Theorem ushgredgedg 26121

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 2622	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		ushgredgedg.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	ushgrf 25958	. . . 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 3687	. . 3 ⊢ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ⊆ dom 𝐼
6		f1ores 6151	. . 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 694	. 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 2623	. . . . . 6 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → (𝐼‘𝑥) = (𝐼‘𝑥))
12	10, 11	mpteq12dva 4732	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥)) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↦ (𝐼‘𝑥)))
13	8, 12	syl5eq 2668	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↦ (𝐼‘𝑥)))
14		f1f 6101	. . . . . . . 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 6250	. . . . . 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 2628	. . . 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 2656	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}))
21		ushgruhgr 25964	. . . . . . . . 9 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph )
22		eqid 2622	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
23	22	uhgrfun 25961	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
24	21, 23	syl 17	. . . . . . . 8 ⊢ (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
25	2	funeqi 5909	. . . . . . . 8 ⊢ (Fun 𝐼 ↔ Fun (iEdg‘𝐺))
26	24, 25	sylibr 224	. . . . . . 7 ⊢ (𝐺 ∈ USHGraph → Fun 𝐼)
27	26	adantr 481	. . . . . 6 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → Fun 𝐼)
28		dfimafn 6245	. . . . . 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 694	. . . . 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 6191	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝐼‘𝑖) = (𝐼‘𝑗))
31	30	eleq2d 2687	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝑁 ∈ (𝐼‘𝑖) ↔ 𝑁 ∈ (𝐼‘𝑗)))
32	31	elrab 3363	. . . . . . . . . 10 ⊢ (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↔ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)))
33		simpl 473	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) → 𝑗 ∈ dom 𝐼)
34		fvelrn 6352	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝐼 ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ∈ ran 𝐼)
35	2	eqcomi 2631	. . . . . . . . . . . . . . . . . . 19 ⊢ (iEdg‘𝐺) = 𝐼
36	35	rneqi 5352	. . . . . . . . . . . . . . . . . 18 ⊢ ran (iEdg‘𝐺) = ran 𝐼
37	36	eleq2i 2693	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼‘𝑗) ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran 𝐼)
38	34, 37	sylibr 224	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐼 ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
39	27, 33, 38	syl2an 494	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗))) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
40	39	3adant3 1081	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
41		eleq1 2689	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝐼‘𝑗) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
42	41	eqcoms 2630	. . . . . . . . . . . . . . 15 ⊢ ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
43	42	3ad2ant3 1084	. . . . . . . . . . . . . 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 247	. . . . . . . . . . . . 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 25941	. . . . . . . . . . . . . . . . . 18 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
47	46	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
48	45, 47	syl5eq 2668	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
49	48	eleq2d 2687	. . . . . . . . . . . . . . 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 1082	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
52	44, 51	mpbird 247	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → 𝑓 ∈ 𝐸)
53		eleq2 2690	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼‘𝑗) = 𝑓 → (𝑁 ∈ (𝐼‘𝑗) ↔ 𝑁 ∈ 𝑓))
54	53	biimpcd 239	. . . . . . . . . . . . . . 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 1256	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → 𝑁 ∈ 𝑓)
58	52, 57	jca 554	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓))
59	58	3exp 1264	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) → ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓))))
60	32, 59	syl5bi 232	. . . . . . . . 9 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} → ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓))))
61	60	rexlimdv 3030	. . . . . . . 8 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓)))
62		funfn 5918	. . . . . . . . . . . . . . 15 ⊢ (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
63	62	biimpi 206	. . . . . . . . . . . . . 14 ⊢ (Fun (iEdg‘𝐺) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
64	24, 63	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
65		fvelrnb 6243	. . . . . . . . . . . . 13 ⊢ ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
66	64, 65	syl 17	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
67	35	dmeqi 5325	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ dom (iEdg‘𝐺) = dom 𝐼
68	67	eleq2i 2693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ dom (iEdg‘𝐺) ↔ 𝑗 ∈ dom 𝐼)
69	68	biimpi 206	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ dom (iEdg‘𝐺) → 𝑗 ∈ dom 𝐼)
70	69	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑗 ∈ dom 𝐼)
71	70	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ dom 𝐼)
72	35	fveq1i 6192	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((iEdg‘𝐺)‘𝑗) = (𝐼‘𝑗)
73	72	eqeq2i 2634	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 = ((iEdg‘𝐺)‘𝑗) ↔ 𝑓 = (𝐼‘𝑗))
74	73	biimpi 206	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = ((iEdg‘𝐺)‘𝑗) → 𝑓 = (𝐼‘𝑗))
75	74	eqcoms 2630	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → 𝑓 = (𝐼‘𝑗))
76	75	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑁 ∈ 𝑓 ↔ 𝑁 ∈ (𝐼‘𝑗)))
77	76	biimpcd 239	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ 𝑓 → (((iEdg‘𝐺)‘𝑗) = 𝑓 → 𝑁 ∈ (𝐼‘𝑗)))
78	77	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) → (((iEdg‘𝐺)‘𝑗) = 𝑓 → 𝑁 ∈ (𝐼‘𝑗)))
79	78	adantld 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑁 ∈ (𝐼‘𝑗)))
80	79	imp 445	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑁 ∈ (𝐼‘𝑗))
81	71, 80	jca 554	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)))
82	81, 32	sylibr 224	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)})
83	72	eqeq1i 2627	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 ↔ (𝐼‘𝑗) = 𝑓)
84	83	biimpi 206	. . . . . . . . . . . . . . . . . . 19 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼‘𝑗) = 𝑓)
85	84	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝐼‘𝑗) = 𝑓)
86	85	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝐼‘𝑗) = 𝑓)
87	82, 86	jca 554	. . . . . . . . . . . . . . . 16 ⊢ (((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 450	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ∧ (𝐼‘𝑗) = 𝑓)))
89	88	reximdv2 3014	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓))
90	89	ex 450	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USHGraph → (𝑁 ∈ 𝑓 → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓)))
91	90	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)))
92	66, 91	sylbid 230	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) → (𝑁 ∈ 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓)))
93	49, 92	sylbid 230	. . . . . . . . . 10 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ 𝐸 → (𝑁 ∈ 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓)))
94	93	impd 447	. . . . . . . . 9 ⊢ (𝐺 ∈ USHGraph → ((𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓))
95	94	adantr 481	. . . . . . . 8 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓))
96	61, 95	impbid 202	. . . . . . 7 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓 ↔ (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓)))
97		vex 3203	. . . . . . . 8 ⊢ 𝑓 ∈ V
98		eqeq2 2633	. . . . . . . . 9 ⊢ (𝑒 = 𝑓 → ((𝐼‘𝑗) = 𝑒 ↔ (𝐼‘𝑗) = 𝑓))
99	98	rexbidv 3052	. . . . . . . 8 ⊢ (𝑒 = 𝑓 → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑒 ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓))
100	97, 99	elab 3350	. . . . . . 7 ⊢ (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑒} ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓)
101		eleq2 2690	. . . . . . . 8 ⊢ (𝑒 = 𝑓 → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ 𝑓))
102		ushgredgedg.b	. . . . . . . 8 ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}
103	101, 102	elrab2 3366	. . . . . . 7 ⊢ (𝑓 ∈ 𝐵 ↔ (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓))
104	96, 100, 103	3bitr4g 303	. . . . . 6 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑒} ↔ 𝑓 ∈ 𝐵))
105	104	eqrdv 2620	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑒} = 𝐵)
106	29, 105	eqtrd 2656	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}) = 𝐵)
107	106	eqcomd 2628	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐵 = (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}))
108	20, 10, 107	f1oeq123d 6133	. 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)})))
109	7, 108	mpbird 247	1 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {cab 2608  ∃wrex 2913  {crab 2916   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177   ↦ cmpt 4729  dom cdm 5114  ran crn 5115   ↾ cres 5116   “ cima 5117  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951   USHGraph cushgr 25952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-edg 25940  df-uhgr 25953  df-ushgr 25954

This theorem is referenced by:  usgredgedg  26122  vtxdushgrfvedglem  26385