝐺 ISubGr 𝐶) = (𝐺 ISubGr 𝐶)
112		isubgr3stgr.i	. . . . . . . . . . 11 ⊢ 𝐼 = (Edg‘(𝐺 ISubGr 𝐶))
113	29, 64, 111, 112	isubgredg 48371	. . . . . . . . . 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 603	. . . . . . . . 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 259	. . . . . . . 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 2841	. . . . . . 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 414	. . . . . 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 3942	. . . . . . 7 ⊢ (𝐽 = {0, 𝑦} → (𝐽 ⊆ (0...𝑁) ↔ {0, 𝑦} ⊆ (0...𝑁)))
119		imaeq2 6015	. . . . . . . 8 ⊢ (𝐽 = {0, 𝑦} → (^◡𝐹 “ 𝐽) = (^◡𝐹 “ {0, 𝑦}))
120	119	eleq1d 2826	. . . . . . 7 ⊢ (𝐽 = {0, 𝑦} → ((^◡𝐹 “ 𝐽) ∈ 𝐼 ↔ (^◡𝐹 “ {0, 𝑦}) ∈ 𝐼))
121	118, 120	imbi12d 346	. . . . . 6 ⊢ (𝐽 = {0, 𝑦} → ((𝐽 ⊆ (0...𝑁) → (^◡𝐹 “ 𝐽) ∈ 𝐼) ↔ ({0, 𝑦} ⊆ (0...𝑁) → (^◡𝐹 “ {0, 𝑦}) ∈ 𝐼)))
122	117, 121	syl5ibrcom 249	. . . . 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 3142	. . . 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 413	. . 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 242	. 2 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → (𝐽 ∈ (Edg‘(StarGr‘𝑁)) → (^◡𝐹 “ 𝐽) ∈ 𝐼))
126	125	3impia 1124	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 208 ∧ wa 397 ∨ wo 854 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∃wrex 3065 Vcvv 3433 ⊆ wss 3885 {cpr 4560 ↦ cmpt 5156 ^◡ccnv 5620 “ cima 5624 Fn wfn 6484 ⟶wf 6485 –1-1→wf1 6486 –1-1-onto→wf1o 6488 ‘cfv 6489 (class class class)co 7360 0cc0 11033 1c1 11034 ℕcn 12169 ℕ₀cn0 12432 ...cfz 13456 Vtxcvtx 29087 Edgcedg 29138 UHGraphcuhgr 29147 UPGraphcupgr 29171 USGraphcusgr 29240 NeighbVtx cnbgr 29423 ClNeighbVtx cclnbgr 48323 ISubGr cisubgr 48365 StarGrcstgr 48456

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-dju 9820 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-xnn0 12506 df-z 12520 df-dec 12640 df-uz 12784 df-fz 13457 df-hash 14288 df-struct 17112 df-slot 17147 df-ndx 17159 df-base 17175 df-edgf 29080 df-vtx 29089 df-iedg 29090 df-edg 29139 df-uhgr 29149 df-upgr 29173 df-uspgr 29241 df-usgr 29242 df-nbgr 29424 df-clnbgr 48324 df-isubgr 48366 df-stgr 48457

This theorem is referenced by: isubgr3stgrlem8 48478

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