Theorem usgredg2v 40454

Description: In a simple graph, the mapping of edges having a fixed endpoint to the other vertex of the edge is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)

Hypotheses

Ref	Expression
usgredg2v.v	⊢ 𝑉 = (Vtx‘𝐺)
usgredg2v.e	⊢ 𝐸 = (iEdg‘𝐺)
usgredg2v.a	⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}
usgredg2v.f	⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}))

Assertion

Ref	Expression
usgredg2v	⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1→𝑉)

Distinct variable groups:   𝑥,𝐸,𝑧   𝑧,𝐺   𝑥,𝑁,𝑧   𝑧,𝑉   𝑦,𝐴   𝑦,𝐸,𝑥,𝑧   𝑦,𝐺   𝑦,𝑁   𝑦,𝑉

Allowed substitution hints:   𝐴(𝑥,𝑧)   𝐹(𝑥,𝑦,𝑧)   𝐺(𝑥)   𝑉(𝑥)

Proof of Theorem usgredg2v

Dummy variables 𝑤 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgredg2v.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2		usgredg2v.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
3		usgredg2v.a	. . . . 5 ⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}
4	1, 2, 3	usgredg2vlem1 40452	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
5	4	ralrimiva 2949	. . 3 ⊢ (𝐺 ∈ USGraph → ∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
6	5	adantr 480	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
7	2	usgrf1 40402	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → 𝐸:dom 𝐸–1-1→ran 𝐸)
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐸:dom 𝐸–1-1→ran 𝐸)
9		elrabi 3328	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} → 𝑦 ∈ dom 𝐸)
10	9, 3	eleq2s 2706	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ dom 𝐸)
11		elrabi 3328	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} → 𝑤 ∈ dom 𝐸)
12	11, 3	eleq2s 2706	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐴 → 𝑤 ∈ dom 𝐸)
13	10, 12	anim12i 588	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑦 ∈ dom 𝐸 ∧ 𝑤 ∈ dom 𝐸))
14		f1fveq 6420	. . . . . . . 8 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ (𝑦 ∈ dom 𝐸 ∧ 𝑤 ∈ dom 𝐸)) → ((𝐸‘𝑦) = (𝐸‘𝑤) ↔ 𝑦 = 𝑤))
15	8, 13, 14	syl2an 493	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝐸‘𝑦) = (𝐸‘𝑤) ↔ 𝑦 = 𝑤))
16	15	bicomd 212	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑦 = 𝑤 ↔ (𝐸‘𝑦) = (𝐸‘𝑤)))
17	16	notbid 307	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (¬ 𝑦 = 𝑤 ↔ ¬ (𝐸‘𝑦) = (𝐸‘𝑤)))
18		simpl 472	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ USGraph )
19		simpl 472	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → 𝑦 ∈ 𝐴)
20	18, 19	anim12i 588	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝐴))
21		preq1 4212	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑧 → {𝑢, 𝑁} = {𝑧, 𝑁})
22	21	eqeq2d 2620	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → ((𝐸‘𝑦) = {𝑢, 𝑁} ↔ (𝐸‘𝑦) = {𝑧, 𝑁}))
23	22	cbvriotav 6522	. . . . . . . . 9 ⊢ (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁})
24	1, 2, 3	usgredg2vlem2 40453	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝐴) → ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) → (𝐸‘𝑦) = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}), 𝑁}))
25	20, 23, 24	mpisyl 21	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐸‘𝑦) = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}), 𝑁})
26		simpr 476	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ 𝐴)
27	18, 26	anim12i 588	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐺 ∈ USGraph ∧ 𝑤 ∈ 𝐴))
28	21	eqeq2d 2620	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → ((𝐸‘𝑤) = {𝑢, 𝑁} ↔ (𝐸‘𝑤) = {𝑧, 𝑁}))
29	28	cbvriotav 6522	. . . . . . . . 9 ⊢ (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})
30	1, 2, 3	usgredg2vlem2 40453	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ 𝑤 ∈ 𝐴) → ((℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}) → (𝐸‘𝑤) = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}), 𝑁}))
31	27, 29, 30	mpisyl 21	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐸‘𝑤) = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}), 𝑁})
32	25, 31	eqeq12d 2625	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝐸‘𝑦) = (𝐸‘𝑤) ↔ {(℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}), 𝑁} = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}), 𝑁}))
33	32	notbid 307	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (¬ (𝐸‘𝑦) = (𝐸‘𝑤) ↔ ¬ {(℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}), 𝑁} = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}), 𝑁}))
34		riotaex 6515	. . . . . . . . . . . 12 ⊢ (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) ∈ V
35	34	a1i 11	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑉 → (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) ∈ V)
36		id 22	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ 𝑉)
37		riotaex 6515	. . . . . . . . . . . 12 ⊢ (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∈ V
38	37	a1i 11	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑉 → (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∈ V)
39		preq12bg 4326	. . . . . . . . . . 11 ⊢ ((((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) ∈ V ∧ 𝑁 ∈ 𝑉) ∧ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∈ V ∧ 𝑁 ∈ 𝑉)) → ({(℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}), 𝑁} = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}), 𝑁} ↔ (((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁})))))
40	35, 36, 38, 36, 39	syl22anc 1319	. . . . . . . . . 10 ⊢ (𝑁 ∈ 𝑉 → ({(℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}), 𝑁} = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}), 𝑁} ↔ (((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁})))))
41	40	notbid 307	. . . . . . . . 9 ⊢ (𝑁 ∈ 𝑉 → (¬ {(℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}), 𝑁} = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}), 𝑁} ↔ ¬ (((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁})))))
42	41	adantl 481	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (¬ {(℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}), 𝑁} = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}), 𝑁} ↔ ¬ (((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁})))))
43		ioran 510	. . . . . . . . . . 11 ⊢ (¬ (((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}))) ↔ (¬ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∧ ¬ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}))))
44		ianor 508	. . . . . . . . . . . . 13 ⊢ (¬ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ↔ (¬ (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∨ ¬ 𝑁 = 𝑁))
45	23, 29	eqeq12i 2624	. . . . . . . . . . . . . . . . 17 ⊢ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ↔ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}))
46	45	notbii 309	. . . . . . . . . . . . . . . 16 ⊢ (¬ (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ↔ ¬ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}))
47	46	biimpi 205	. . . . . . . . . . . . . . 15 ⊢ (¬ (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) → ¬ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}))
48	47	a1d 25	. . . . . . . . . . . . . 14 ⊢ (¬ (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) → (𝐺 ∈ USGraph → ¬ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})))
49		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ 𝑁 = 𝑁
50	49	pm2.24i 145	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑁 = 𝑁 → (𝐺 ∈ USGraph → ¬ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})))
51	48, 50	jaoi 393	. . . . . . . . . . . . 13 ⊢ ((¬ (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∨ ¬ 𝑁 = 𝑁) → (𝐺 ∈ USGraph → ¬ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})))
52	44, 51	sylbi 206	. . . . . . . . . . . 12 ⊢ (¬ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) → (𝐺 ∈ USGraph → ¬ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})))
53	52	adantr 480	. . . . . . . . . . 11 ⊢ ((¬ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∧ ¬ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}))) → (𝐺 ∈ USGraph → ¬ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})))
54	43, 53	sylbi 206	. . . . . . . . . 10 ⊢ (¬ (((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}))) → (𝐺 ∈ USGraph → ¬ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})))
55	54	com12 32	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → (¬ (((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}))) → ¬ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})))
56	55	adantr 480	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (¬ (((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}))) → ¬ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})))
57	42, 56	sylbid 229	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (¬ {(℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}), 𝑁} = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}), 𝑁} → ¬ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})))
58	57	adantr 480	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (¬ {(℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}), 𝑁} = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}), 𝑁} → ¬ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})))
59	33, 58	sylbid 229	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (¬ (𝐸‘𝑦) = (𝐸‘𝑤) → ¬ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})))
60	17, 59	sylbid 229	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (¬ 𝑦 = 𝑤 → ¬ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})))
61	60	con4d 113	. . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}) → 𝑦 = 𝑤))
62	61	ralrimivva 2954	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑦 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}) → 𝑦 = 𝑤))
63		usgredg2v.f	. . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}))
64		fveq2 6103	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝐸‘𝑦) = (𝐸‘𝑤))
65	64	eqeq1d 2612	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝐸‘𝑦) = {𝑧, 𝑁} ↔ (𝐸‘𝑤) = {𝑧, 𝑁}))
66	65	riotabidv 6513	. . 3 ⊢ (𝑦 = 𝑤 → (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}))
67	63, 66	f1mpt 6419	. 2 ⊢ (𝐹:𝐴–1-1→𝑉 ↔ (∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}) → 𝑦 = 𝑤)))
68	6, 62, 67	sylanbrc 695	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1→𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  {cpr 4127   ↦ cmpt 4643  dom cdm 5038  ran crn 5039  –1-1→wf1 5801  ‘cfv 5804  ℩crio 6510  Vtxcvtx 25673  iEdgciedg 25674   USGraph cusgr 40379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-umgr 25750  df-edga 25793  df-usgr 40381

This theorem is referenced by:  usgredgleord  40455