Theorem ushgredgedgloop 29869

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⟶1-1onto𝐵)

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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		ushgredgedgloop.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	ushgrf 29698	. . . 4 ⊢ (𝐺 ∈ USHGraph → 𝐼:dom 𝐼⟶1-1(𝒫 (Vtx‘𝐺) ∖ {∅}))
4	3	adantr 486	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐼:dom 𝐼⟶1-1(𝒫 (Vtx‘𝐺) ∖ {∅}))
5		ssrab2 4008	. . 3 ⊢ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ⊆ dom 𝐼
6		f1ores 6869	. . 3 ⊢ ((𝐼:dom 𝐼⟶1-1(𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ⊆ dom 𝐼) → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}):{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}⟶1-1onto(𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}))
7	4, 5, 6	sylancl 598	. 2 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}):{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}⟶1-1onto(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 11	. . . . . 6 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐴 = {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}})
11		eqidd 2764	. . . . . 6 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → (𝐼‘𝑥) = (𝐼‘𝑥))
12	10, 11	mpteq12dva 5197	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥)) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ↦ (𝐼‘𝑥)))
13	8, 12	eqtrid 2810	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ↦ (𝐼‘𝑥)))
14		f1f 6808	. . . . . . 7 ⊢ (𝐼:dom 𝐼⟶1-1(𝒫 (Vtx‘𝐺) ∖ {∅}) → 𝐼:dom 𝐼⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
15	3, 14	syl 18	. . . . . 6 ⊢ (𝐺 ∈ USHGraph → 𝐼:dom 𝐼⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
16	5	a1i 11	. . . . . 6 ⊢ (𝐺 ∈ USHGraph → {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ⊆ dom 𝐼)
17	15, 16	feqresmpt 6985	. . . . 5 ⊢ (𝐺 ∈ USHGraph → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ↦ (𝐼‘𝑥)))
18	17	adantr 486	. . . 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 2801	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}))
20		ushgruhgr 29704	. . . . . . . 8 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
21		eqid 2763	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
22	21	uhgrfun 29701	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
23	20, 22	syl 18	. . . . . . 7 ⊢ (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
24	2	funeqi 6588	. . . . . . 7 ⊢ (Fun 𝐼 ↔ Fun (iEdg‘𝐺))
25	23, 24	sylibr 237	. . . . . 6 ⊢ (𝐺 ∈ USHGraph → Fun 𝐼)
26	25	adantr 486	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → Fun 𝐼)
27		dfimafn 6978	. . . . 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 598	. . . 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 6925	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝐼‘𝑖) = {𝑁} ↔ (𝐼‘𝑗) = {𝑁}))
30	29	elrab 3648	. . . . . . . . 9 ⊢ (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ↔ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}))
31		simpl 488	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) → 𝑗 ∈ dom 𝐼)
32		fvelrn 7108	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐼 ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ∈ ran 𝐼)
33	2	eqcomi 2772	. . . . . . . . . . . . . . . . 17 ⊢ (iEdg‘𝐺) = 𝐼
34	33	rneqi 5931	. . . . . . . . . . . . . . . 16 ⊢ ran (iEdg‘𝐺) = ran 𝐼
35	32, 34	eleqtrrdi 2875	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐼 ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
36	26, 31, 35	syl2an 608	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁})) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
37	36	3adant3 1150	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
38		eleq1 2852	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝐼‘𝑗) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
39	38	eqcoms 2771	. . . . . . . . . . . . . 14 ⊢ ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
40	39	3ad2ant3 1153	. . . . . . . . . . . . 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 260	. . . . . . . . . . . 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		edgval 29684	. . . . . . . . . . . . . . . . 17 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
44	43	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ USHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
45	42, 44	eqtrid 2810	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
46	45	eleq2d 2850	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
47	46	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
48	47	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
49	41, 48	mpbird 260	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → 𝑓 ∈ 𝐸)
50		eqeq1 2767	. . . . . . . . . . . . . . 15 ⊢ ((𝐼‘𝑗) = 𝑓 → ((𝐼‘𝑗) = {𝑁} ↔ 𝑓 = {𝑁}))
51	50	biimpcd 252	. . . . . . . . . . . . . 14 ⊢ ((𝐼‘𝑗) = {𝑁} → ((𝐼‘𝑗) = 𝑓 → 𝑓 = {𝑁}))
52	51	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) → ((𝐼‘𝑗) = 𝑓 → 𝑓 = {𝑁}))
53	52	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) → ((𝐼‘𝑗) = 𝑓 → 𝑓 = {𝑁})))
54	53	3imp 1128	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → 𝑓 = {𝑁})
55	49, 54	jca 521	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}))
56	55	3exp 1137	. . . . . . . . 9 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) → ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}))))
57	30, 56	biimtrid 245	. . . . . . . 8 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} → ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}))))
58	57	rexlimdv 3165	. . . . . . 7 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁})))
59	23	funfnd 6599	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
60		fvelrnb 6976	. . . . . . . . . . . 12 ⊢ ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
61	59, 60	syl 18	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
62	33	dmeqi 5898	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom (iEdg‘𝐺) = dom 𝐼
63	62	eleq2i 2856	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ dom (iEdg‘𝐺) ↔ 𝑗 ∈ dom 𝐼)
64	63	biimpi 219	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ dom (iEdg‘𝐺) → 𝑗 ∈ dom 𝐼)
65	64	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑗 ∈ dom 𝐼)
66	65	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ dom 𝐼)
67	33	fveq1i 6917	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((iEdg‘𝐺)‘𝑗) = (𝐼‘𝑗)
68	67	eqeq2i 2776	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = ((iEdg‘𝐺)‘𝑗) ↔ 𝑓 = (𝐼‘𝑗))
69	68	biimpi 219	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = ((iEdg‘𝐺)‘𝑗) → 𝑓 = (𝐼‘𝑗))
70	69	eqcoms 2771	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → 𝑓 = (𝐼‘𝑗))
71	70	eqeq1d 2765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑓 = {𝑁} ↔ (𝐼‘𝑗) = {𝑁}))
72	71	biimpcd 252	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = {𝑁} → (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼‘𝑗) = {𝑁}))
73	72	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) → (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼‘𝑗) = {𝑁}))
74	73	adantld 496	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝐼‘𝑗) = {𝑁}))
75	74	imp 412	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝐼‘𝑗) = {𝑁})
76	66, 75	jca 521	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}))
77	76, 30	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}})
78	67	eqeq1i 2768	. . . . . . . . . . . . . . . . . . 19 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 ↔ (𝐼‘𝑗) = 𝑓)
79	78	biimpi 219	. . . . . . . . . . . . . . . . . 18 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼‘𝑗) = 𝑓)
80	79	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝐼‘𝑗) = 𝑓)
81	80	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝐼‘𝑗) = 𝑓)
82	77, 81	jca 521	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ∧ (𝐼‘𝑗) = 𝑓))
83	82	ex 418	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ∧ (𝐼‘𝑗) = 𝑓)))
84	83	reximdv2 3176	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓))
85	84	ex 418	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USHGraph → (𝑓 = {𝑁} → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓)))
86	85	com23 87	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USHGraph → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑓 = {𝑁} → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓)))
87	61, 86	sylbid 243	. . . . . . . . . 10 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) → (𝑓 = {𝑁} → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓)))
88	46, 87	sylbid 243	. . . . . . . . 9 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ 𝐸 → (𝑓 = {𝑁} → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓)))
89	88	impd 416	. . . . . . . 8 ⊢ (𝐺 ∈ USHGraph → ((𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓))
90	89	adantr 486	. . . . . . 7 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓))
91	58, 90	impbid 215	. . . . . 6 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓 ↔ (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁})))
92		vex 3458	. . . . . . 7 ⊢ 𝑓 ∈ V
93		eqeq2 2775	. . . . . . . 8 ⊢ (𝑒 = 𝑓 → ((𝐼‘𝑗) = 𝑒 ↔ (𝐼‘𝑗) = 𝑓))
94	93	rexbidv 3190	. . . . . . 7 ⊢ (𝑒 = 𝑓 → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑒 ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓))
95	92, 94	elab 3636	. . . . . 6 ⊢ (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑒} ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓)
96		eqeq1 2767	. . . . . . 7 ⊢ (𝑒 = 𝑓 → (𝑒 = {𝑁} ↔ 𝑓 = {𝑁}))
97		ushgredgedgloop.b	. . . . . . 7 ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑁}}
98	96, 97	elrab2 3652	. . . . . 6 ⊢ (𝑓 ∈ 𝐵 ↔ (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}))
99	91, 95, 98	3bitr4g 317	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑒} ↔ 𝑓 ∈ 𝐵))
100	99	eqrdv 2761	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑒} = 𝐵)
101	28, 100	eqtr2d 2799	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐵 = (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}))
102	19, 10, 101	f1oeq123d 6848	. 2 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝐹:𝐴⟶1-1onto𝐵 ↔ (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}):{𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}⟶1-1onto(𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}})))
103	7, 102	mpbird 260	1 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴⟶1-1onto𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090  {crab 3416   ⊆ wss 3900   ∖ cdif 4048  ∅c0 4281  𝒫 cpw 4560  {csn 4588   ↦ cmpt 5192  dom cdm 5658  ran crn 5659   ↾ cres 5660   “ cima 5661  Fun wfun 6560   Fn wfn 6561  ⟶wf 6562  ⟶1-1wf1 6563  ⟶1-1ontowf1o 6565  ‘cfv 6566  Vtxcvtx 29631  iEdgciedg 29632  Edgcedg 29682  UHGraphcuhgr 29691  USHGraphcushgr 29692

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-edg 29683  df-uhgr 29693  df-ushgr 29694

This theorem is used by:  vtxdushgrfvedg  30128