Theorem ushgredgedgloop 28756

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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2731	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		ushgredgedgloop.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	ushgrf 28591	. . . 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 4077	. . 3 ⊢ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ⊆ dom 𝐼
6		f1ores 6847	. . 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}}–1-1-onto→(𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}))
7	4, 5, 6	sylancl 585	. 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 11	. . . . . 6 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐴 = {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}})
11		eqidd 2732	. . . . . 6 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → (𝐼‘𝑥) = (𝐼‘𝑥))
12	10, 11	mpteq12dva 5237	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥)) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ↦ (𝐼‘𝑥)))
13	8, 12	eqtrid 2783	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ↦ (𝐼‘𝑥)))
14		f1f 6787	. . . . . . 7 ⊢ (𝐼:dom 𝐼–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}) → 𝐼:dom 𝐼⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
15	3, 14	syl 17	. . . . . 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 6961	. . . . 5 ⊢ (𝐺 ∈ USHGraph → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ↦ (𝐼‘𝑥)))
18	17	adantr 480	. . . 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 2774	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}))
20		ushgruhgr 28597	. . . . . . . 8 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
21		eqid 2731	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
22	21	uhgrfun 28594	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
23	20, 22	syl 17	. . . . . . 7 ⊢ (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
24	2	funeqi 6569	. . . . . . 7 ⊢ (Fun 𝐼 ↔ Fun (iEdg‘𝐺))
25	23, 24	sylibr 233	. . . . . 6 ⊢ (𝐺 ∈ USHGraph → Fun 𝐼)
26	25	adantr 480	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → Fun 𝐼)
27		dfimafn 6954	. . . . 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 585	. . . 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 6900	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝐼‘𝑖) = {𝑁} ↔ (𝐼‘𝑗) = {𝑁}))
30	29	elrab 3683	. . . . . . . . 9 ⊢ (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ↔ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}))
31		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) → 𝑗 ∈ dom 𝐼)
32		fvelrn 7078	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐼 ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ∈ ran 𝐼)
33	2	eqcomi 2740	. . . . . . . . . . . . . . . . 17 ⊢ (iEdg‘𝐺) = 𝐼
34	33	rneqi 5936	. . . . . . . . . . . . . . . 16 ⊢ ran (iEdg‘𝐺) = ran 𝐼
35	32, 34	eleqtrrdi 2843	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐼 ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
36	26, 31, 35	syl2an 595	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁})) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
37	36	3adant3 1131	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
38		eleq1 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝐼‘𝑗) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
39	38	eqcoms 2739	. . . . . . . . . . . . . 14 ⊢ ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
40	39	3ad2ant3 1134	. . . . . . . . . . . . 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 257	. . . . . . . . . . . 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 28577	. . . . . . . . . . . . . . . . 17 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
44	43	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ USHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
45	42, 44	eqtrid 2783	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
46	45	eleq2d 2818	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
47	46	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
48	47	3ad2ant1 1132	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
49	41, 48	mpbird 257	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → 𝑓 ∈ 𝐸)
50		eqeq1 2735	. . . . . . . . . . . . . . 15 ⊢ ((𝐼‘𝑗) = 𝑓 → ((𝐼‘𝑗) = {𝑁} ↔ 𝑓 = {𝑁}))
51	50	biimpcd 248	. . . . . . . . . . . . . 14 ⊢ ((𝐼‘𝑗) = {𝑁} → ((𝐼‘𝑗) = 𝑓 → 𝑓 = {𝑁}))
52	51	adantl 481	. . . . . . . . . . . . 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 1110	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → 𝑓 = {𝑁})
55	49, 54	jca 511	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}))
56	55	3exp 1118	. . . . . . . . 9 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) → ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}))))
57	30, 56	biimtrid 241	. . . . . . . 8 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} → ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}))))
58	57	rexlimdv 3152	. . . . . . 7 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁})))
59	23	funfnd 6579	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
60		fvelrnb 6952	. . . . . . . . . . . 12 ⊢ ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
61	59, 60	syl 17	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
62	33	dmeqi 5904	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom (iEdg‘𝐺) = dom 𝐼
63	62	eleq2i 2824	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ dom (iEdg‘𝐺) ↔ 𝑗 ∈ dom 𝐼)
64	63	biimpi 215	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ dom (iEdg‘𝐺) → 𝑗 ∈ dom 𝐼)
65	64	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑗 ∈ dom 𝐼)
66	65	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ dom 𝐼)
67	33	fveq1i 6892	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((iEdg‘𝐺)‘𝑗) = (𝐼‘𝑗)
68	67	eqeq2i 2744	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = ((iEdg‘𝐺)‘𝑗) ↔ 𝑓 = (𝐼‘𝑗))
69	68	biimpi 215	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = ((iEdg‘𝐺)‘𝑗) → 𝑓 = (𝐼‘𝑗))
70	69	eqcoms 2739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → 𝑓 = (𝐼‘𝑗))
71	70	eqeq1d 2733	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑓 = {𝑁} ↔ (𝐼‘𝑗) = {𝑁}))
72	71	biimpcd 248	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = {𝑁} → (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼‘𝑗) = {𝑁}))
73	72	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) → (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼‘𝑗) = {𝑁}))
74	73	adantld 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝐼‘𝑗) = {𝑁}))
75	74	imp 406	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝐼‘𝑗) = {𝑁})
76	66, 75	jca 511	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}))
77	76, 30	sylibr 233	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}})
78	67	eqeq1i 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 ↔ (𝐼‘𝑗) = 𝑓)
79	78	biimpi 215	. . . . . . . . . . . . . . . . . 18 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼‘𝑗) = 𝑓)
80	79	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝐼‘𝑗) = 𝑓)
81	80	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝐼‘𝑗) = 𝑓)
82	77, 81	jca 511	. . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . 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 3163	. . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 86	. . . . . . . . . . 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 239	. . . . . . . . . 10 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) → (𝑓 = {𝑁} → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓)))
88	46, 87	sylbid 239	. . . . . . . . 9 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ 𝐸 → (𝑓 = {𝑁} → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓)))
89	88	impd 410	. . . . . . . 8 ⊢ (𝐺 ∈ USHGraph → ((𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓))
90	89	adantr 480	. . . . . . 7 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓))
91	58, 90	impbid 211	. . . . . 6 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓 ↔ (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁})))
92		vex 3477	. . . . . . 7 ⊢ 𝑓 ∈ V
93		eqeq2 2743	. . . . . . . 8 ⊢ (𝑒 = 𝑓 → ((𝐼‘𝑗) = 𝑒 ↔ (𝐼‘𝑗) = 𝑓))
94	93	rexbidv 3177	. . . . . . 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 3668	. . . . . 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 2735	. . . . . . 7 ⊢ (𝑒 = 𝑓 → (𝑒 = {𝑁} ↔ 𝑓 = {𝑁}))
97		ushgredgedgloop.b	. . . . . . 7 ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑁}}
98	96, 97	elrab2 3686	. . . . . 6 ⊢ (𝑓 ∈ 𝐵 ↔ (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}))
99	91, 95, 98	3bitr4g 314	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑒} ↔ 𝑓 ∈ 𝐵))
100	99	eqrdv 2729	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑒} = 𝐵)
101	28, 100	eqtr2d 2772	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐵 = (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}))
102	19, 10, 101	f1oeq123d 6827	. 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}})))
103	7, 102	mpbird 257	1 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  {cab 2708  ∃wrex 3069  {crab 3431   ∖ cdif 3945   ⊆ wss 3948  ∅c0 4322  𝒫 cpw 4602  {csn 4628   ↦ cmpt 5231  dom cdm 5676  ran crn 5677   ↾ cres 5678   “ cima 5679  Fun wfun 6537   Fn wfn 6538  ⟶wf 6539  –1-1→wf1 6540  –1-1-onto→wf1o 6542  ‘cfv 6543  Vtxcvtx 28524  iEdgciedg 28525  Edgcedg 28575  UHGraphcuhgr 28584  USHGraphcushgr 28585

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 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-edg 28576  df-uhgr 28586  df-ushgr 28587

This theorem is referenced by:  vtxdushgrfvedg  29015