Theorem ushgredgedg 27704

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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		ushgredgedg.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	ushgrf 27541	. . . 4 ⊢ (𝐺 ∈ USHGraph → 𝐼:dom 𝐼–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
4	3	adantr 481	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐼:dom 𝐼–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
5		ssrab2 4023	. . 3 ⊢ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ⊆ dom 𝐼
6		f1ores 6765	. . 3 ⊢ ((𝐼:dom 𝐼–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ {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 586	. 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 11	. . . . . 6 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐴 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)})
11		eqidd 2738	. . . . . 6 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → (𝐼‘𝑥) = (𝐼‘𝑥))
12	10, 11	mpteq12dva 5174	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥)) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↦ (𝐼‘𝑥)))
13	8, 12	eqtrid 2789	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↦ (𝐼‘𝑥)))
14		f1f 6705	. . . . . . . 8 ⊢ (𝐼:dom 𝐼–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}) → 𝐼:dom 𝐼⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
15	3, 14	syl 17	. . . . . . 7 ⊢ (𝐺 ∈ USHGraph → 𝐼:dom 𝐼⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
16	5	a1i 11	. . . . . . 7 ⊢ (𝐺 ∈ USHGraph → {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ⊆ dom 𝐼)
17	15, 16	feqresmpt 6875	. . . . . 6 ⊢ (𝐺 ∈ USHGraph → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↦ (𝐼‘𝑥)))
18	17	adantr 481	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↦ (𝐼‘𝑥)))
19	18	eqcomd 2743	. . . 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 2777	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}))
21		ushgruhgr 27547	. . . . . . . . 9 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
22		eqid 2737	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
23	22	uhgrfun 27544	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
24	21, 23	syl 17	. . . . . . . 8 ⊢ (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
25	2	funeqi 6489	. . . . . . . 8 ⊢ (Fun 𝐼 ↔ Fun (iEdg‘𝐺))
26	24, 25	sylibr 233	. . . . . . 7 ⊢ (𝐺 ∈ USHGraph → Fun 𝐼)
27	26	adantr 481	. . . . . 6 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → Fun 𝐼)
28		dfimafn 6869	. . . . . 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 586	. . . . 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 6809	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝐼‘𝑖) = (𝐼‘𝑗))
31	30	eleq2d 2823	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝑁 ∈ (𝐼‘𝑖) ↔ 𝑁 ∈ (𝐼‘𝑗)))
32	31	elrab 3633	. . . . . . . . . 10 ⊢ (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↔ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)))
33		simpl 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) → 𝑗 ∈ dom 𝐼)
34		fvelrn 6991	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝐼 ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ∈ ran 𝐼)
35	2	eqcomi 2746	. . . . . . . . . . . . . . . . . . 19 ⊢ (iEdg‘𝐺) = 𝐼
36	35	rneqi 5863	. . . . . . . . . . . . . . . . . 18 ⊢ ran (iEdg‘𝐺) = ran 𝐼
37	36	eleq2i 2829	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼‘𝑗) ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran 𝐼)
38	34, 37	sylibr 233	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐼 ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
39	27, 33, 38	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗))) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
40	39	3adant3 1131	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
41		eleq1 2825	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝐼‘𝑗) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
42	41	eqcoms 2745	. . . . . . . . . . . . . . 15 ⊢ ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
43	42	3ad2ant3 1134	. . . . . . . . . . . . . 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 256	. . . . . . . . . . . . 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		edgval 27527	. . . . . . . . . . . . . . . . . 18 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
47	46	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
48	45, 47	eqtrid 2789	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
49	48	eleq2d 2823	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
50	49	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
51	50	3ad2ant1 1132	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
52	44, 51	mpbird 256	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → 𝑓 ∈ 𝐸)
53		eleq2 2826	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼‘𝑗) = 𝑓 → (𝑁 ∈ (𝐼‘𝑗) ↔ 𝑁 ∈ 𝑓))
54	53	biimpcd 248	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (𝐼‘𝑗) → ((𝐼‘𝑗) = 𝑓 → 𝑁 ∈ 𝑓))
55	54	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) → ((𝐼‘𝑗) = 𝑓 → 𝑁 ∈ 𝑓))
56	55	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) → ((𝐼‘𝑗) = 𝑓 → 𝑁 ∈ 𝑓)))
57	56	3imp 1110	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → 𝑁 ∈ 𝑓)
58	52, 57	jca 512	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓))
59	58	3exp 1118	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) → ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓))))
60	32, 59	biimtrid 241	. . . . . . . . 9 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} → ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓))))
61	60	rexlimdv 3147	. . . . . . . 8 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓)))
62	24	funfnd 6499	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
63		fvelrnb 6867	. . . . . . . . . . . . 13 ⊢ ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
64	62, 63	syl 17	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
65	35	dmeqi 5831	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ dom (iEdg‘𝐺) = dom 𝐼
66	65	eleq2i 2829	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ dom (iEdg‘𝐺) ↔ 𝑗 ∈ dom 𝐼)
67	66	biimpi 215	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ dom (iEdg‘𝐺) → 𝑗 ∈ dom 𝐼)
68	67	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑗 ∈ dom 𝐼)
69	68	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ dom 𝐼)
70	35	fveq1i 6810	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((iEdg‘𝐺)‘𝑗) = (𝐼‘𝑗)
71	70	eqeq2i 2750	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 = ((iEdg‘𝐺)‘𝑗) ↔ 𝑓 = (𝐼‘𝑗))
72	71	biimpi 215	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = ((iEdg‘𝐺)‘𝑗) → 𝑓 = (𝐼‘𝑗))
73	72	eqcoms 2745	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → 𝑓 = (𝐼‘𝑗))
74	73	eleq2d 2823	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑁 ∈ 𝑓 ↔ 𝑁 ∈ (𝐼‘𝑗)))
75	74	biimpcd 248	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ 𝑓 → (((iEdg‘𝐺)‘𝑗) = 𝑓 → 𝑁 ∈ (𝐼‘𝑗)))
76	75	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) → (((iEdg‘𝐺)‘𝑗) = 𝑓 → 𝑁 ∈ (𝐼‘𝑗)))
77	76	adantld 491	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑁 ∈ (𝐼‘𝑗)))
78	77	imp 407	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑁 ∈ (𝐼‘𝑗))
79	69, 78	jca 512	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)))
80	79, 32	sylibr 233	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)})
81	70	eqeq1i 2742	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 ↔ (𝐼‘𝑗) = 𝑓)
82	81	biimpi 215	. . . . . . . . . . . . . . . . . . 19 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼‘𝑗) = 𝑓)
83	82	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝐼‘𝑗) = 𝑓)
84	83	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝐼‘𝑗) = 𝑓)
85	80, 84	jca 512	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ∧ (𝐼‘𝑗) = 𝑓))
86	85	ex 413	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ∧ (𝐼‘𝑗) = 𝑓)))
87	86	reximdv2 3158	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓))
88	87	ex 413	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USHGraph → (𝑁 ∈ 𝑓 → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓)))
89	88	com23 86	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USHGraph → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑁 ∈ 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓)))
90	64, 89	sylbid 239	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) → (𝑁 ∈ 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓)))
91	49, 90	sylbid 239	. . . . . . . . . 10 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ 𝐸 → (𝑁 ∈ 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓)))
92	91	impd 411	. . . . . . . . 9 ⊢ (𝐺 ∈ USHGraph → ((𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓))
93	92	adantr 481	. . . . . . . 8 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓))
94	61, 93	impbid 211	. . . . . . 7 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓 ↔ (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓)))
95		vex 3445	. . . . . . . 8 ⊢ 𝑓 ∈ V
96		eqeq2 2749	. . . . . . . . 9 ⊢ (𝑒 = 𝑓 → ((𝐼‘𝑗) = 𝑒 ↔ (𝐼‘𝑗) = 𝑓))
97	96	rexbidv 3172	. . . . . . . 8 ⊢ (𝑒 = 𝑓 → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑒 ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓))
98	95, 97	elab 3618	. . . . . . 7 ⊢ (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑒} ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓)
99		eleq2 2826	. . . . . . . 8 ⊢ (𝑒 = 𝑓 → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ 𝑓))
100		ushgredgedg.b	. . . . . . . 8 ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}
101	99, 100	elrab2 3636	. . . . . . 7 ⊢ (𝑓 ∈ 𝐵 ↔ (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓))
102	94, 98, 101	3bitr4g 313	. . . . . 6 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑒} ↔ 𝑓 ∈ 𝐵))
103	102	eqrdv 2735	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑒} = 𝐵)
104	29, 103	eqtrd 2777	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}) = 𝐵)
105	104	eqcomd 2743	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐵 = (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}))
106	20, 10, 105	f1oeq123d 6745	. 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)})))
107	7, 106	mpbird 256	1 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  {cab 2714  ∃wrex 3071  {crab 3404   ∖ cdif 3893   ⊆ wss 3896  ∅c0 4266  𝒫 cpw 4543  {csn 4569   ↦ cmpt 5168  dom cdm 5605  ran crn 5606   ↾ cres 5607   “ cima 5608  Fun wfun 6457   Fn wfn 6458  ⟶wf 6459  –1-1→wf1 6460  –1-1-onto→wf1o 6462  ‘cfv 6463  Vtxcvtx 27474  iEdgciedg 27475  Edgcedg 27525  UHGraphcuhgr 27534  USHGraphcushgr 27535

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pr 5365  ax-un 7626

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-sbc 3726  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-fv 6471  df-edg 27526  df-uhgr 27536  df-ushgr 27537

This theorem is referenced by:  usgredgedg  27705  vtxdushgrfvedglem  27964