isubgr3stgrlem7 - Mathbox for Alexander van der Vekens

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Theorem isubgr3stgrlem7 48661

Description: Lemma 7 for isubgr3stgr 48664. (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 48646	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (𝐽 ∈ (Edg‘(StarGr‘𝑁)) ↔ (𝐽 ⊆ (0...𝑁) ∧ ∃𝑦 ∈ (1...𝑁)𝐽 = {0, 𝑦})))
3	1, 2	mp1i 14	. . 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 11200	. . . . . . . . . . . . 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 4787	. . . . . . . . . . . 12 ⊢ ((0 ∈ V ∧ 𝑦 ∈ (1...𝑁)) → ((0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁)) ↔ {0, 𝑦} ⊆ (0...𝑁)))
7	5, 6	sylan 591	. . . . . . . . . . 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 6834	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → ^◡𝐹:𝑊–1-1-onto→𝐶)
9		f1ofn 6822	. . . . . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Vtx‘𝑆) = (Vtx‘(StarGr‘𝑁))
13		stgrvtx 48643	. . . . . . . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑊 = (0...𝑁)
16	15	fneq2i 6634	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡𝐹 Fn 𝑊 ↔ ^◡𝐹 Fn (0...𝑁))
17	9, 16	sylib 221	. . . . . . . . . . . . . . . . 17 ⊢ (^◡𝐹:𝑊–1-1-onto→𝐶 → ^◡𝐹 Fn (0...𝑁))
18	8, 17	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → ^◡𝐹 Fn (0...𝑁))
19	18	ad2antrl 740	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → ^◡𝐹 Fn (0...𝑁))
20	19	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ^◡𝐹 Fn (0...𝑁))
21	20	anim1i 626	. . . . . . . . . . . . 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 1109	. . . . . . . . . . . . 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 237	. . . . . . . . . . . 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 417	. . . . . . . . . . 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 263	. . . . . . . . . 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 411	. . . . . . . . 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 6965	. . . . . . . . 9 ⊢ ((^◡𝐹 Fn (0...𝑁) ∧ 0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁)) → (^◡𝐹 “ {0, 𝑦}) = {(^◡𝐹‘0), (^◡𝐹‘𝑦)})
28	26, 27	syl 18	. . . . . . . 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 48518	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (𝐺 ClNeighbVtx 𝑋))
31	30	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝐺 ClNeighbVtx 𝑋))
32		isubgr3stgr.c	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)
33	31, 32	eleqtrrdi 2880	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐶)
34		simpl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) → 𝐹:𝐶–1-1-onto→𝑊)
35	33, 34	anim12ci 625	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → (𝐹:𝐶–1-1-onto→𝑊 ∧ 𝑋 ∈ 𝐶))
36		simprr 784	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → (𝐹‘𝑋) = 0)
37	35, 36	jca 520	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 7277	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝐶–1-1-onto→𝑊 ∧ 𝑋 ∈ 𝐶) → ((𝐹‘𝑋) = 0 → (^◡𝐹‘0) = 𝑋))
40	39	imp 411	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ 𝑋 ∈ 𝐶) ∧ (𝐹‘𝑋) = 0) → (^◡𝐹‘0) = 𝑋)
41	38, 40	syl 18	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (^◡𝐹‘0) = 𝑋)
42	30, 32	eleqtrrdi 2880	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝐶)
43	42	ad3antlr 743	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → 𝑋 ∈ 𝐶)
44		f1of 6821	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡𝐹:𝑊–1-1-onto→𝐶 → ^◡𝐹:𝑊⟶𝐶)
45	8, 44	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → ^◡𝐹:𝑊⟶𝐶)
46	45	ad2antrl 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → ^◡𝐹:𝑊⟶𝐶)
47	46	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ^◡𝐹:𝑊⟶𝐶)
48		fz1ssfz0 13651	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑁) ⊆ (0...𝑁)
49	48	sseli 3939	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ (0...𝑁))
50	49, 15	eleqtrrdi 2880	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ 𝑊)
51	50	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → 𝑦 ∈ 𝑊)
52	47, 51	ffvelcdmd 7081	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (^◡𝐹‘𝑦) ∈ 𝐶)
53	43, 52	jca 520	. . . . . . . . . . . . 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 2861	. . . . . . . . . . . . . . . . 17 ⊢ ((^◡𝐹‘𝑦) ∈ 𝐶 ↔ (^◡𝐹‘𝑦) ∈ (𝐺 ClNeighbVtx 𝑋))
55		usgrupgr 29476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)
56	55	anim1i 626	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉))
57	56	ad2antrr 738	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉))
58	29	clnbgrssvtx 48520	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉
59	32, 58	eqsstri 3989	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐶 ⊆ 𝑉
60	59, 52	sselid 3941	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (^◡𝐹‘𝑦) ∈ 𝑉)
61		df-3an 1103	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉 ∧ (^◡𝐹‘𝑦) ∈ 𝑉) ↔ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉) ∧ (^◡𝐹‘𝑦) ∈ 𝑉))
62	57, 60, 61	sylanbrc 594	. . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . 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 48523	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉 ∧ (^◡𝐹‘𝑦) ∈ 𝑉) → ((^◡𝐹‘𝑦) ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((^◡𝐹‘𝑦) = 𝑋 ∨ {(^◡𝐹‘𝑦), 𝑋} ∈ 𝐸)))
66	63, 65	syl 18	. . . . . . . . . . . . . . . . 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 286	. . . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((^◡𝐹‘0) = 𝑋 → ((^◡𝐹‘𝑦) = (^◡𝐹‘0) ↔ (^◡𝐹‘𝑦) = 𝑋))
69	68	adantl 486	. . . . . . . . . . . . . . . . . . 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 6820	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (^◡𝐹:𝑊–1-1-onto→𝐶 → ^◡𝐹:𝑊–1-1→𝐶)
71	8, 70	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → ^◡𝐹:𝑊–1-1→𝐶)
72	71	ad2antrl 740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → ^◡𝐹:𝑊–1-1→𝐶)
73		0elfz 13652	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ₀ → 0 ∈ (0...𝑁))
74	1, 73	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ (0...𝑁)
75	74, 15	eleqtrri 2868	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ 𝑊
76	50, 75	jctir 529	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (1...𝑁) → (𝑦 ∈ 𝑊 ∧ 0 ∈ 𝑊))
77		f1veqaeq 7255	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((^◡𝐹:𝑊–1-1→𝐶 ∧ (𝑦 ∈ 𝑊 ∧ 0 ∈ 𝑊)) → ((^◡𝐹‘𝑦) = (^◡𝐹‘0) → 𝑦 = 0))
78	72, 76, 77	syl2an 607	. . . . . . . . . . . . . . . . . . . . 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 13581	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
80		nnne0 12270	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0)
81		eqneqall 2975	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 0 → (𝑦 ≠ 0 → {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸))
82	80, 81	syl5com 32	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℕ → (𝑦 = 0 → {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸))
83	79, 82	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (1...𝑁) → (𝑦 = 0 → {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸))
84	83	adantl 486	. . . . . . . . . . . . . . . . . . . . 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 48	. . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . 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 263	. . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . 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 4701	. . . . . . . . . . . . . . . . . . . 20 ⊢ {(^◡𝐹‘𝑦), 𝑋} = {𝑋, (^◡𝐹‘𝑦)}
90	89	eleq1i 2860	. . . . . . . . . . . . . . . . . . 19 ⊢ ({(^◡𝐹‘𝑦), 𝑋} ∈ 𝐸 ↔ {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸)
91	90	biimpi 219	. . . . . . . . . . . . . . . . . 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 872	. . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . 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 4789	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝐶 ∧ (^◡𝐹‘𝑦) ∈ 𝐶) → {𝑋, (^◡𝐹‘𝑦)} ⊆ 𝐶)
97	96	adantl 486	. . . . . . . . . . . . . 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 520	. . . . . . . . . . . . 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 701	. . . . . . . . . . . 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 4702	. . . . . . . . . . . . . . 15 ⊢ ((^◡𝐹‘0) = 𝑋 → {(^◡𝐹‘0), (^◡𝐹‘𝑦)} = {𝑋, (^◡𝐹‘𝑦)})
101	100	eleq1d 2854	. . . . . . . . . . . . . 14 ⊢ ((^◡𝐹‘0) = 𝑋 → ({(^◡𝐹‘0), (^◡𝐹‘𝑦)} ∈ 𝐸 ↔ {𝑋, (^◡𝐹‘𝑦)} ∈ 𝐸))
102	100	sseq1d 3974	. . . . . . . . . . . . . 14 ⊢ ((^◡𝐹‘0) = 𝑋 → ({(^◡𝐹‘0), (^◡𝐹‘𝑦)} ⊆ 𝐶 ↔ {𝑋, (^◡𝐹‘𝑦)} ⊆ 𝐶))
103	101, 102	anbi12d 643	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 260	. . . . . . . . . . 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 699	. . . . . . . . . 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 485	. . . . . . . . 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 29477	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)
109	108	ad3antrrr 742	. . . . . . . . . 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 2769	. . . . . . . . . . 11 ⊢ (𝐺 ISubGr 𝐶) = (𝐺 ISubGr 𝐶)
112		isubgr3stgr.i	. . . . . . . . . . 11 ⊢ 𝐼 = (Edg‘(𝐺 ISubGr 𝐶))
113	29, 64, 111, 112	isubgredg 48555	. . . . . . . . . 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 607	. . . . . . . . 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 260	. . . . . . . 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 2869	. . . . . . 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 417	. . . . . 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 3968	. . . . . . 7 ⊢ (𝐽 = {0, 𝑦} → (𝐽 ⊆ (0...𝑁) ↔ {0, 𝑦} ⊆ (0...𝑁)))
119		imaeq2 6059	. . . . . . . 8 ⊢ (𝐽 = {0, 𝑦} → (^◡𝐹 “ 𝐽) = (^◡𝐹 “ {0, 𝑦}))
120	119	eleq1d 2854	. . . . . . 7 ⊢ (𝐽 = {0, 𝑦} → ((^◡𝐹 “ 𝐽) ∈ 𝐼 ↔ (^◡𝐹 “ {0, 𝑦}) ∈ 𝐼))
121	118, 120	imbi12d 347	. . . . . 6 ⊢ (𝐽 = {0, 𝑦} → ((𝐽 ⊆ (0...𝑁) → (^◡𝐹 “ 𝐽) ∈ 𝐼) ↔ ({0, 𝑦} ⊆ (0...𝑁) → (^◡𝐹 “ {0, 𝑦}) ∈ 𝐼)))
122	117, 121	syl5ibrcom 250	. . . . 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 3172	. . . 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 416	. . 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 243	. 2 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → (𝐽 ∈ (Edg‘(StarGr‘𝑁)) → (^◡𝐹 “ 𝐽) ∈ 𝐼))
126	125	3impia 1133	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 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∃wrex 3095 Vcvv 3461 ⊆ wss 3911 {cpr 4594 ↦ cmpt 5194 ^◡ccnv 5661 “ cima 5665 Fn wfn 6532 ⟶wf 6533 –1-1→wf1 6534 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7411 0cc0 11100 1c1 11101 ℕcn 12233 ℕ₀cn0 12504 ...cfz 13535 Vtxcvtx 29287 Edgcedg 29338 UHGraphcuhgr 29347 UPGraphcupgr 29371 USGraphcusgr 29440 NeighbVtx cnbgr 29623 ClNeighbVtx cclnbgr 48507 ISubGr cisubgr 48549 StarGrcstgr 48640

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-reu 3376 df-rab 3423 df-v 3463 df-sbc 3752 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-oadd 8457 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-dju 9887 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-xnn0 12578 df-z 12592 df-dec 12712 df-uz 12863 df-fz 13536 df-hash 14367 df-struct 17207 df-slot 17242 df-ndx 17254 df-base 17270 df-edgf 29280 df-vtx 29289 df-iedg 29290 df-edg 29339 df-uhgr 29349 df-upgr 29373 df-uspgr 29441 df-usgr 29442 df-nbgr 29624 df-clnbgr 48508 df-isubgr 48550 df-stgr 48641

This theorem is referenced by: isubgr3stgrlem8 48662

W3C validator