Theorem ushgredgedgloopOLD 26535

Description: Obsolete version of ushgredgedgloop 26534 as of 6-Jul-2022. (Contributed by AV, 11-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
ushgredgedgloopOLD	⊢ ((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 ushgredgedgloopOLD

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

Step	Hyp	Ref	Expression
1		eqid 2825	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		ushgredgedgloopOLD.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	ushgrf 26368	. . . 4 ⊢ (𝐺 ∈ USHGraph → 𝐼:dom 𝐼–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
4	3	adantr 474	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐼:dom 𝐼–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
5		ssrab2 3914	. . 3 ⊢ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ⊆ dom 𝐼
6		f1ores 6396	. . 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 580	. 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		ushgredgedgloopOLD.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥))
9		ushgredgedgloopOLD.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 2826	. . . . . 6 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑥 ∈ 𝐴) → (𝐼‘𝑥) = (𝐼‘𝑥))
12	10, 11	mpteq12dva 4957	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥)) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ↦ (𝐼‘𝑥)))
13	8, 12	syl5eq 2873	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ↦ (𝐼‘𝑥)))
14		f1f 6342	. . . . . . 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 6501	. . . . 5 ⊢ (𝐺 ∈ USHGraph → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ↦ (𝐼‘𝑥)))
18	17	adantr 474	. . . 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 2864	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}))
20		ushgruhgr 26374	. . . . . . . 8 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
21		eqid 2825	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
22	21	uhgrfun 26371	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
23	20, 22	syl 17	. . . . . . 7 ⊢ (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
24	2	funeqi 6148	. . . . . . 7 ⊢ (Fun 𝐼 ↔ Fun (iEdg‘𝐺))
25	23, 24	sylibr 226	. . . . . 6 ⊢ (𝐺 ∈ USHGraph → Fun 𝐼)
26	25	adantr 474	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → Fun 𝐼)
27		dfimafn 6496	. . . . 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 580	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}) = {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑒})
29		fveq2 6437	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝐼‘𝑖) = (𝐼‘𝑗))
30	29	eqeq1d 2827	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝐼‘𝑖) = {𝑁} ↔ (𝐼‘𝑗) = {𝑁}))
31	30	elrab 3585	. . . . . . . . 9 ⊢ (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ↔ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}))
32		simpl 476	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) → 𝑗 ∈ dom 𝐼)
33		fvelrn 6606	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐼 ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ∈ ran 𝐼)
34	2	eqcomi 2834	. . . . . . . . . . . . . . . . 17 ⊢ (iEdg‘𝐺) = 𝐼
35	34	rneqi 5588	. . . . . . . . . . . . . . . 16 ⊢ ran (iEdg‘𝐺) = ran 𝐼
36	33, 35	syl6eleqr 2917	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐼 ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
37	26, 32, 36	syl2an 589	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁})) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
38	37	3adant3 1166	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → (𝐼‘𝑗) ∈ ran (iEdg‘𝐺))
39		eleq1 2894	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝐼‘𝑗) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
40	39	eqcoms 2833	. . . . . . . . . . . . . 14 ⊢ ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
41	40	3ad2ant3 1169	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼‘𝑗) ∈ ran (iEdg‘𝐺)))
42	38, 41	mpbird 249	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → 𝑓 ∈ ran (iEdg‘𝐺))
43		ushgredgedgloopOLD.e	. . . . . . . . . . . . . . . 16 ⊢ 𝐸 = (Edg‘𝐺)
44		edgval 26354	. . . . . . . . . . . . . . . . 17 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
45	44	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ USHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
46	43, 45	syl5eq 2873	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
47	46	eleq2d 2892	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
48	47	adantr 474	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
49	48	3ad2ant1 1167	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → (𝑓 ∈ 𝐸 ↔ 𝑓 ∈ ran (iEdg‘𝐺)))
50	42, 49	mpbird 249	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → 𝑓 ∈ 𝐸)
51		eqeq1 2829	. . . . . . . . . . . . . . 15 ⊢ ((𝐼‘𝑗) = 𝑓 → ((𝐼‘𝑗) = {𝑁} ↔ 𝑓 = {𝑁}))
52	51	biimpcd 241	. . . . . . . . . . . . . 14 ⊢ ((𝐼‘𝑗) = {𝑁} → ((𝐼‘𝑗) = 𝑓 → 𝑓 = {𝑁}))
53	52	adantl 475	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) → ((𝐼‘𝑗) = 𝑓 → 𝑓 = {𝑁}))
54	53	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) → ((𝐼‘𝑗) = 𝑓 → 𝑓 = {𝑁})))
55	54	3imp 1141	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → 𝑓 = {𝑁})
56	50, 55	jca 507	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) ∧ (𝐼‘𝑗) = 𝑓) → (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}))
57	56	3exp 1152	. . . . . . . . 9 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}) → ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}))))
58	31, 57	syl5bi 234	. . . . . . . 8 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} → ((𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}))))
59	58	rexlimdv 3239	. . . . . . 7 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓 → (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁})))
60		funfn 6157	. . . . . . . . . . . . . 14 ⊢ (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
61	60	biimpi 208	. . . . . . . . . . . . 13 ⊢ (Fun (iEdg‘𝐺) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
62	23, 61	syl 17	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
63		fvelrnb 6494	. . . . . . . . . . . 12 ⊢ ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
64	62, 63	syl 17	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
65	34	dmeqi 5561	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom (iEdg‘𝐺) = dom 𝐼
66	65	eleq2i 2898	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ dom (iEdg‘𝐺) ↔ 𝑗 ∈ dom 𝐼)
67	66	biimpi 208	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ dom (iEdg‘𝐺) → 𝑗 ∈ dom 𝐼)
68	67	adantr 474	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑗 ∈ dom 𝐼)
69	68	adantl 475	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ dom 𝐼)
70	34	fveq1i 6438	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((iEdg‘𝐺)‘𝑗) = (𝐼‘𝑗)
71	70	eqeq2i 2837	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = ((iEdg‘𝐺)‘𝑗) ↔ 𝑓 = (𝐼‘𝑗))
72	71	biimpi 208	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = ((iEdg‘𝐺)‘𝑗) → 𝑓 = (𝐼‘𝑗))
73	72	eqcoms 2833	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → 𝑓 = (𝐼‘𝑗))
74	73	eqeq1d 2827	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑓 = {𝑁} ↔ (𝐼‘𝑗) = {𝑁}))
75	74	biimpcd 241	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = {𝑁} → (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼‘𝑗) = {𝑁}))
76	75	adantl 475	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) → (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼‘𝑗) = {𝑁}))
77	76	adantld 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝐼‘𝑗) = {𝑁}))
78	77	imp 397	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝐼‘𝑗) = {𝑁})
79	69, 78	jca 507	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ dom 𝐼 ∧ (𝐼‘𝑗) = {𝑁}))
80	79, 31	sylibr 226	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}})
81	70	eqeq1i 2830	. . . . . . . . . . . . . . . . . . 19 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 ↔ (𝐼‘𝑗) = 𝑓)
82	81	biimpi 208	. . . . . . . . . . . . . . . . . 18 ⊢ (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼‘𝑗) = 𝑓)
83	82	adantl 475	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝐼‘𝑗) = 𝑓)
84	83	adantl 475	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝐼‘𝑗) = 𝑓)
85	80, 84	jca 507	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ∧ (𝐼‘𝑗) = 𝑓))
86	85	ex 403	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} ∧ (𝐼‘𝑗) = 𝑓)))
87	86	reximdv2 3222	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓))
88	87	ex 403	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USHGraph → (𝑓 = {𝑁} → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓)))
89	88	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)))
90	64, 89	sylbid 232	. . . . . . . . . 10 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) → (𝑓 = {𝑁} → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓)))
91	47, 90	sylbid 232	. . . . . . . . 9 ⊢ (𝐺 ∈ USHGraph → (𝑓 ∈ 𝐸 → (𝑓 = {𝑁} → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓)))
92	91	impd 400	. . . . . . . 8 ⊢ (𝐺 ∈ USHGraph → ((𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓))
93	92	adantr 474	. . . . . . 7 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓))
94	59, 93	impbid 204	. . . . . 6 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓 ↔ (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁})))
95		vex 3417	. . . . . . 7 ⊢ 𝑓 ∈ V
96		eqeq2 2836	. . . . . . . 8 ⊢ (𝑒 = 𝑓 → ((𝐼‘𝑗) = 𝑒 ↔ (𝐼‘𝑗) = 𝑓))
97	96	rexbidv 3262	. . . . . . 7 ⊢ (𝑒 = 𝑓 → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑒 ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓))
98	95, 97	elab 3571	. . . . . 6 ⊢ (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑒} ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑓)
99		eqeq1 2829	. . . . . . 7 ⊢ (𝑒 = 𝑓 → (𝑒 = {𝑁} ↔ 𝑓 = {𝑁}))
100		ushgredgedgloopOLD.b	. . . . . . 7 ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑁}}
101	99, 100	elrab2 3589	. . . . . 6 ⊢ (𝑓 ∈ 𝐵 ↔ (𝑓 ∈ 𝐸 ∧ 𝑓 = {𝑁}))
102	94, 98, 101	3bitr4g 306	. . . . 5 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑒} ↔ 𝑓 ∈ 𝐵))
103	102	eqrdv 2823	. . . 4 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} (𝐼‘𝑗) = 𝑒} = 𝐵)
104	28, 103	eqtr2d 2862	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐵 = (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}}))
105	19, 10, 104	f1oeq123d 6377	. 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}})))
106	7, 105	mpbird 249	1 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  {cab 2811  ∃wrex 3118  {crab 3121   ∖ cdif 3795   ⊆ wss 3798  ∅c0 4146  𝒫 cpw 4380  {csn 4399   ↦ cmpt 4954  dom cdm 5346  ran crn 5347   ↾ cres 5348   “ cima 5349  Fun wfun 6121   Fn wfn 6122  ⟶wf 6123  –1-1→wf1 6124  –1-1-onto→wf1o 6126  ‘cfv 6127  Vtxcvtx 26301  iEdgciedg 26302  Edgcedg 26352  UHGraphcuhgr 26361  USHGraphcushgr 26362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-edg 26353  df-uhgr 26363  df-ushgr 26364

This theorem is referenced by: (None)