Theorem ushgredgedg 27499

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 2738	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		ushgredgedg.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	ushgrf 27336	. . . 4 ⊢ (𝐺 ∈ USHGraph → 𝐼:dom 𝐼–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
4	3	adantr 480	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐼:dom 𝐼–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
5		ssrab2 4009	. . 3 ⊢ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ⊆ dom 𝐼
6		f1ores 6714	. . 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 585	. 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 2739	. . . . . 6 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → (𝐼‘𝑥) = (𝐼‘𝑥))
12	10, 11	mpteq12dva 5159	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥)) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↦ (𝐼‘𝑥)))
13	8, 12	syl5eq 2791	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↦ (𝐼‘𝑥)))
14		f1f 6654	. . . . . . . 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 6820	. . . . . 6 ⊢ (𝐺 ∈ USHGraph → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↦ (𝐼‘𝑥)))
18	17	adantr 480	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↦ (𝐼‘𝑥)))
19	18	eqcomd 2744	. . . 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 2778	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}))
21		ushgruhgr 27342	. . . . . . . . 9 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
22		eqid 2738	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
23	22	uhgrfun 27339	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
24	21, 23	syl 17	. . . . . . . 8 ⊢ (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
25	2	funeqi 6439	. . . . . . . 8 ⊢ (Fun 𝐼 ↔ Fun (iEdg‘𝐺))
26	24, 25	sylibr 233	. . . . . . 7 ⊢ (𝐺 ∈ USHGraph → Fun 𝐼)
27	26	adantr 480	. . . . . 6 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → Fun 𝐼)
28		dfimafn 6814	. . . . . 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 585	. . . . 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 6756	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝐼‘𝑖) = (𝐼‘𝑗))
31	30	eleq2d 2824	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝑁 ∈ (𝐼‘𝑖) ↔ 𝑁 ∈ (𝐼‘𝑗)))
32	31	elrab 3617	. . . . . . . . . 10 ⊢ (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} ↔ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)))
33		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) → 𝑗 ∈ dom 𝐼)
34		fvelrn 6936	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝐼 ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ∈ ran 𝐼)
35	2	eqcomi 2747	. . . . . . . . . . . . . . . . . . 19 ⊢ (iEdg‘𝐺) = 𝐼
36	35	rneqi 5835	. . . . . . . . . . . . . . . . . 18 ⊢ ran (iEdg‘𝐺) = ran 𝐼
37	36	eleq2i 2830	. . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗))) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
40	39	3adant3 1130	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
41		eleq1 2826	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝐼‘𝑗) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
42	41	eqcoms 2746	. . . . . . . . . . . . . . 15 ⊢ ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
43	42	3ad2ant3 1133	. . . . . . . . . . . . . 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 27322	. . . . . . . . . . . . . . . . . 18 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
47	46	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
48	45, 47	syl5eq 2791	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
49	48	eleq2d 2824	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
50	49	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
51	50	3ad2ant1 1131	. . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 1109	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → 𝑁 ∈ 𝑓)
58	52, 57	jca 511	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) ∧ (𝐼‘𝑗) = 𝑓) → (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓))
59	58	3exp 1117	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐼 ∧ 𝑁 ∈ (𝐼‘𝑗)) → ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓))))
60	32, 59	syl5bi 241	. . . . . . . . 9 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} → ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓))))
61	60	rexlimdv 3211	. . . . . . . 8 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓)))
62	24	funfnd 6449	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
63		fvelrnb 6812	. . . . . . . . . . . . 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 5802	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ dom (iEdg‘𝐺) = dom 𝐼
66	65	eleq2i 2830	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ dom (iEdg‘𝐺) ↔ 𝑗 ∈ dom 𝐼)
67	66	biimpi 215	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ dom (iEdg‘𝐺) → 𝑗 ∈ dom 𝐼)
68	67	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑗 ∈ dom 𝐼)
69	68	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ dom 𝐼)
70	35	fveq1i 6757	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((iEdg‘𝐺)‘𝑗) = (𝐼‘𝑗)
71	70	eqeq2i 2751	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 = ((iEdg‘𝐺)‘𝑗) ↔ 𝑓 = (𝐼‘𝑗))
72	71	biimpi 215	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = ((iEdg‘𝐺)‘𝑗) → 𝑓 = (𝐼‘𝑗))
73	72	eqcoms 2746	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → 𝑓 = (𝐼‘𝑗))
74	73	eleq2d 2824	. . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) → (((iEdg‘𝐺)‘𝑗) = 𝑓 → 𝑁 ∈ (𝐼‘𝑗)))
77	76	adantld 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑁 ∈ (𝐼‘𝑗)))
78	77	imp 406	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑁 ∈ (𝐼‘𝑗))
79	69, 78	jca 511	. . . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 ↔ (𝐼‘𝑗) = 𝑓)
82	81	biimpi 215	. . . . . . . . . . . . . . . . . . 19 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼‘𝑗) = 𝑓)
83	82	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝐼‘𝑗) = 𝑓)
84	83	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑓) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝐼‘𝑗) = 𝑓)
85	80, 84	jca 511	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 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 3198	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 410	. . . . . . . . 9 ⊢ (𝐺 ∈ USHGraph → ((𝑓 ∈ 𝐸 ∧ 𝑁 ∈ 𝑓) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑓))
93	92	adantr 480	. . . . . . . 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 3426	. . . . . . . 8 ⊢ 𝑓 ∈ V
96		eqeq2 2750	. . . . . . . . 9 ⊢ (𝑒 = 𝑓 → ((𝐼‘𝑗) = 𝑒 ↔ (𝐼‘𝑗) = 𝑓))
97	96	rexbidv 3225	. . . . . . . 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 3602	. . . . . . 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 2827	. . . . . . . 8 ⊢ (𝑒 = 𝑓 → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ 𝑓))
100		ushgredgedg.b	. . . . . . . 8 ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}
101	99, 100	elrab2 3620	. . . . . . 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 2736	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} (𝐼‘𝑗) = 𝑒} = 𝐵)
104	29, 103	eqtrd 2778	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}) = 𝐵)
105	104	eqcomd 2744	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐵 = (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)}))
106	20, 10, 105	f1oeq123d 6694	. 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 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  {cab 2715  ∃wrex 3064  {crab 3067   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  {csn 4558   ↦ cmpt 5153  dom cdm 5580  ran crn 5581   ↾ cres 5582   “ cima 5583  Fun wfun 6412   Fn wfn 6413  ⟶wf 6414  –1-1→wf1 6415  –1-1-onto→wf1o 6417  ‘cfv 6418  Vtxcvtx 27269  iEdgciedg 27270  Edgcedg 27320  UHGraphcuhgr 27329  USHGraphcushgr 27330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-edg 27321  df-uhgr 27331  df-ushgr 27332

This theorem is referenced by:  usgredgedg  27500  vtxdushgrfvedglem  27759