Theorem ushgredgedgloop 16108

Description: In a simple hypergraph there is a 1-1 onto mapping between the indexed edges being loops at a fixed vertex 𝑁 and the set of loops at this vertex 𝑁. (Contributed by AV, 11-Dec-2020.) (Revised by AV, 6-Jul-2022.)

Hypotheses

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

Assertion

Ref	Expression
ushgredgedgloop	⊢ ((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 ushgredgedgloop

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

Step	Hyp	Ref	Expression
1		eqid 2230	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		ushgredgedgloop.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	ushgrfm 15954	. . . 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 3311	. . 3 ⊢ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ⊆ dom 𝐼
6		f1ores 5601	. . 3 ⊢ ((𝐼:dom 𝐼–1-1→{𝑝 ∈ 𝒫 (Vtx‘𝐺) ∣ ∃𝑤 𝑤 ∈ 𝑝} ∧ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ⊆ dom 𝐼) → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}):{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}–1-1-onto→(𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}))
7	4, 5, 6	sylancl 413	. 2 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}):{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}–1-1-onto→(𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}))
8		ushgredgedgloop.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥))
9		ushgredgedgloop.a	. . . . . . 7 ⊢ 𝐴 = {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}
10	9	a1i 9	. . . . . 6 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐴 = {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}})
11		eqidd 2231	. . . . . 6 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → (𝐼‘𝑥) = (𝐼‘𝑥))
12	10, 11	mpteq12dva 4171	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥)) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ↦ (𝐼‘𝑥)))
13	8, 12	eqtrid 2275	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ↦ (𝐼‘𝑥)))
14		f1f 5545	. . . . . . 7 ⊢ (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	. . . . . 6 ⊢ (𝐺 ∈ USHGraph → 𝐼:dom 𝐼⟶{𝑝 ∈ 𝒫 (Vtx‘𝐺) ∣ ∃𝑤 𝑤 ∈ 𝑝})
16	5	a1i 9	. . . . . 6 ⊢ (𝐺 ∈ USHGraph → {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ⊆ dom 𝐼)
17	15, 16	feqresmpt 5703	. . . . 5 ⊢ (𝐺 ∈ USHGraph → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ↦ (𝐼‘𝑥)))
18	17	adantr 276	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ↦ (𝐼‘𝑥)))
19	13, 18	eqtr4d 2266	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}))
20		ushgruhgr 15960	. . . . . . . 8 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
21		eqid 2230	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
22	21	uhgrfun 15957	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
23	20, 22	syl 14	. . . . . . 7 ⊢ (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
24	2	funeqi 5349	. . . . . . 7 ⊢ (Fun 𝐼 ↔ Fun (iEdg‘𝐺))
25	23, 24	sylibr 134	. . . . . 6 ⊢ (𝐺 ∈ USHGraph → Fun 𝐼)
26	25	adantr 276	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → Fun 𝐼)
27		dfimafn 5697	. . . . 5 ⊢ ((Fun 𝐼 ∧ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ⊆ dom 𝐼) → (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}) = {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑒})
28	26, 5, 27	sylancl 413	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}) = {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑒})
29		fveqeq2 5651	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝐼‘𝑖) = {𝑁} ↔ (𝐼‘𝑗) = {𝑁}))
30	29	elrab 2961	. . . . . . . . 9 ⊢ (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ↔ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}))
31		simpl 109	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) → 𝑗 ∈ dom 𝐼)
32		fvelrn 5781	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐼 ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ∈ ran 𝐼)
33	2	eqcomi 2234	. . . . . . . . . . . . . . . . 17 ⊢ (iEdg‘𝐺) = 𝐼
34	33	rneqi 4962	. . . . . . . . . . . . . . . 16 ⊢ ran (iEdg‘𝐺) = ran 𝐼
35	32, 34	eleqtrrdi 2324	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐼 ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
36	26, 31, 35	syl2an 289	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁})) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
37	36	3adant3 1043	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
38		eleq1 2293	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝐼‘𝑗) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
39	38	eqcoms 2233	. . . . . . . . . . . . . 14 ⊢ ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
40	39	3ad2ant3 1046	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
41	37, 40	mpbird 167	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → 𝑓 ∈ ran (iEdg‘𝐺))
42		ushgredgedgloop.e	. . . . . . . . . . . . . . . 16 ⊢ 𝐸 = (Edg‘𝐺)
43		edgvalg 15939	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ USHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
44	42, 43	eqtrid 2275	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
45	44	eleq2d 2300	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
46	45	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
47	46	3ad2ant1 1044	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
48	41, 47	mpbird 167	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → 𝑓 ∈ 𝐸)
49		eqeq1 2237	. . . . . . . . . . . . . . 15 ⊢ ((𝐼‘𝑗) = 𝑓 → ((𝐼‘𝑗) = {𝑁} ↔ 𝑓 = {𝑁}))
50	49	biimpcd 159	. . . . . . . . . . . . . 14 ⊢ ((𝐼‘𝑗) = {𝑁} → ((𝐼‘𝑗) = 𝑓 → 𝑓 = {𝑁}))
51	50	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) → ((𝐼‘𝑗) = 𝑓 → 𝑓 = {𝑁}))
52	51	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) → ((𝐼‘𝑗) = 𝑓 → 𝑓 = {𝑁})))
53	52	3imp 1219	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → 𝑓 = {𝑁})
54	48, 53	jca 306	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}))
55	54	3exp 1228	. . . . . . . . 9 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) → ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}))))
56	30, 55	biimtrid 152	. . . . . . . 8 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} → ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}))))
57	56	rexlimdv 2648	. . . . . . 7 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁})))
58	23	funfnd 5359	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
59		fvelrnb 5696	. . . . . . . . . . . 12 ⊢ ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
60	58, 59	syl 14	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
61	33	dmeqi 4934	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom (iEdg‘𝐺) = dom 𝐼
62	61	eleq2i 2297	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ dom (iEdg‘𝐺) ↔ 𝑗 ∈ dom 𝐼)
63	62	biimpi 120	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ dom (iEdg‘𝐺) → 𝑗 ∈ dom 𝐼)
64	63	adantr 276	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑗 ∈ dom 𝐼)
65	64	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ dom 𝐼)
66	33	fveq1i 5643	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((iEdg‘𝐺)‘𝑗) = (𝐼‘𝑗)
67	66	eqeq2i 2241	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = ((iEdg‘𝐺)‘𝑗) ↔ 𝑓 = (𝐼‘𝑗))
68	67	biimpi 120	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = ((iEdg‘𝐺)‘𝑗) → 𝑓 = (𝐼‘𝑗))
69	68	eqcoms 2233	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → 𝑓 = (𝐼‘𝑗))
70	69	eqeq1d 2239	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑓 = {𝑁} ↔ (𝐼‘𝑗) = {𝑁}))
71	70	biimpcd 159	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = {𝑁} → (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼‘𝑗) = {𝑁}))
72	71	adantl 277	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) → (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼‘𝑗) = {𝑁}))
73	72	adantld 278	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝐼‘𝑗) = {𝑁}))
74	73	imp 124	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝐼‘𝑗) = {𝑁})
75	65, 74	jca 306	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}))
76	75, 30	sylibr 134	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}})
77	66	eqeq1i 2238	. . . . . . . . . . . . . . . . . . 19 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 ↔ (𝐼‘𝑗) = 𝑓)
78	77	biimpi 120	. . . . . . . . . . . . . . . . . 18 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼‘𝑗) = 𝑓)
79	78	adantl 277	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝐼‘𝑗) = 𝑓)
80	79	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝐼‘𝑗) = 𝑓)
81	76, 80	jca 306	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ∧ (𝐼‘𝑗) = 𝑓))
82	81	ex 115	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ∧ (𝐼‘𝑗) = 𝑓)))
83	82	reximdv2 2630	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓))
84	83	ex 115	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USHGraph → (𝑓 = {𝑁} → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓)))
85	84	com23 78	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USHGraph → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑓 = {𝑁} → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓)))
86	60, 85	sylbid 150	. . . . . . . . . 10 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) → (𝑓 = {𝑁} → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓)))
87	45, 86	sylbid 150	. . . . . . . . 9 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ 𝐸 → (𝑓 = {𝑁} → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓)))
88	87	impd 254	. . . . . . . 8 ⊢ (𝐺 ∈ USHGraph → ((𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓))
89	88	adantr 276	. . . . . . 7 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓))
90	57, 89	impbid 129	. . . . . 6 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓 ↔ (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁})))
91		vex 2804	. . . . . . 7 ⊢ 𝑓 ∈ V
92		eqeq2 2240	. . . . . . . 8 ⊢ (𝑒 = 𝑓 → ((𝐼‘𝑗) = 𝑒 ↔ (𝐼‘𝑗) = 𝑓))
93	92	rexbidv 2532	. . . . . . 7 ⊢ (𝑒 = 𝑓 → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑒 ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓))
94	91, 93	elab 2949	. . . . . 6 ⊢ (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑒} ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓)
95		eqeq1 2237	. . . . . . 7 ⊢ (𝑒 = 𝑓 → (𝑒 = {𝑁} ↔ 𝑓 = {𝑁}))
96		ushgredgedgloop.b	. . . . . . 7 ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑁}}
97	95, 96	elrab2 2964	. . . . . 6 ⊢ (𝑓 ∈ 𝐵 ↔ (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}))
98	90, 94, 97	3bitr4g 223	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑒} ↔ 𝑓 ∈ 𝐵))
99	98	eqrdv 2228	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑒} = 𝐵)
100	28, 99	eqtr2d 2264	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐵 = (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}))
101	19, 10, 100	f1oeq123d 5580	. 2 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}):{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}–1-1-onto→(𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}})))
102	7, 101	mpbird 167	1 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004   = wceq 1397  ∃wex 1540   ∈ wcel 2201  {cab 2216  ∃wrex 2510  {crab 2513   ⊆ wss 3199  𝒫 cpw 3653  {csn 3670   ↦ cmpt 4151  dom cdm 4727  ran crn 4728   ↾ cres 4729   “ cima 4730  Fun wfun 5322   Fn wfn 5323  ⟶wf 5324  –1-1→wf1 5325  –1-1-onto→wf1o 5327  ‘cfv 5328  Vtxcvtx 15892  iEdgciedg 15893  Edgcedg 15937  UHGraphcuhgr 15947  USHGraphcushgr 15948

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-cnre 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-sub 8357  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-9 9214  df-n0 9408  df-dec 9617  df-ndx 13108  df-slot 13109  df-base 13111  df-edgf 15885  df-vtx 15894  df-iedg 15895  df-edg 15938  df-uhgrm 15949  df-ushgrm 15950

This theorem is referenced by:  vtxduspgrfvedgfi  16181