Theorem usgredg2v 26522

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 26520	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
5	4	ralrimiva 3174	. . 3 ⊢ (𝐺 ∈ USGraph → ∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
6	5	adantr 474	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
7	2	usgrf1 26470	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → 𝐸:dom 𝐸–1-1→ran 𝐸)
8	7	adantr 474	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐸:dom 𝐸–1-1→ran 𝐸)
9		elrabi 3579	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} → 𝑦 ∈ dom 𝐸)
10	9, 3	eleq2s 2923	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ dom 𝐸)
11		elrabi 3579	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} → 𝑤 ∈ dom 𝐸)
12	11, 3	eleq2s 2923	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐴 → 𝑤 ∈ dom 𝐸)
13	10, 12	anim12i 608	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑦 ∈ dom 𝐸 ∧ 𝑤 ∈ dom 𝐸))
14		f1fveq 6773	. . . . . . . 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 591	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝐸‘𝑦) = (𝐸‘𝑤) ↔ 𝑦 = 𝑤))
16	15	bicomd 215	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑦 = 𝑤 ↔ (𝐸‘𝑦) = (𝐸‘𝑤)))
17	16	notbid 310	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (¬ 𝑦 = 𝑤 ↔ ¬ (𝐸‘𝑦) = (𝐸‘𝑤)))
18		simpl 476	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ USGraph)
19		simpl 476	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → 𝑦 ∈ 𝐴)
20	18, 19	anim12i 608	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝐴))
21		preq1 4485	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑧 → {𝑢, 𝑁} = {𝑧, 𝑁})
22	21	eqeq2d 2834	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → ((𝐸‘𝑦) = {𝑢, 𝑁} ↔ (𝐸‘𝑦) = {𝑧, 𝑁}))
23	22	cbvriotav 6876	. . . . . . . . 9 ⊢ (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁})
24	1, 2, 3	usgredg2vlem2 26521	. . . . . . . . 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 479	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ 𝐴)
27	18, 26	anim12i 608	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐺 ∈ USGraph ∧ 𝑤 ∈ 𝐴))
28	21	eqeq2d 2834	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → ((𝐸‘𝑤) = {𝑢, 𝑁} ↔ (𝐸‘𝑤) = {𝑧, 𝑁}))
29	28	cbvriotav 6876	. . . . . . . . 9 ⊢ (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})
30	1, 2, 3	usgredg2vlem2 26521	. . . . . . . . 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 2839	. . . . . . 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 310	. . . . . 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 6869	. . . . . . . . . . . 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 6869	. . . . . . . . . . . 12 ⊢ (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∈ V
38	37	a1i 11	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑉 → (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∈ V)
39		preq12bg 4600	. . . . . . . . . . 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 874	. . . . . . . . . 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 310	. . . . . . . . 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 475	. . . . . . . 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 1013	. . . . . . . . . . 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 1011	. . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . . . . 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 312	. . . . . . . . . . . . . . . 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 208	. . . . . . . . . . . . . . 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 2824	. . . . . . . . . . . . . . 15 ⊢ 𝑁 = 𝑁
50	49	pm2.24i 148	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑁 = 𝑁 → (𝐺 ∈ USGraph → ¬ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})))
51	48, 50	jaoi 890	. . . . . . . . . . . . 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 209	. . . . . . . . . . . 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 474	. . . . . . . . . . 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 209	. . . . . . . . . 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 474	. . . . . . . 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 232	. . . . . . 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 474	. . . . . 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 232	. . . . 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 232	. . . 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 115	. . 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 3179	. 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		fveqeq2 6441	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝐸‘𝑦) = {𝑧, 𝑁} ↔ (𝐸‘𝑤) = {𝑧, 𝑁}))
65	64	riotabidv 6867	. . 3 ⊢ (𝑦 = 𝑤 → (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}))
66	63, 65	f1mpt 6772	. 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)))
67	6, 62, 66	sylanbrc 580	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1→𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 880   = wceq 1658   ∈ wcel 2166  ∀wral 3116  {crab 3120  Vcvv 3413  {cpr 4398   ↦ cmpt 4951  dom cdm 5341  ran crn 5342  –1-1→wf1 6119  ‘cfv 6122  ℩crio 6864  Vtxcvtx 26293  iEdgciedg 26294  USGraphcusgr 26447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208  ax-cnex 10307  ax-resscn 10308  ax-1cn 10309  ax-icn 10310  ax-addcl 10311  ax-addrcl 10312  ax-mulcl 10313  ax-mulrcl 10314  ax-mulcom 10315  ax-addass 10316  ax-mulass 10317  ax-distr 10318  ax-i2m1 10319  ax-1ne0 10320  ax-1rid 10321  ax-rnegex 10322  ax-rrecex 10323  ax-cnre 10324  ax-pre-lttri 10325  ax-pre-lttrn 10326  ax-pre-ltadd 10327  ax-pre-mulgt0 10328

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-nel 3102  df-ral 3121  df-rex 3122  df-reu 3123  df-rmo 3124  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-pss 3813  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-tp 4401  df-op 4403  df-uni 4658  df-int 4697  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-tr 4975  df-id 5249  df-eprel 5254  df-po 5262  df-so 5263  df-fr 5300  df-we 5302  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-pred 5919  df-ord 5965  df-on 5966  df-lim 5967  df-suc 5968  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-riota 6865  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-om 7326  df-1st 7427  df-2nd 7428  df-wrecs 7671  df-recs 7733  df-rdg 7771  df-1o 7825  df-2o 7826  df-oadd 7829  df-er 8008  df-en 8222  df-dom 8223  df-sdom 8224  df-fin 8225  df-card 9077  df-cda 9304  df-pnf 10392  df-mnf 10393  df-xr 10394  df-ltxr 10395  df-le 10396  df-sub 10586  df-neg 10587  df-nn 11350  df-2 11413  df-n0 11618  df-z 11704  df-uz 11968  df-fz 12619  df-hash 13410  df-edg 26345  df-umgr 26380  df-usgr 26449

This theorem is referenced by:  usgriedgleord  26523