Theorem ushgredgedg 16450

Description: In a simple hypergraph there is a 1-1 onto mapping between the indexed edges containing a fixed vertex and the set of edges containing this vertex. (Contributed by AV, 11-Dec-2020.)

Hypotheses

Ref	Expression
ushgredgedg.e	⊢ 𝐸 = (Edg‘𝐺)
ushgredgedg.i	⊢ 𝐼 = (iEdg‘𝐺)
ushgredgedg.v	⊢ 𝑉 = (Vtx‘𝐺)
ushgredgedg.a	⊢ 𝐴 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}
ushgredgedg.b	⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}
ushgredgedg.f	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥))

Assertion

Ref	Expression
ushgredgedg	⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵)

Distinct variable groups:   𝐵,𝑒   𝑒,𝐸,𝑖   𝑒,𝐺,𝑖,𝑥   𝑒,𝐼,𝑖,𝑥   𝑒,𝑁,𝑖,𝑥   𝑒,𝑉,𝑖,𝑥

Allowed substitution hints:   𝐴(𝑥,𝑒,𝑖)   𝐵(𝑥,𝑖)   𝐸(𝑥)   𝐹(𝑥,𝑒,𝑖)

Proof of Theorem ushgredgedg

Dummy variables 𝑓 𝑗 𝑝 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2238	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		ushgredgedg.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	ushgrfm 16298	. . . 4 ⊢ (𝐺 ∈ USHGraph → 𝐼:dom 𝐼–1-1→{𝑝 ∈ 𝒫 (Vtx‘𝐺) ∣ ∃𝑤 𝑤 ∈ 𝑝})
4	3	adantr 276	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐼:dom 𝐼–1-1→{𝑝 ∈ 𝒫 (Vtx‘𝐺) ∣ ∃𝑤 𝑤 ∈ 𝑝})
5		ssrab2 3333	. . 3 ⊢ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ⊆ dom 𝐼
6		f1ores 5652	. . 3 ⊢ ((𝐼:dom 𝐼–1-1→{𝑝 ∈ 𝒫 (Vtx‘𝐺) ∣ ∃𝑤 𝑤 ∈ 𝑝} ∧ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ⊆ dom 𝐼) → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}):{𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}–1-1-onto→(𝐼 “ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}))
7	4, 5, 6	sylancl 417	. 2 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}):{𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}–1-1-onto→(𝐼 “ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}))
8		ushgredgedg.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥))
9		ushgredgedg.a	. . . . . . 7 ⊢ 𝐴 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}
10	9	a1i 9	. . . . . 6 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐴 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)})
11		eqidd 2239	. . . . . 6 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → (𝐼‘𝑥) = (𝐼‘𝑥))
12	10, 11	mpteq12dva 4210	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥)) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↦ (𝐼‘𝑥)))
13	8, 12	eqtrid 2283	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↦ (𝐼‘𝑥)))
14		f1f 5596	. . . . . . . 8 ⊢ (𝐼:dom 𝐼–1-1→{𝑝 ∈ 𝒫 (Vtx‘𝐺) ∣ ∃𝑤 𝑤 ∈ 𝑝} → 𝐼:dom 𝐼⟶{𝑝 ∈ 𝒫 (Vtx‘𝐺) ∣ ∃𝑤 𝑤 ∈ 𝑝})
15	3, 14	syl 14	. . . . . . 7 ⊢ (𝐺 ∈ USHGraph → 𝐼:dom 𝐼⟶{𝑝 ∈ 𝒫 (Vtx‘𝐺) ∣ ∃𝑤 𝑤 ∈ 𝑝})
16	5	a1i 9	. . . . . . 7 ⊢ (𝐺 ∈ USHGraph → {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ⊆ dom 𝐼)
17	15, 16	feqresmpt 5754	. . . . . 6 ⊢ (𝐺 ∈ USHGraph → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↦ (𝐼‘𝑥)))
18	17	adantr 276	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↦ (𝐼‘𝑥)))
19	18	eqcomd 2244	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↦ (𝐼‘𝑥)) = (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}))
20	13, 19	eqtrd 2271	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}))
21		ushgruhgr 16304	. . . . . . . . 9 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
22		eqid 2238	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
23	22	uhgrfun 16301	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
24	21, 23	syl 14	. . . . . . . 8 ⊢ (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
25	2	funeqi 5396	. . . . . . . 8 ⊢ (Fun 𝐼 ↔ Fun (iEdg‘𝐺))
26	24, 25	sylibr 134	. . . . . . 7 ⊢ (𝐺 ∈ USHGraph → Fun 𝐼)
27	26	adantr 276	. . . . . 6 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → Fun 𝐼)
28		dfimafn 5748	. . . . . 6 ⊢ ((Fun 𝐼 ∧ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ⊆ dom 𝐼) → (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}) = {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑒})
29	27, 5, 28	sylancl 417	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}) = {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑒})
30		fveq2 5693	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝐼‘𝑖) = (𝐼‘𝑗))
31	30	eleq2d 2308	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝑁 ∈ (𝐼‘𝑖) ↔ 𝑁 ∈ (𝐼‘𝑗)))
32	31	elrab 2982	. . . . . . . . . 10 ⊢ (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↔ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)))
33		simpl 109	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) → 𝑗 ∈ dom 𝐼)
34		fvelrn 5833	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝐼 ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ∈ ran 𝐼)
35	2	eqcomi 2242	. . . . . . . . . . . . . . . . . . 19 ⊢ (iEdg‘𝐺) = 𝐼
36	35	rneqi 5008	. . . . . . . . . . . . . . . . . 18 ⊢ ran (iEdg‘𝐺) = ran 𝐼
37	36	eleq2i 2305	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼‘𝑗) ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran 𝐼)
38	34, 37	sylibr 134	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐼 ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
39	27, 33, 38	syl2an 289	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗))) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
40	39	3adant3 1048	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
41		eleq1 2301	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝐼‘𝑗) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
42	41	eqcoms 2241	. . . . . . . . . . . . . . 15 ⊢ ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
43	42	3ad2ant3 1051	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
44	40, 43	mpbird 167	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → 𝑓 ∈ ran (iEdg‘𝐺))
45		ushgredgedg.e	. . . . . . . . . . . . . . . . 17 ⊢ 𝐸 = (Edg‘𝐺)
46		edgvalg 16283	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
47	45, 46	eqtrid 2283	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
48	47	eleq2d 2308	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
49	48	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
50	49	3ad2ant1 1049	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
51	44, 50	mpbird 167	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → 𝑓 ∈ 𝐸)
52		eleq2 2302	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼‘𝑗) = 𝑓 → (𝑁 ∈ (𝐼‘𝑗) ↔ 𝑁 ∈ 𝑓))
53	52	biimpcd 159	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (𝐼‘𝑗) → ((𝐼‘𝑗) = 𝑓 → 𝑁 ∈ 𝑓))
54	53	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) → ((𝐼‘𝑗) = 𝑓 → 𝑁 ∈ 𝑓))
55	54	a1i 9	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) → ((𝐼‘𝑗) = 𝑓 → 𝑁 ∈ 𝑓)))
56	55	3imp 1224	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → 𝑁 ∈ 𝑓)
57	51, 56	jca 306	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓))
58	57	3exp 1233	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) → ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓))))
59	32, 58	biimtrid 152	. . . . . . . . 9 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} → ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓))))
60	59	rexlimdv 2667	. . . . . . . 8 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓)))
61	24	funfnd 5406	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
62		fvelrnb 5747	. . . . . . . . . . . . 13 ⊢ ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
63	61, 62	syl 14	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
64	35	dmeqi 4980	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ dom (iEdg‘𝐺) = dom 𝐼
65	64	eleq2i 2305	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ dom (iEdg‘𝐺) ↔ 𝑗 ∈ dom 𝐼)
66	65	biimpi 120	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ dom (iEdg‘𝐺) → 𝑗 ∈ dom 𝐼)
67	66	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑗 ∈ dom 𝐼)
68	67	adantl 277	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ dom 𝐼)
69	35	fveq1i 5694	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((iEdg‘𝐺)‘𝑗) = (𝐼‘𝑗)
70	69	eqeq2i 2249	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 = ((iEdg‘𝐺)‘𝑗) ↔ 𝑓 = (𝐼‘𝑗))
71	70	biimpi 120	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = ((iEdg‘𝐺)‘𝑗) → 𝑓 = (𝐼‘𝑗))
72	71	eqcoms 2241	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → 𝑓 = (𝐼‘𝑗))
73	72	eleq2d 2308	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑁 ∈ 𝑓 ↔ 𝑁 ∈ (𝐼‘𝑗)))
74	73	biimpcd 159	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ 𝑓 → (((iEdg‘𝐺)‘𝑗) = 𝑓 → 𝑁 ∈ (𝐼‘𝑗)))
75	74	adantl 277	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) → (((iEdg‘𝐺)‘𝑗) = 𝑓 → 𝑁 ∈ (𝐼‘𝑗)))
76	75	adantld 278	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑁 ∈ (𝐼‘𝑗)))
77	76	imp 124	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑁 ∈ (𝐼‘𝑗))
78	68, 77	jca 306	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)))
79	78, 32	sylibr 134	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)})
80	69	eqeq1i 2246	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 ↔ (𝐼‘𝑗) = 𝑓)
81	80	biimpi 120	. . . . . . . . . . . . . . . . . . 19 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼‘𝑗) = 𝑓)
82	81	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝐼‘𝑗) = 𝑓)
83	82	adantl 277	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝐼‘𝑗) = 𝑓)
84	79, 83	jca 306	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ∧ (𝐼‘𝑗) = 𝑓))
85	84	ex 115	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ∧ (𝐼‘𝑗) = 𝑓)))
86	85	reximdv2 2649	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓))
87	86	ex 115	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USHGraph → (𝑁 ∈ 𝑓 → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓)))
88	87	com23 78	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USHGraph → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑁 ∈ 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓)))
89	63, 88	sylbid 150	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) → (𝑁 ∈ 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓)))
90	48, 89	sylbid 150	. . . . . . . . . 10 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ 𝐸 → (𝑁 ∈ 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓)))
91	90	impd 254	. . . . . . . . 9 ⊢ (𝐺 ∈ USHGraph → ((𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓))
92	91	adantr 276	. . . . . . . 8 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓))
93	60, 92	impbid 129	. . . . . . 7 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓 ↔ (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓)))
94		vex 2824	. . . . . . . 8 ⊢ 𝑓 ∈ V
95		eqeq2 2248	. . . . . . . . 9 ⊢ (𝑒 = 𝑓 → ((𝐼‘𝑗) = 𝑒 ↔ (𝐼‘𝑗) = 𝑓))
96	95	rexbidv 2551	. . . . . . . 8 ⊢ (𝑒 = 𝑓 → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑒 ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓))
97	94, 96	elab 2970	. . . . . . 7 ⊢ (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑒} ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓)
98		eleq2 2302	. . . . . . . 8 ⊢ (𝑒 = 𝑓 → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ 𝑓))
99		ushgredgedg.b	. . . . . . . 8 ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}
100	98, 99	elrab2 2985	. . . . . . 7 ⊢ (𝑓 ∈ 𝐵 ↔ (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓))
101	93, 97, 100	3bitr4g 223	. . . . . 6 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑒} ↔ 𝑓 ∈ 𝐵))
102	101	eqrdv 2236	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑒} = 𝐵)
103	29, 102	eqtrd 2271	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}) = 𝐵)
104	103	eqcomd 2244	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐵 = (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}))
105	20, 10, 104	f1oeq123d 5631	. 2 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}):{𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}–1-1-onto→(𝐼 “ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)})))
106	7, 105	mpbird 167	1 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {cab 2224  ∃wrex 2529  {crab 2532   ⊆ wss 3220  𝒫 cpw 3688   ↦ cmpt 4190  dom cdm 4772  ran crn 4773   ↾ cres 4774   “ cima 4775  Fun wfun 5369   Fn wfn 5370  ⟶wf 5371  –1-1→wf1 5372  –1-1-onto→wf1o 5374  ‘cfv 5375  Vtxcvtx 16236  iEdgciedg 16237  Edgcedg 16281  UHGraphcuhgr 16291  USHGraphcushgr 16292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-sub 8493  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-dec 9761  df-ndx 13338  df-slot 13339  df-base 13341  df-edgf 16229  df-vtx 16238  df-iedg 16239  df-edg 16282  df-uhgrm 16293  df-ushgrm 16294

This theorem is referenced by:  usgredgedg  16451  vtxduspgrfvedgfilem  16524