Theorem usgredg2v 29304

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 29302	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
5	4	ralrimiva 3129	. . 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 29249	. . . . . . . . 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 3643	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} → 𝑦 ∈ dom 𝐸)
10	9, 3	eleq2s 2855	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ dom 𝐸)
11		elrabi 3643	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} → 𝑤 ∈ dom 𝐸)
12	11, 3	eleq2s 2855	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐴 → 𝑤 ∈ dom 𝐸)
13	10, 12	anim12i 614	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑦 ∈ dom 𝐸 ∧ 𝑤 ∈ dom 𝐸))
14		f1fveq 7210	. . . . . . . 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 597	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝐸‘𝑦) = (𝐸‘𝑤) ↔ 𝑦 = 𝑤))
16	15	bicomd 223	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑦 = 𝑤 ↔ (𝐸‘𝑦) = (𝐸‘𝑤)))
17	16	notbid 318	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (¬ 𝑦 = 𝑤 ↔ ¬ (𝐸‘𝑦) = (𝐸‘𝑤)))
18		simpl 482	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ USGraph)
19		simpl 482	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → 𝑦 ∈ 𝐴)
20	18, 19	anim12i 614	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝐴))
21		preq1 4691	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑧 → {𝑢, 𝑁} = {𝑧, 𝑁})
22	21	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → ((𝐸‘𝑦) = {𝑢, 𝑁} ↔ (𝐸‘𝑦) = {𝑧, 𝑁}))
23	22	cbvriotavw 7327	. . . . . . . . 9 ⊢ (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁})
24	1, 2, 3	usgredg2vlem2 29303	. . . . . . . . 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		an3 660	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐺 ∈ USGraph ∧ 𝑤 ∈ 𝐴))
27	21	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → ((𝐸‘𝑤) = {𝑢, 𝑁} ↔ (𝐸‘𝑤) = {𝑧, 𝑁}))
28	27	cbvriotavw 7327	. . . . . . . . 9 ⊢ (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})
29	1, 2, 3	usgredg2vlem2 29303	. . . . . . . . 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 21	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐸‘𝑤) = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}), 𝑁})
31	25, 30	eqeq12d 2753	. . . . . . 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 318	. . . . . 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 7321	. . . . . . . . . . . 12 ⊢ (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) ∈ V
34	33	a1i 11	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑉 → (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) ∈ V)
35		id 22	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ 𝑉)
36		riotaex 7321	. . . . . . . . . . . 12 ⊢ (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∈ V
37	36	a1i 11	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑉 → (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∈ V)
38		preq12bg 4810	. . . . . . . . . . 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 839	. . . . . . . . . 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 318	. . . . . . . . 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 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})))))
42		ioran 986	. . . . . . . . . . 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 984	. . . . . . . . . . . . 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 2755	. . . . . . . . . . . . . . . . 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 320	. . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . 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 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})))
48		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ 𝑁 = 𝑁
49	48	pm2.24i 150	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑁 = 𝑁 → (𝐺 ∈ USGraph → ¬ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})))
50	47, 49	jaoi 858	. . . . . . . . . . . . 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 217	. . . . . . . . . . . 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 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})))
53	42, 52	sylbi 217	. . . . . . . . . 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 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})))
55	54	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})))
56	41, 55	sylbid 240	. . . . . . 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 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})))
58	32, 57	sylbid 240	. . . . 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 240	. . . 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 115	. . 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 3180	. 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 6844	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝐸‘𝑦) = {𝑧, 𝑁} ↔ (𝐸‘𝑤) = {𝑧, 𝑁}))
64	63	riotabidv 7319	. . 3 ⊢ (𝑦 = 𝑤 → (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}))
65	62, 64	f1mpt 7209	. 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)))
66	6, 61, 65	sylanbrc 584	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1→𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3400  Vcvv 3441  {cpr 4583   ↦ cmpt 5180  dom cdm 5625  ran crn 5626  –1-1→wf1 6490  ‘cfv 6493  ℩crio 7316  Vtxcvtx 29073  iEdgciedg 29074  USGraphcusgr 29226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-dju 9817  df-card 9855  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-n0 12406  df-z 12493  df-uz 12756  df-fz 13428  df-hash 14258  df-edg 29125  df-umgr 29160  df-usgr 29228

This theorem is referenced by:  usgriedgleord  29305