Theorem nbgraf1olem5 25974

Description: Lemma 5 for nbgraf1o 25976. The mapping of neighbors to edge indices is a one-to-one onto function. (Contributed by Alexander van der Vekens, 19-Dec-2017.)

Hypotheses

Ref	Expression
nbgraf1o.n	⊢ 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)
nbgraf1o.i	⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑖)}
nbgraf1o.f	⊢ 𝐹 = (𝑛 ∈ 𝑁 ↦ (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛}))

Assertion

Ref	Expression
nbgraf1olem5	⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → 𝐹:𝑁–1-1-onto→𝐼)

Distinct variable groups:   𝑖,𝐸,𝑛   𝑈,𝑖,𝑛   𝑖,𝑉,𝑛   𝑖,𝐼,𝑛   𝑛,𝑁

Allowed substitution hints:   𝐹(𝑖,𝑛)   𝑁(𝑖)

Proof of Theorem nbgraf1olem5

Dummy variables 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nbgraf1o.n	. . . 4 ⊢ 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)
2		nbgraf1o.i	. . . 4 ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑖)}
3		nbgraf1o.f	. . . 4 ⊢ 𝐹 = (𝑛 ∈ 𝑁 ↦ (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛}))
4	1, 2, 3	nbgraf1olem2 25971	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → 𝐹:𝑁⟶𝐼)
5		ffn 5958	. . 3 ⊢ (𝐹:𝑁⟶𝐼 → 𝐹 Fn 𝑁)
6	4, 5	syl 17	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → 𝐹 Fn 𝑁)
7	1, 2, 3	nbgraf1olem3 25972	. . . . . . . . . . 11 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉 ∧ 𝑛 ∈ 𝑁) → (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛}) = (◡𝐸‘{𝑈, 𝑛}))
8	7	3expa 1257	. . . . . . . . . 10 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑛 ∈ 𝑁) → (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛}) = (◡𝐸‘{𝑈, 𝑛}))
9	8	eqeq2d 2620	. . . . . . . . 9 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑛 ∈ 𝑁) → (𝑗 = (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛}) ↔ 𝑗 = (◡𝐸‘{𝑈, 𝑛})))
10		usgraf1o 25887	. . . . . . . . . . . . . 14 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1-onto→ran 𝐸)
11	10	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → 𝐸:dom 𝐸–1-1-onto→ran 𝐸)
12	11	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑛 ∈ 𝑁) → 𝐸:dom 𝐸–1-1-onto→ran 𝐸)
13		nbgraeledg 25959	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑉 USGrph 𝐸 → (𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ↔ {𝑛, 𝑈} ∈ ran 𝐸))
14		prcom 4211	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑛, 𝑈} = {𝑈, 𝑛}
15	14	eleq1i 2679	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑛, 𝑈} ∈ ran 𝐸 ↔ {𝑈, 𝑛} ∈ ran 𝐸)
16	13, 15	syl6bb 275	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 USGrph 𝐸 → (𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ↔ {𝑈, 𝑛} ∈ ran 𝐸))
17	16	biimpcd 238	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) → (𝑉 USGrph 𝐸 → {𝑈, 𝑛} ∈ ran 𝐸))
18	17	a1d 25	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) → (𝑈 ∈ 𝑉 → (𝑉 USGrph 𝐸 → {𝑈, 𝑛} ∈ ran 𝐸)))
19	18, 1	eleq2s 2706	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑁 → (𝑈 ∈ 𝑉 → (𝑉 USGrph 𝐸 → {𝑈, 𝑛} ∈ ran 𝐸)))
20	19	com13 86	. . . . . . . . . . . . 13 ⊢ (𝑉 USGrph 𝐸 → (𝑈 ∈ 𝑉 → (𝑛 ∈ 𝑁 → {𝑈, 𝑛} ∈ ran 𝐸)))
21	20	imp31 447	. . . . . . . . . . . 12 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑛 ∈ 𝑁) → {𝑈, 𝑛} ∈ ran 𝐸)
22		f1ocnvdm 6440	. . . . . . . . . . . 12 ⊢ ((𝐸:dom 𝐸–1-1-onto→ran 𝐸 ∧ {𝑈, 𝑛} ∈ ran 𝐸) → (◡𝐸‘{𝑈, 𝑛}) ∈ dom 𝐸)
23	12, 21, 22	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑛 ∈ 𝑁) → (◡𝐸‘{𝑈, 𝑛}) ∈ dom 𝐸)
24		prid1g 4239	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ {𝑈, 𝑛})
25	24	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → 𝑈 ∈ {𝑈, 𝑛})
26	25	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑛 ∈ 𝑁) → 𝑈 ∈ {𝑈, 𝑛})
27	1	eleq2i 2680	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑁 ↔ 𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈))
28		nbgracnvfv 25969	. . . . . . . . . . . . . 14 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈)) → (𝐸‘(◡𝐸‘{𝑈, 𝑛})) = {𝑈, 𝑛})
29	27, 28	sylan2b 491	. . . . . . . . . . . . 13 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑛 ∈ 𝑁) → (𝐸‘(◡𝐸‘{𝑈, 𝑛})) = {𝑈, 𝑛})
30	29	adantlr 747	. . . . . . . . . . . 12 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑛 ∈ 𝑁) → (𝐸‘(◡𝐸‘{𝑈, 𝑛})) = {𝑈, 𝑛})
31	26, 30	eleqtrrd 2691	. . . . . . . . . . 11 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑛 ∈ 𝑁) → 𝑈 ∈ (𝐸‘(◡𝐸‘{𝑈, 𝑛})))
32	23, 31	jca 553	. . . . . . . . . 10 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑛 ∈ 𝑁) → ((◡𝐸‘{𝑈, 𝑛}) ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘(◡𝐸‘{𝑈, 𝑛}))))
33		eleq1 2676	. . . . . . . . . . 11 ⊢ (𝑗 = (◡𝐸‘{𝑈, 𝑛}) → (𝑗 ∈ dom 𝐸 ↔ (◡𝐸‘{𝑈, 𝑛}) ∈ dom 𝐸))
34		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑗 = (◡𝐸‘{𝑈, 𝑛}) → (𝐸‘𝑗) = (𝐸‘(◡𝐸‘{𝑈, 𝑛})))
35	34	eleq2d 2673	. . . . . . . . . . 11 ⊢ (𝑗 = (◡𝐸‘{𝑈, 𝑛}) → (𝑈 ∈ (𝐸‘𝑗) ↔ 𝑈 ∈ (𝐸‘(◡𝐸‘{𝑈, 𝑛}))))
36	33, 35	anbi12d 743	. . . . . . . . . 10 ⊢ (𝑗 = (◡𝐸‘{𝑈, 𝑛}) → ((𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗)) ↔ ((◡𝐸‘{𝑈, 𝑛}) ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘(◡𝐸‘{𝑈, 𝑛})))))
37	32, 36	syl5ibrcom 236	. . . . . . . . 9 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑛 ∈ 𝑁) → (𝑗 = (◡𝐸‘{𝑈, 𝑛}) → (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))))
38	9, 37	sylbid 229	. . . . . . . 8 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑛 ∈ 𝑁) → (𝑗 = (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛}) → (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))))
39	38	rexlimdva 3013	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → (∃𝑛 ∈ 𝑁 𝑗 = (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛}) → (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))))
40		simpl 472	. . . . . . . . . . 11 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → 𝑉 USGrph 𝐸)
41	40	adantr 480	. . . . . . . . . 10 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) → 𝑉 USGrph 𝐸)
42		simpl 472	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗)) → 𝑗 ∈ dom 𝐸)
43	42	adantl 481	. . . . . . . . . 10 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) → 𝑗 ∈ dom 𝐸)
44		simprr 792	. . . . . . . . . 10 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) → 𝑈 ∈ (𝐸‘𝑗))
45		usgraedg4 25916	. . . . . . . . . 10 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗)) → ∃𝑛 ∈ 𝑉 (𝐸‘𝑗) = {𝑈, 𝑛})
46	41, 43, 44, 45	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) → ∃𝑛 ∈ 𝑉 (𝐸‘𝑗) = {𝑈, 𝑛})
47		usgrafun 25878	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑉 USGrph 𝐸 → Fun 𝐸)
48		fvelrn 6260	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun 𝐸 ∧ 𝑗 ∈ dom 𝐸) → (𝐸‘𝑗) ∈ ran 𝐸)
49	48	ex 449	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Fun 𝐸 → (𝑗 ∈ dom 𝐸 → (𝐸‘𝑗) ∈ ran 𝐸))
50	47, 49	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑉 USGrph 𝐸 → (𝑗 ∈ dom 𝐸 → (𝐸‘𝑗) ∈ ran 𝐸))
51	50	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → (𝑗 ∈ dom 𝐸 → (𝐸‘𝑗) ∈ ran 𝐸))
52	51	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ dom 𝐸 → ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → (𝐸‘𝑗) ∈ ran 𝐸))
53	52	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗)) → ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → (𝐸‘𝑗) ∈ ran 𝐸))
54	53	impcom 445	. . . . . . . . . . . . . . . 16 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) → (𝐸‘𝑗) ∈ ran 𝐸)
55	54	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) ∧ (𝑛 ∈ 𝑉 ∧ (𝐸‘𝑗) = {𝑈, 𝑛})) → (𝐸‘𝑗) ∈ ran 𝐸)
56		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑈, 𝑛} = (𝐸‘𝑗) → {𝑈, 𝑛} = (𝐸‘𝑗))
57	56	eqcoms 2618	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸‘𝑗) = {𝑈, 𝑛} → {𝑈, 𝑛} = (𝐸‘𝑗))
58	14, 57	syl5eq 2656	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸‘𝑗) = {𝑈, 𝑛} → {𝑛, 𝑈} = (𝐸‘𝑗))
59	58	eleq1d 2672	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸‘𝑗) = {𝑈, 𝑛} → ({𝑛, 𝑈} ∈ ran 𝐸 ↔ (𝐸‘𝑗) ∈ ran 𝐸))
60	59	ad2antll 761	. . . . . . . . . . . . . . 15 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) ∧ (𝑛 ∈ 𝑉 ∧ (𝐸‘𝑗) = {𝑈, 𝑛})) → ({𝑛, 𝑈} ∈ ran 𝐸 ↔ (𝐸‘𝑗) ∈ ran 𝐸))
61	55, 60	mpbird 246	. . . . . . . . . . . . . 14 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) ∧ (𝑛 ∈ 𝑉 ∧ (𝐸‘𝑗) = {𝑈, 𝑛})) → {𝑛, 𝑈} ∈ ran 𝐸)
62	13	ad3antrrr 762	. . . . . . . . . . . . . 14 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) ∧ (𝑛 ∈ 𝑉 ∧ (𝐸‘𝑗) = {𝑈, 𝑛})) → (𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ↔ {𝑛, 𝑈} ∈ ran 𝐸))
63	61, 62	mpbird 246	. . . . . . . . . . . . 13 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) ∧ (𝑛 ∈ 𝑉 ∧ (𝐸‘𝑗) = {𝑈, 𝑛})) → 𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈))
64	63, 27	sylibr 223	. . . . . . . . . . . 12 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) ∧ (𝑛 ∈ 𝑉 ∧ (𝐸‘𝑗) = {𝑈, 𝑛})) → 𝑛 ∈ 𝑁)
65		f1ocnvfv1 6432	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸:dom 𝐸–1-1-onto→ran 𝐸 ∧ 𝑗 ∈ dom 𝐸) → (◡𝐸‘(𝐸‘𝑗)) = 𝑗)
66	11, 42, 65	syl2an 493	. . . . . . . . . . . . . . 15 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) → (◡𝐸‘(𝐸‘𝑗)) = 𝑗)
67	66	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) ∧ (𝑛 ∈ 𝑉 ∧ (𝐸‘𝑗) = {𝑈, 𝑛})) → (◡𝐸‘(𝐸‘𝑗)) = 𝑗)
68		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑈, 𝑛} = (𝐸‘𝑗) → (◡𝐸‘{𝑈, 𝑛}) = (◡𝐸‘(𝐸‘𝑗)))
69	68	eqcoms 2618	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸‘𝑗) = {𝑈, 𝑛} → (◡𝐸‘{𝑈, 𝑛}) = (◡𝐸‘(𝐸‘𝑗)))
70	69	eqeq1d 2612	. . . . . . . . . . . . . . 15 ⊢ ((𝐸‘𝑗) = {𝑈, 𝑛} → ((◡𝐸‘{𝑈, 𝑛}) = 𝑗 ↔ (◡𝐸‘(𝐸‘𝑗)) = 𝑗))
71	70	ad2antll 761	. . . . . . . . . . . . . 14 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) ∧ (𝑛 ∈ 𝑉 ∧ (𝐸‘𝑗) = {𝑈, 𝑛})) → ((◡𝐸‘{𝑈, 𝑛}) = 𝑗 ↔ (◡𝐸‘(𝐸‘𝑗)) = 𝑗))
72	67, 71	mpbird 246	. . . . . . . . . . . . 13 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) ∧ (𝑛 ∈ 𝑉 ∧ (𝐸‘𝑗) = {𝑈, 𝑛})) → (◡𝐸‘{𝑈, 𝑛}) = 𝑗)
73		eqcom 2617	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛}) ↔ (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛}) = 𝑗)
74		simplll 794	. . . . . . . . . . . . . . 15 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) ∧ (𝑛 ∈ 𝑉 ∧ (𝐸‘𝑗) = {𝑈, 𝑛})) → 𝑉 USGrph 𝐸)
75		simpllr 795	. . . . . . . . . . . . . . 15 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) ∧ (𝑛 ∈ 𝑉 ∧ (𝐸‘𝑗) = {𝑈, 𝑛})) → 𝑈 ∈ 𝑉)
76	7	eqeq1d 2612	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉 ∧ 𝑛 ∈ 𝑁) → ((℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛}) = 𝑗 ↔ (◡𝐸‘{𝑈, 𝑛}) = 𝑗))
77	74, 75, 64, 76	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) ∧ (𝑛 ∈ 𝑉 ∧ (𝐸‘𝑗) = {𝑈, 𝑛})) → ((℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛}) = 𝑗 ↔ (◡𝐸‘{𝑈, 𝑛}) = 𝑗))
78	73, 77	syl5bb 271	. . . . . . . . . . . . 13 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) ∧ (𝑛 ∈ 𝑉 ∧ (𝐸‘𝑗) = {𝑈, 𝑛})) → (𝑗 = (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛}) ↔ (◡𝐸‘{𝑈, 𝑛}) = 𝑗))
79	72, 78	mpbird 246	. . . . . . . . . . . 12 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) ∧ (𝑛 ∈ 𝑉 ∧ (𝐸‘𝑗) = {𝑈, 𝑛})) → 𝑗 = (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛}))
80	64, 79	jca 553	. . . . . . . . . . 11 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) ∧ (𝑛 ∈ 𝑉 ∧ (𝐸‘𝑗) = {𝑈, 𝑛})) → (𝑛 ∈ 𝑁 ∧ 𝑗 = (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛})))
81	80	ex 449	. . . . . . . . . 10 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) → ((𝑛 ∈ 𝑉 ∧ (𝐸‘𝑗) = {𝑈, 𝑛}) → (𝑛 ∈ 𝑁 ∧ 𝑗 = (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛}))))
82	81	reximdv2 2997	. . . . . . . . 9 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) → (∃𝑛 ∈ 𝑉 (𝐸‘𝑗) = {𝑈, 𝑛} → ∃𝑛 ∈ 𝑁 𝑗 = (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛})))
83	46, 82	mpd 15	. . . . . . . 8 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))) → ∃𝑛 ∈ 𝑁 𝑗 = (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛}))
84	83	ex 449	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → ((𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗)) → ∃𝑛 ∈ 𝑁 𝑗 = (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛})))
85	39, 84	impbid 201	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → (∃𝑛 ∈ 𝑁 𝑗 = (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛}) ↔ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))))
86	85	abbidv 2728	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → {𝑗 ∣ ∃𝑛 ∈ 𝑁 𝑗 = (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛})} = {𝑗 ∣ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))})
87		eleq1 2676	. . . . . . 7 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ dom 𝐸 ↔ 𝑖 ∈ dom 𝐸))
88		fveq2 6103	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → (𝐸‘𝑗) = (𝐸‘𝑖))
89	88	eleq2d 2673	. . . . . . 7 ⊢ (𝑗 = 𝑖 → (𝑈 ∈ (𝐸‘𝑗) ↔ 𝑈 ∈ (𝐸‘𝑖)))
90	87, 89	anbi12d 743	. . . . . 6 ⊢ (𝑗 = 𝑖 → ((𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗)) ↔ (𝑖 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑖))))
91	90	cbvabv 2734	. . . . 5 ⊢ {𝑗 ∣ (𝑗 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑗))} = {𝑖 ∣ (𝑖 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑖))}
92	86, 91	syl6eq 2660	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → {𝑗 ∣ ∃𝑛 ∈ 𝑁 𝑗 = (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛})} = {𝑖 ∣ (𝑖 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑖))})
93		df-rab 2905	. . . 4 ⊢ {𝑖 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑖)} = {𝑖 ∣ (𝑖 ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘𝑖))}
94	92, 93	syl6eqr 2662	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → {𝑗 ∣ ∃𝑛 ∈ 𝑁 𝑗 = (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛})} = {𝑖 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑖)})
95	3	rnmpt 5292	. . 3 ⊢ ran 𝐹 = {𝑗 ∣ ∃𝑛 ∈ 𝑁 𝑗 = (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛})}
96	94, 95, 2	3eqtr4g 2669	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → ran 𝐹 = 𝐼)
97	1, 2, 3	nbgraf1olem4 25973	. . . . . . . 8 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝑁) → (𝐹‘𝑥) = (◡𝐸‘{𝑈, 𝑥}))
98	97	3expa 1257	. . . . . . 7 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑥 ∈ 𝑁) → (𝐹‘𝑥) = (◡𝐸‘{𝑈, 𝑥}))
99	98	adantr 480	. . . . . 6 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → (𝐹‘𝑥) = (◡𝐸‘{𝑈, 𝑥}))
100		simplll 794	. . . . . . 7 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → 𝑉 USGrph 𝐸)
101		simpllr 795	. . . . . . 7 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → 𝑈 ∈ 𝑉)
102		simpr 476	. . . . . . 7 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁)
103	1, 2, 3	nbgraf1olem4 25973	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉 ∧ 𝑦 ∈ 𝑁) → (𝐹‘𝑦) = (◡𝐸‘{𝑈, 𝑦}))
104	100, 101, 102, 103	syl3anc 1318	. . . . . 6 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → (𝐹‘𝑦) = (◡𝐸‘{𝑈, 𝑦}))
105	99, 104	eqeq12d 2625	. . . . 5 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (◡𝐸‘{𝑈, 𝑥}) = (◡𝐸‘{𝑈, 𝑦})))
106		usgraf1 25889	. . . . . . 7 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1→ran 𝐸)
107	106	ad3antrrr 762	. . . . . 6 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → 𝐸:dom 𝐸–1-1→ran 𝐸)
108	1	eleq2i 2680	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑁 ↔ 𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈))
109		nbgraeledg 25959	. . . . . . . . . . . 12 ⊢ (𝑉 USGrph 𝐸 → (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ↔ {𝑥, 𝑈} ∈ ran 𝐸))
110		prcom 4211	. . . . . . . . . . . . 13 ⊢ {𝑥, 𝑈} = {𝑈, 𝑥}
111	110	eleq1i 2679	. . . . . . . . . . . 12 ⊢ ({𝑥, 𝑈} ∈ ran 𝐸 ↔ {𝑈, 𝑥} ∈ ran 𝐸)
112	109, 111	syl6bb 275	. . . . . . . . . . 11 ⊢ (𝑉 USGrph 𝐸 → (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ↔ {𝑈, 𝑥} ∈ ran 𝐸))
113	112	biimpd 218	. . . . . . . . . 10 ⊢ (𝑉 USGrph 𝐸 → (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) → {𝑈, 𝑥} ∈ ran 𝐸))
114	108, 113	syl5bi 231	. . . . . . . . 9 ⊢ (𝑉 USGrph 𝐸 → (𝑥 ∈ 𝑁 → {𝑈, 𝑥} ∈ ran 𝐸))
115	114	adantr 480	. . . . . . . 8 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → (𝑥 ∈ 𝑁 → {𝑈, 𝑥} ∈ ran 𝐸))
116	115	imp 444	. . . . . . 7 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑥 ∈ 𝑁) → {𝑈, 𝑥} ∈ ran 𝐸)
117	116	adantr 480	. . . . . 6 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → {𝑈, 𝑥} ∈ ran 𝐸)
118	1	eleq2i 2680	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑁 ↔ 𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈))
119		nbgraeledg 25959	. . . . . . . . . . . 12 ⊢ (𝑉 USGrph 𝐸 → (𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ↔ {𝑦, 𝑈} ∈ ran 𝐸))
120		prcom 4211	. . . . . . . . . . . . 13 ⊢ {𝑦, 𝑈} = {𝑈, 𝑦}
121	120	eleq1i 2679	. . . . . . . . . . . 12 ⊢ ({𝑦, 𝑈} ∈ ran 𝐸 ↔ {𝑈, 𝑦} ∈ ran 𝐸)
122	119, 121	syl6bb 275	. . . . . . . . . . 11 ⊢ (𝑉 USGrph 𝐸 → (𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ↔ {𝑈, 𝑦} ∈ ran 𝐸))
123	122	biimpd 218	. . . . . . . . . 10 ⊢ (𝑉 USGrph 𝐸 → (𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) → {𝑈, 𝑦} ∈ ran 𝐸))
124	118, 123	syl5bi 231	. . . . . . . . 9 ⊢ (𝑉 USGrph 𝐸 → (𝑦 ∈ 𝑁 → {𝑈, 𝑦} ∈ ran 𝐸))
125	124	adantr 480	. . . . . . . 8 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → (𝑦 ∈ 𝑁 → {𝑈, 𝑦} ∈ ran 𝐸))
126	125	adantr 480	. . . . . . 7 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑥 ∈ 𝑁) → (𝑦 ∈ 𝑁 → {𝑈, 𝑦} ∈ ran 𝐸))
127	126	imp 444	. . . . . 6 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → {𝑈, 𝑦} ∈ ran 𝐸)
128		f1ocnvfvrneq 6441	. . . . . . 7 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ ({𝑈, 𝑥} ∈ ran 𝐸 ∧ {𝑈, 𝑦} ∈ ran 𝐸)) → ((◡𝐸‘{𝑈, 𝑥}) = (◡𝐸‘{𝑈, 𝑦}) → {𝑈, 𝑥} = {𝑈, 𝑦}))
129		vex 3176	. . . . . . . 8 ⊢ 𝑥 ∈ V
130		vex 3176	. . . . . . . 8 ⊢ 𝑦 ∈ V
131	129, 130	preqr2 4321	. . . . . . 7 ⊢ ({𝑈, 𝑥} = {𝑈, 𝑦} → 𝑥 = 𝑦)
132	128, 131	syl6 34	. . . . . 6 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ ({𝑈, 𝑥} ∈ ran 𝐸 ∧ {𝑈, 𝑦} ∈ ran 𝐸)) → ((◡𝐸‘{𝑈, 𝑥}) = (◡𝐸‘{𝑈, 𝑦}) → 𝑥 = 𝑦))
133	107, 117, 127, 132	syl12anc 1316	. . . . 5 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → ((◡𝐸‘{𝑈, 𝑥}) = (◡𝐸‘{𝑈, 𝑦}) → 𝑥 = 𝑦))
134	105, 133	sylbid 229	. . . 4 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑥 ∈ 𝑁) ∧ 𝑦 ∈ 𝑁) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
135	134	ralrimiva 2949	. . 3 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑥 ∈ 𝑁) → ∀𝑦 ∈ 𝑁 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
136	135	ralrimiva 2949	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
137		dff1o6 6431	. 2 ⊢ (𝐹:𝑁–1-1-onto→𝐼 ↔ (𝐹 Fn 𝑁 ∧ ran 𝐹 = 𝐼 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
138	6, 96, 136, 137	syl3anbrc 1239	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → 𝐹:𝑁–1-1-onto→𝐼)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  {crab 2900  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ^◡ccnv 5037  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  ℩crio 6510  (class class class)co 6549   USGrph cusg 25859   Neighbors cnbgra 25946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-usgra 25862  df-nbgra 25949

This theorem is referenced by:  nbgraf1o0  25975