Theorem usgraedg4 25916

Description: The value of the "edge function" of a graph is a set containing two elements (the endvertices of the corresponding edge). (Contributed by Alexander van der Vekens, 18-Dec-2017.)

Assertion

Ref	Expression
usgraedg4	⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})

Distinct variable groups:   𝑦,𝐸   𝑦,𝑉   𝑦,𝑋   𝑦,𝑌

Proof of Theorem usgraedg4

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgraf0 25877	. . 3 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1→{𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2})
2		f1f 6014	. . 3 ⊢ (𝐸:dom 𝐸–1-1→{𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} → 𝐸:dom 𝐸⟶{𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2})
3		ffvelrn 6265	. . . . 5 ⊢ ((𝐸:dom 𝐸⟶{𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} ∧ 𝑋 ∈ dom 𝐸) → (𝐸‘𝑋) ∈ {𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2})
4		fveq2 6103	. . . . . . . 8 ⊢ (𝑦 = (𝐸‘𝑋) → (#‘𝑦) = (#‘(𝐸‘𝑋)))
5	4	eqeq1d 2612	. . . . . . 7 ⊢ (𝑦 = (𝐸‘𝑋) → ((#‘𝑦) = 2 ↔ (#‘(𝐸‘𝑋)) = 2))
6	5	elrab 3331	. . . . . 6 ⊢ ((𝐸‘𝑋) ∈ {𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} ↔ ((𝐸‘𝑋) ∈ 𝒫 𝑉 ∧ (#‘(𝐸‘𝑋)) = 2))
7		hash2pr 13108	. . . . . . 7 ⊢ (((𝐸‘𝑋) ∈ 𝒫 𝑉 ∧ (#‘(𝐸‘𝑋)) = 2) → ∃𝑎∃𝑏(𝐸‘𝑋) = {𝑎, 𝑏})
8		19.42vv 1907	. . . . . . . . . . 11 ⊢ (∃𝑎∃𝑏(𝑌 ∈ (𝐸‘𝑋) ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ↔ (𝑌 ∈ (𝐸‘𝑋) ∧ ∃𝑎∃𝑏(𝐸‘𝑋) = {𝑎, 𝑏}))
9		eleq2 2677	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸‘𝑋) = {𝑎, 𝑏} → (𝑌 ∈ (𝐸‘𝑋) ↔ 𝑌 ∈ {𝑎, 𝑏}))
10		elpri 4145	. . . . . . . . . . . . . . . . 17 ⊢ (𝑌 ∈ {𝑎, 𝑏} → (𝑌 = 𝑎 ∨ 𝑌 = 𝑏))
11		andir 908	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑌 = 𝑎 ∨ 𝑌 = 𝑏) ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ↔ ((𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ∨ (𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏})))
12	11	biimpi 205	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑌 = 𝑎 ∨ 𝑌 = 𝑏) ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) → ((𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ∨ (𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏})))
13	12	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑌 = 𝑎 ∨ 𝑌 = 𝑏) → ((𝐸‘𝑋) = {𝑎, 𝑏} → ((𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ∨ (𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}))))
14	10, 13	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 ∈ {𝑎, 𝑏} → ((𝐸‘𝑋) = {𝑎, 𝑏} → ((𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ∨ (𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}))))
15	9, 14	syl6bi 242	. . . . . . . . . . . . . . 15 ⊢ ((𝐸‘𝑋) = {𝑎, 𝑏} → (𝑌 ∈ (𝐸‘𝑋) → ((𝐸‘𝑋) = {𝑎, 𝑏} → ((𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ∨ (𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏})))))
16	15	pm2.43a 52	. . . . . . . . . . . . . 14 ⊢ ((𝐸‘𝑋) = {𝑎, 𝑏} → (𝑌 ∈ (𝐸‘𝑋) → ((𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ∨ (𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}))))
17	16	impcom 445	. . . . . . . . . . . . 13 ⊢ ((𝑌 ∈ (𝐸‘𝑋) ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) → ((𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ∨ (𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏})))
18	17	2eximi 1753	. . . . . . . . . . . 12 ⊢ (∃𝑎∃𝑏(𝑌 ∈ (𝐸‘𝑋) ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) → ∃𝑎∃𝑏((𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ∨ (𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏})))
19		19.43 1799	. . . . . . . . . . . . . . 15 ⊢ (∃𝑏((𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ∨ (𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏})) ↔ (∃𝑏(𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ∨ ∃𝑏(𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏})))
20	19	exbii 1764	. . . . . . . . . . . . . 14 ⊢ (∃𝑎∃𝑏((𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ∨ (𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏})) ↔ ∃𝑎(∃𝑏(𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ∨ ∃𝑏(𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏})))
21		19.43 1799	. . . . . . . . . . . . . 14 ⊢ (∃𝑎(∃𝑏(𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ∨ ∃𝑏(𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏})) ↔ (∃𝑎∃𝑏(𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ∨ ∃𝑎∃𝑏(𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏})))
22	20, 21	bitri 263	. . . . . . . . . . . . 13 ⊢ (∃𝑎∃𝑏((𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ∨ (𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏})) ↔ (∃𝑎∃𝑏(𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ∨ ∃𝑎∃𝑏(𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏})))
23		preq2 4213	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑦 → {𝑎, 𝑏} = {𝑎, 𝑦})
24	23	eqeq2d 2620	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑦 → ((𝐸‘𝑋) = {𝑎, 𝑏} ↔ (𝐸‘𝑋) = {𝑎, 𝑦}))
25	24	anbi2d 736	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → ((𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ↔ (𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑦})))
26	25	cbvexv 2263	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑏(𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ↔ ∃𝑦(𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑦}))
27		preq1 4212	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑌 → {𝑎, 𝑦} = {𝑌, 𝑦})
28	27	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑌 = 𝑎 → {𝑎, 𝑦} = {𝑌, 𝑦})
29	28	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑌 = 𝑎 → ((𝐸‘𝑋) = {𝑎, 𝑦} ↔ (𝐸‘𝑋) = {𝑌, 𝑦}))
30	29	biimpa 500	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑦}) → (𝐸‘𝑋) = {𝑌, 𝑦})
31		vex 3176	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑎 ∈ V
32		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑌 = 𝑎 → (𝑌 ∈ V ↔ 𝑎 ∈ V))
33	31, 32	mpbiri 247	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑌 = 𝑎 → 𝑌 ∈ V)
34	33	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑦}) → 𝑌 ∈ V)
35		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐸‘𝑋) = {𝑌, 𝑦} → ((𝐸‘𝑋) ∈ 𝒫 𝑉 ↔ {𝑌, 𝑦} ∈ 𝒫 𝑉))
36		prex 4836	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ {𝑌, 𝑦} ∈ V
37	36	elpw 4114	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ({𝑌, 𝑦} ∈ 𝒫 𝑉 ↔ {𝑌, 𝑦} ⊆ 𝑉)
38		vex 3176	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 𝑦 ∈ V
39		prssg 4290	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑌 ∈ V ∧ 𝑦 ∈ V) → ((𝑌 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ↔ {𝑌, 𝑦} ⊆ 𝑉))
40	39	bicomd 212	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑌 ∈ V ∧ 𝑦 ∈ V) → ({𝑌, 𝑦} ⊆ 𝑉 ↔ (𝑌 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
41	38, 40	mpan2 703	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑌 ∈ V → ({𝑌, 𝑦} ⊆ 𝑉 ↔ (𝑌 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
42		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
43	41, 42	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑌 ∈ V → ({𝑌, 𝑦} ⊆ 𝑉 → 𝑦 ∈ 𝑉))
44	43	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ({𝑌, 𝑦} ⊆ 𝑉 → (𝑌 ∈ V → 𝑦 ∈ 𝑉))
45	37, 44	sylbi 206	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({𝑌, 𝑦} ∈ 𝒫 𝑉 → (𝑌 ∈ V → 𝑦 ∈ 𝑉))
46	35, 45	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐸‘𝑋) = {𝑌, 𝑦} → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → (𝑌 ∈ V → 𝑦 ∈ 𝑉)))
47	46	com23 84	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐸‘𝑋) = {𝑌, 𝑦} → (𝑌 ∈ V → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → 𝑦 ∈ 𝑉)))
48	47	imp 444	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐸‘𝑋) = {𝑌, 𝑦} ∧ 𝑌 ∈ V) → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → 𝑦 ∈ 𝑉))
49	48	imp 444	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐸‘𝑋) = {𝑌, 𝑦} ∧ 𝑌 ∈ V) ∧ (𝐸‘𝑋) ∈ 𝒫 𝑉) → 𝑦 ∈ 𝑉)
50		simpll 786	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐸‘𝑋) = {𝑌, 𝑦} ∧ 𝑌 ∈ V) ∧ (𝐸‘𝑋) ∈ 𝒫 𝑉) → (𝐸‘𝑋) = {𝑌, 𝑦})
51	49, 50	jca 553	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐸‘𝑋) = {𝑌, 𝑦} ∧ 𝑌 ∈ V) ∧ (𝐸‘𝑋) ∈ 𝒫 𝑉) → (𝑦 ∈ 𝑉 ∧ (𝐸‘𝑋) = {𝑌, 𝑦}))
52	51	ex 449	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐸‘𝑋) = {𝑌, 𝑦} ∧ 𝑌 ∈ V) → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → (𝑦 ∈ 𝑉 ∧ (𝐸‘𝑋) = {𝑌, 𝑦})))
53	30, 34, 52	syl2anc 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑦}) → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → (𝑦 ∈ 𝑉 ∧ (𝐸‘𝑋) = {𝑌, 𝑦})))
54	53	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸‘𝑋) ∈ 𝒫 𝑉 → ((𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑦}) → (𝑦 ∈ 𝑉 ∧ (𝐸‘𝑋) = {𝑌, 𝑦})))
55	54	eximdv 1833	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸‘𝑋) ∈ 𝒫 𝑉 → (∃𝑦(𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑦}) → ∃𝑦(𝑦 ∈ 𝑉 ∧ (𝐸‘𝑋) = {𝑌, 𝑦})))
56		df-rex 2902	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦} ↔ ∃𝑦(𝑦 ∈ 𝑉 ∧ (𝐸‘𝑋) = {𝑌, 𝑦}))
57	55, 56	syl6ibr 241	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸‘𝑋) ∈ 𝒫 𝑉 → (∃𝑦(𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑦}) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
58	57	com12 32	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑦(𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑦}) → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
59	26, 58	sylbi 206	. . . . . . . . . . . . . . 15 ⊢ (∃𝑏(𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
60	59	exlimiv 1845	. . . . . . . . . . . . . 14 ⊢ (∃𝑎∃𝑏(𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
61		excom 2029	. . . . . . . . . . . . . . 15 ⊢ (∃𝑎∃𝑏(𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ↔ ∃𝑏∃𝑎(𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}))
62		preq1 4212	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑦 → {𝑎, 𝑏} = {𝑦, 𝑏})
63	62	eqeq2d 2620	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑦 → ((𝐸‘𝑋) = {𝑎, 𝑏} ↔ (𝐸‘𝑋) = {𝑦, 𝑏}))
64	63	anbi2d 736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑦 → ((𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ↔ (𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑦, 𝑏})))
65	64	cbvexv 2263	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑎(𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ↔ ∃𝑦(𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑦, 𝑏}))
66		preq2 4213	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 = 𝑌 → {𝑦, 𝑏} = {𝑦, 𝑌})
67	66	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑌 = 𝑏 → {𝑦, 𝑏} = {𝑦, 𝑌})
68	67	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑌 = 𝑏 → ((𝐸‘𝑋) = {𝑦, 𝑏} ↔ (𝐸‘𝑋) = {𝑦, 𝑌}))
69	68	biimpa 500	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑦, 𝑏}) → (𝐸‘𝑋) = {𝑦, 𝑌})
70		vex 3176	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑏 ∈ V
71		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑌 = 𝑏 → (𝑌 ∈ V ↔ 𝑏 ∈ V))
72	70, 71	mpbiri 247	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑌 = 𝑏 → 𝑌 ∈ V)
73	72	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑦, 𝑏}) → 𝑌 ∈ V)
74		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐸‘𝑋) = {𝑦, 𝑌} → ((𝐸‘𝑋) ∈ 𝒫 𝑉 ↔ {𝑦, 𝑌} ∈ 𝒫 𝑉))
75		prex 4836	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ {𝑦, 𝑌} ∈ V
76	75	elpw 4114	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({𝑦, 𝑌} ∈ 𝒫 𝑉 ↔ {𝑦, 𝑌} ⊆ 𝑉)
77		prssg 4290	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑦 ∈ V ∧ 𝑌 ∈ V) → ((𝑦 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ↔ {𝑦, 𝑌} ⊆ 𝑉))
78	77	bicomd 212	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 ∈ V ∧ 𝑌 ∈ V) → ({𝑦, 𝑌} ⊆ 𝑉 ↔ (𝑦 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
79	38, 78	mpan 702	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑌 ∈ V → ({𝑦, 𝑌} ⊆ 𝑉 ↔ (𝑦 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)))
80		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑦 ∈ 𝑉)
81	79, 80	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑌 ∈ V → ({𝑦, 𝑌} ⊆ 𝑉 → 𝑦 ∈ 𝑉))
82	81	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({𝑦, 𝑌} ⊆ 𝑉 → (𝑌 ∈ V → 𝑦 ∈ 𝑉))
83	76, 82	sylbi 206	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ({𝑦, 𝑌} ∈ 𝒫 𝑉 → (𝑌 ∈ V → 𝑦 ∈ 𝑉))
84	74, 83	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐸‘𝑋) = {𝑦, 𝑌} → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → (𝑌 ∈ V → 𝑦 ∈ 𝑉)))
85	84	com23 84	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐸‘𝑋) = {𝑦, 𝑌} → (𝑌 ∈ V → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → 𝑦 ∈ 𝑉)))
86	85	imp 444	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐸‘𝑋) = {𝑦, 𝑌} ∧ 𝑌 ∈ V) → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → 𝑦 ∈ 𝑉))
87		prcom 4211	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ {𝑦, 𝑌} = {𝑌, 𝑦}
88	87	eqeq2i 2622	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐸‘𝑋) = {𝑦, 𝑌} ↔ (𝐸‘𝑋) = {𝑌, 𝑦})
89	88	biimpi 205	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐸‘𝑋) = {𝑦, 𝑌} → (𝐸‘𝑋) = {𝑌, 𝑦})
90	89	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐸‘𝑋) = {𝑦, 𝑌} ∧ 𝑌 ∈ V) → (𝐸‘𝑋) = {𝑌, 𝑦})
91	86, 90	jctird 565	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐸‘𝑋) = {𝑦, 𝑌} ∧ 𝑌 ∈ V) → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → (𝑦 ∈ 𝑉 ∧ (𝐸‘𝑋) = {𝑌, 𝑦})))
92	69, 73, 91	syl2anc 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑦, 𝑏}) → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → (𝑦 ∈ 𝑉 ∧ (𝐸‘𝑋) = {𝑌, 𝑦})))
93	92	com12 32	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸‘𝑋) ∈ 𝒫 𝑉 → ((𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑦, 𝑏}) → (𝑦 ∈ 𝑉 ∧ (𝐸‘𝑋) = {𝑌, 𝑦})))
94	93	eximdv 1833	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸‘𝑋) ∈ 𝒫 𝑉 → (∃𝑦(𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑦, 𝑏}) → ∃𝑦(𝑦 ∈ 𝑉 ∧ (𝐸‘𝑋) = {𝑌, 𝑦})))
95	94, 56	syl6ibr 241	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸‘𝑋) ∈ 𝒫 𝑉 → (∃𝑦(𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑦, 𝑏}) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
96	95	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑦(𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑦, 𝑏}) → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
97	65, 96	sylbi 206	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑎(𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
98	97	exlimiv 1845	. . . . . . . . . . . . . . 15 ⊢ (∃𝑏∃𝑎(𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
99	61, 98	sylbi 206	. . . . . . . . . . . . . 14 ⊢ (∃𝑎∃𝑏(𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
100	60, 99	jaoi 393	. . . . . . . . . . . . 13 ⊢ ((∃𝑎∃𝑏(𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ∨ ∃𝑎∃𝑏(𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏})) → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
101	22, 100	sylbi 206	. . . . . . . . . . . 12 ⊢ (∃𝑎∃𝑏((𝑌 = 𝑎 ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) ∨ (𝑌 = 𝑏 ∧ (𝐸‘𝑋) = {𝑎, 𝑏})) → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
102	18, 101	syl 17	. . . . . . . . . . 11 ⊢ (∃𝑎∃𝑏(𝑌 ∈ (𝐸‘𝑋) ∧ (𝐸‘𝑋) = {𝑎, 𝑏}) → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
103	8, 102	sylbir 224	. . . . . . . . . 10 ⊢ ((𝑌 ∈ (𝐸‘𝑋) ∧ ∃𝑎∃𝑏(𝐸‘𝑋) = {𝑎, 𝑏}) → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
104	103	ex 449	. . . . . . . . 9 ⊢ (𝑌 ∈ (𝐸‘𝑋) → (∃𝑎∃𝑏(𝐸‘𝑋) = {𝑎, 𝑏} → ((𝐸‘𝑋) ∈ 𝒫 𝑉 → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})))
105	104	com13 86	. . . . . . . 8 ⊢ ((𝐸‘𝑋) ∈ 𝒫 𝑉 → (∃𝑎∃𝑏(𝐸‘𝑋) = {𝑎, 𝑏} → (𝑌 ∈ (𝐸‘𝑋) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})))
106	105	adantr 480	. . . . . . 7 ⊢ (((𝐸‘𝑋) ∈ 𝒫 𝑉 ∧ (#‘(𝐸‘𝑋)) = 2) → (∃𝑎∃𝑏(𝐸‘𝑋) = {𝑎, 𝑏} → (𝑌 ∈ (𝐸‘𝑋) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})))
107	7, 106	mpd 15	. . . . . 6 ⊢ (((𝐸‘𝑋) ∈ 𝒫 𝑉 ∧ (#‘(𝐸‘𝑋)) = 2) → (𝑌 ∈ (𝐸‘𝑋) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
108	6, 107	sylbi 206	. . . . 5 ⊢ ((𝐸‘𝑋) ∈ {𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} → (𝑌 ∈ (𝐸‘𝑋) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
109	3, 108	syl 17	. . . 4 ⊢ ((𝐸:dom 𝐸⟶{𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} ∧ 𝑋 ∈ dom 𝐸) → (𝑌 ∈ (𝐸‘𝑋) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦}))
110	109	ex 449	. . 3 ⊢ (𝐸:dom 𝐸⟶{𝑦 ∈ 𝒫 𝑉 ∣ (#‘𝑦) = 2} → (𝑋 ∈ dom 𝐸 → (𝑌 ∈ (𝐸‘𝑋) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})))
111	1, 2, 110	3syl 18	. 2 ⊢ (𝑉 USGrph 𝐸 → (𝑋 ∈ dom 𝐸 → (𝑌 ∈ (𝐸‘𝑋) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})))
112	111	3imp 1249	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ dom 𝐸 ∧ 𝑌 ∈ (𝐸‘𝑋)) → ∃𝑦 ∈ 𝑉 (𝐸‘𝑋) = {𝑌, 𝑦})

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  {cpr 4127   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  2c2 10947  #chash 12979   USGrph cusg 25859

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-usgra 25862

This theorem is referenced by:  usgraedgreu  25917  nbgraf1olem5  25974