Theorem usgredg2v 29835

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⟶1-1𝑉)

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 29833	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
5	4	ralrimiva 3158	. . 3 ⊢ (𝐺 ∈ USGraph → ∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
6	5	adantr 486	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
7	2	usgrf1 29780	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → 𝐸:dom 𝐸⟶1-1ran 𝐸)
8	7	adantr 486	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐸:dom 𝐸⟶1-1ran 𝐸)
9		elrabi 3644	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} → 𝑦 ∈ dom 𝐸)
10	9, 3	eleq2s 2882	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ dom 𝐸)
11		elrabi 3644	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} → 𝑤 ∈ dom 𝐸)
12	11, 3	eleq2s 2882	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐴 → 𝑤 ∈ dom 𝐸)
13	10, 12	anim12i 625	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑦 ∈ dom 𝐸 ∧ 𝑤 ∈ dom 𝐸))
14		f1fveq 7286	. . . . . . . 8 ⊢ ((𝐸:dom 𝐸⟶1-1ran 𝐸 ∧ (𝑦 ∈ dom 𝐸 ∧ 𝑤 ∈ dom 𝐸)) → ((𝐸‘𝑦) = (𝐸‘𝑤) ↔ 𝑦 = 𝑤))
15	8, 13, 14	syl2an 608	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝐸‘𝑦) = (𝐸‘𝑤) ↔ 𝑦 = 𝑤))
16	15	bicomd 226	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑦 = 𝑤 ↔ (𝐸‘𝑦) = (𝐸‘𝑤)))
17	16	notbid 321	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (¬ 𝑦 = 𝑤 ↔ ¬ (𝐸‘𝑦) = (𝐸‘𝑤)))
18		simpl 488	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ USGraph)
19		simpl 488	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → 𝑦 ∈ 𝐴)
20	18, 19	anim12i 625	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝐴))
21		preq1 4694	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑧 → {𝑢, 𝑁} = {𝑧, 𝑁})
22	21	eqeq2d 2774	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → ((𝐸‘𝑦) = {𝑢, 𝑁} ↔ (𝐸‘𝑦) = {𝑧, 𝑁}))
23	22	cbvriotavw 7407	. . . . . . . . 9 ⊢ (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁})
24	1, 2, 3	usgredg2vlem2 29834	. . . . . . . . 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 22	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐸‘𝑦) = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}), 𝑁})
26		an3 672	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐺 ∈ USGraph ∧ 𝑤 ∈ 𝐴))
27	21	eqeq2d 2774	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → ((𝐸‘𝑤) = {𝑢, 𝑁} ↔ (𝐸‘𝑤) = {𝑧, 𝑁}))
28	27	cbvriotavw 7407	. . . . . . . . 9 ⊢ (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})
29	1, 2, 3	usgredg2vlem2 29834	. . . . . . . . 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}))
30	26, 28, 29	mpisyl 22	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐸‘𝑤) = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}), 𝑁})
31	25, 30	eqeq12d 2779	. . . . . . 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}))
32	31	notbid 321	. . . . . 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}))
33		riotaex 7401	. . . . . . . . . . . 12 ⊢ (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) ∈ V
34	33	a1i 11	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑉 → (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) ∈ V)
35		id 23	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ 𝑉)
36		riotaex 7401	. . . . . . . . . . . 12 ⊢ (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∈ V
37	36	a1i 11	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑉 → (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∈ V)
38		preq12bg 4811	. . . . . . . . . . 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})))))
39	34, 35, 37, 35, 38	syl22anc 852	. . . . . . . . . 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})))))
40	39	notbid 321	. . . . . . . . 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})))))
41	40	adantl 487	. . . . . . . 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})))))
42		ioran 999	. . . . . . . . . . 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}))))
43		ianor 997	. . . . . . . . . . . . 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))
44	23, 28	eqeq12i 2781	. . . . . . . . . . . . . . . . 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}))
45	44	notbii 323	. . . . . . . . . . . . . . . 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}))
46	45	biimpi 219	. . . . . . . . . . . . . . 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}))
47	46	a1d 26	. . . . . . . . . . . . . 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})))
48		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ 𝑁 = 𝑁
49	48	pm2.24i 151	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑁 = 𝑁 → (𝐺 ∈ USGraph → ¬ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})))
50	47, 49	jaoi 871	. . . . . . . . . . . . 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})))
51	43, 50	sylbi 220	. . . . . . . . . . . 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})))
52	51	adantr 486	. . . . . . . . . . 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})))
53	42, 52	sylbi 220	. . . . . . . . . 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})))
54	53	com12 33	. . . . . . . . 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})))
55	54	adantr 486	. . . . . . . 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})))
56	41, 55	sylbid 243	. . . . . . 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})))
57	56	adantr 486	. . . . . 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})))
58	32, 57	sylbid 243	. . . . 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})))
59	17, 58	sylbid 243	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (¬ 𝑦 = 𝑤 → ¬ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})))
60	59	con4d 116	. . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}) → 𝑦 = 𝑤))
61	60	ralrimivva 3209	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑦 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}) → 𝑦 = 𝑤))
62		usgredg2v.f	. . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}))
63		fveqeq2 6914	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝐸‘𝑦) = {𝑧, 𝑁} ↔ (𝐸‘𝑤) = {𝑧, 𝑁}))
64	63	riotabidv 7399	. . 3 ⊢ (𝑦 = 𝑤 → (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}))
65	62, 64	f1mpt 7285	. 2 ⊢ (𝐹:𝐴⟶1-1𝑉 ↔ (∀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)))
66	6, 61, 65	sylanbrc 595	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴⟶1-1𝑉)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  Vcvv 3454  {cpr 4590   ↦ cmpt 5189  dom cdm 5655  ran crn 5656  ⟶1-1wf1 6552  ‘cfv 6555  ℩crio 7396  Vtxcvtx 29601  iEdgciedg 29602  USGraphcusgr 29757

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 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  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-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  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 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-oadd 8505  df-er 8742  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-dju 10006  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-n0 12631  df-z 12718  df-uz 12990  df-fz 13664  df-hash 14499  df-edg 29653  df-umgr 29688  df-usgr 29759

This theorem is used by:  usgriedgleord  29836