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Theorem usgr2pthlem 29799

Description: Lemma for usgr2pth 29800. (Contributed by Alexander van der Vekens, 27-Jan-2018.) (Revised by AV, 5-Jun-2021.)

**Hypotheses**
Ref	Expression
usgr2pthlem.v	⊢ 𝑉 = (Vtx‘𝐺)
usgr2pthlem.i	⊢ 𝐼 = (iEdg‘𝐺)

**Assertion**
Ref	Expression
usgr2pthlem	⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))

Distinct variable groups: 𝑖,𝐹 𝑥,𝐹,𝑦,𝑧 𝑥,𝐺,𝑦,𝑧 𝑖,𝐼 𝑥,𝐼,𝑦,𝑧 𝑃,𝑖 𝑥,𝑃,𝑦,𝑧 𝑥,𝑉,𝑦,𝑧 Allowed substitution hints: 𝐺(𝑖) 𝑉(𝑖)

**Proof of Theorem usgr2pthlem**
Step	Hyp	Ref	Expression
1		0nn0 12568	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ₀
2		2nn0 12570	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ₀
3		0le2 12395	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 2
4		elfz2nn0 13675	. . . . . . . . . . . . . 14 ⊢ (0 ∈ (0...2) ↔ (0 ∈ ℕ₀ ∧ 2 ∈ ℕ₀ ∧ 0 ≤ 2))
5	1, 2, 3, 4	mpbir3an 1341	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0...2)
6		ffvelcdm 7115	. . . . . . . . . . . . 13 ⊢ ((𝑃:(0...2)⟶𝑉 ∧ 0 ∈ (0...2)) → (𝑃‘0) ∈ 𝑉)
7	5, 6	mpan2 690	. . . . . . . . . . . 12 ⊢ (𝑃:(0...2)⟶𝑉 → (𝑃‘0) ∈ 𝑉)
8	7	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → (𝑃‘0) ∈ 𝑉)
9		1nn0 12569	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ₀
10		1le2 12502	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ≤ 2
11		elfz2nn0 13675	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ (0...2) ↔ (1 ∈ ℕ₀ ∧ 2 ∈ ℕ₀ ∧ 1 ≤ 2))
12	9, 2, 10, 11	mpbir3an 1341	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ (0...2)
13		ffvelcdm 7115	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...2)⟶𝑉 ∧ 1 ∈ (0...2)) → (𝑃‘1) ∈ 𝑉)
14	12, 13	mpan2 690	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...2)⟶𝑉 → (𝑃‘1) ∈ 𝑉)
15	14	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → (𝑃‘1) ∈ 𝑉)
16		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) → 𝐺 ∈ USGraph)
17		fvex 6933	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃‘1) ∈ V
18	16, 17	jctir 520	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) → (𝐺 ∈ USGraph ∧ (𝑃‘1) ∈ V))
19		prcom 4757	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ {(𝑃‘0), (𝑃‘1)} = {(𝑃‘1), (𝑃‘0)}
20	19	eqeq2i 2753	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ↔ (𝐼‘(𝐹‘0)) = {(𝑃‘1), (𝑃‘0)})
21	20	biimpi 216	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} → (𝐼‘(𝐹‘0)) = {(𝑃‘1), (𝑃‘0)})
22	21	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝐼‘(𝐹‘0)) = {(𝑃‘1), (𝑃‘0)})
23	22	ad2antlr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) → (𝐼‘(𝐹‘0)) = {(𝑃‘1), (𝑃‘0)})
24		usgr2pthlem.i	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐼 = (iEdg‘𝐺)
25	24	usgrnloopv 29235	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ USGraph ∧ (𝑃‘1) ∈ V) → ((𝐼‘(𝐹‘0)) = {(𝑃‘1), (𝑃‘0)} → (𝑃‘1) ≠ (𝑃‘0)))
26	18, 23, 25	sylc 65	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) → (𝑃‘1) ≠ (𝑃‘0))
27	26	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → (𝑃‘1) ≠ (𝑃‘0))
28	17	elsn 4663	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃‘1) ∈ {(𝑃‘0)} ↔ (𝑃‘1) = (𝑃‘0))
29	28	necon3bbii 2994	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑃‘1) ∈ {(𝑃‘0)} ↔ (𝑃‘1) ≠ (𝑃‘0))
30	27, 29	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → ¬ (𝑃‘1) ∈ {(𝑃‘0)})
31	15, 30	eldifd 3987	. . . . . . . . . . . . . 14 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → (𝑃‘1) ∈ (𝑉 ∖ {(𝑃‘0)}))
32	31	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) ∧ 𝑥 = (𝑃‘0)) → (𝑃‘1) ∈ (𝑉 ∖ {(𝑃‘0)}))
33		sneq 4658	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑃‘0) → {𝑥} = {(𝑃‘0)})
34	33	difeq2d 4149	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑃‘0) → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {(𝑃‘0)}))
35	34	eleq2d 2830	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑃‘0) → ((𝑃‘1) ∈ (𝑉 ∖ {𝑥}) ↔ (𝑃‘1) ∈ (𝑉 ∖ {(𝑃‘0)})))
36	35	adantl 481	. . . . . . . . . . . . 13 ⊢ (((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) ∧ 𝑥 = (𝑃‘0)) → ((𝑃‘1) ∈ (𝑉 ∖ {𝑥}) ↔ (𝑃‘1) ∈ (𝑉 ∖ {(𝑃‘0)})))
37	32, 36	mpbird 257	. . . . . . . . . . . 12 ⊢ (((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) ∧ 𝑥 = (𝑃‘0)) → (𝑃‘1) ∈ (𝑉 ∖ {𝑥}))
38		2re 12367	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ
39	38	leidi 11824	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ≤ 2
40		elfz2nn0 13675	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 ∈ (0...2) ↔ (2 ∈ ℕ₀ ∧ 2 ∈ ℕ₀ ∧ 2 ≤ 2))
41	2, 2, 39, 40	mpbir3an 1341	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ (0...2)
42		ffvelcdm 7115	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...2)⟶𝑉 ∧ 2 ∈ (0...2)) → (𝑃‘2) ∈ 𝑉)
43	41, 42	mpan2 690	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃:(0...2)⟶𝑉 → (𝑃‘2) ∈ 𝑉)
44	43	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → (𝑃‘2) ∈ 𝑉)
45	24	usgrf1 29207	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1→ran 𝐼)
46	45	ad2antlr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → 𝐼:dom 𝐼–1-1→ran 𝐼)
47		simpl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → 𝐹:(0..^2)–1-1→dom 𝐼)
48	47	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → 𝐹:(0..^2)–1-1→dom 𝐼)
49	46, 48	jca 511	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → (𝐼:dom 𝐼–1-1→ran 𝐼 ∧ 𝐹:(0..^2)–1-1→dom 𝐼))
50		2nn 12366	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 2 ∈ ℕ
51		lbfzo0 13756	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
52	50, 51	mpbir 231	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ (0..^2)
53		1lt2 12464	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 < 2
54		elfzo0 13757	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 ∈ (0..^2) ↔ (1 ∈ ℕ₀ ∧ 2 ∈ ℕ ∧ 1 < 2))
55	9, 50, 53, 54	mpbir3an 1341	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ (0..^2)
56	52, 55	pm3.2i 470	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))
57	56	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → (0 ∈ (0..^2) ∧ 1 ∈ (0..^2)))
58		0ne1 12364	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ≠ 1
59	58	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → 0 ≠ 1)
60	49, 57, 59	3jca 1128	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → ((𝐼:dom 𝐼–1-1→ran 𝐼 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2)) ∧ 0 ≠ 1))
61		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
62	61	ad2antrr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
63		2f1fvneq 7297	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼:dom 𝐼–1-1→ran 𝐼 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2)) ∧ 0 ≠ 1) → (((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → {(𝑃‘0), (𝑃‘1)} ≠ {(𝑃‘1), (𝑃‘2)}))
64	60, 62, 63	sylc 65	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → {(𝑃‘0), (𝑃‘1)} ≠ {(𝑃‘1), (𝑃‘2)})
65		necom 3000	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃‘2) ≠ (𝑃‘0) ↔ (𝑃‘0) ≠ (𝑃‘2))
66		fvex 6933	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃‘0) ∈ V
67		fvex 6933	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃‘2) ∈ V
68	66, 17, 67	3pm3.2i 1339	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃‘0) ∈ V ∧ (𝑃‘1) ∈ V ∧ (𝑃‘2) ∈ V)
69		fvexd 6935	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) → (𝑃‘0) ∈ V)
70		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)})
71	70	ad2antlr 726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) → (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)})
72	16, 69, 71	jca31 514	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) → ((𝐺 ∈ USGraph ∧ (𝑃‘0) ∈ V) ∧ (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}))
73	72	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → ((𝐺 ∈ USGraph ∧ (𝑃‘0) ∈ V) ∧ (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}))
74	24	usgrnloopv 29235	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ USGraph ∧ (𝑃‘0) ∈ V) → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} → (𝑃‘0) ≠ (𝑃‘1)))
75	74	imp 406	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 ∈ USGraph ∧ (𝑃‘0) ∈ V) ∧ (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}) → (𝑃‘0) ≠ (𝑃‘1))
76	73, 75	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → (𝑃‘0) ≠ (𝑃‘1))
77		pr1nebg 4882	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑃‘0) ∈ V ∧ (𝑃‘1) ∈ V ∧ (𝑃‘2) ∈ V) ∧ (𝑃‘0) ≠ (𝑃‘1)) → ((𝑃‘0) ≠ (𝑃‘2) ↔ {(𝑃‘0), (𝑃‘1)} ≠ {(𝑃‘1), (𝑃‘2)}))
78	68, 76, 77	sylancr 586	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → ((𝑃‘0) ≠ (𝑃‘2) ↔ {(𝑃‘0), (𝑃‘1)} ≠ {(𝑃‘1), (𝑃‘2)}))
79	65, 78	bitrid 283	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → ((𝑃‘2) ≠ (𝑃‘0) ↔ {(𝑃‘0), (𝑃‘1)} ≠ {(𝑃‘1), (𝑃‘2)}))
80	64, 79	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → (𝑃‘2) ≠ (𝑃‘0))
81		fvexd 6935	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) → (𝑃‘2) ∈ V)
82		prcom 4757	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {(𝑃‘1), (𝑃‘2)} = {(𝑃‘2), (𝑃‘1)}
83	82	eqeq2i 2753	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ↔ (𝐼‘(𝐹‘1)) = {(𝑃‘2), (𝑃‘1)})
84	83	biimpi 216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} → (𝐼‘(𝐹‘1)) = {(𝑃‘2), (𝑃‘1)})
85	84	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝐼‘(𝐹‘1)) = {(𝑃‘2), (𝑃‘1)})
86	85	ad2antlr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) → (𝐼‘(𝐹‘1)) = {(𝑃‘2), (𝑃‘1)})
87	16, 81, 86	jca31 514	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) → ((𝐺 ∈ USGraph ∧ (𝑃‘2) ∈ V) ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘2), (𝑃‘1)}))
88	87	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → ((𝐺 ∈ USGraph ∧ (𝑃‘2) ∈ V) ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘2), (𝑃‘1)}))
89	24	usgrnloopv 29235	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USGraph ∧ (𝑃‘2) ∈ V) → ((𝐼‘(𝐹‘1)) = {(𝑃‘2), (𝑃‘1)} → (𝑃‘2) ≠ (𝑃‘1)))
90	89	imp 406	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ USGraph ∧ (𝑃‘2) ∈ V) ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘2), (𝑃‘1)}) → (𝑃‘2) ≠ (𝑃‘1))
91	88, 90	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → (𝑃‘2) ≠ (𝑃‘1))
92	80, 91	nelprd 4679	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → ¬ (𝑃‘2) ∈ {(𝑃‘0), (𝑃‘1)})
93	44, 92	eldifd 3987	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → (𝑃‘2) ∈ (𝑉 ∖ {(𝑃‘0), (𝑃‘1)}))
94	93	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ ((((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) ∧ 𝑥 = (𝑃‘0)) ∧ 𝑦 = (𝑃‘1)) → (𝑃‘2) ∈ (𝑉 ∖ {(𝑃‘0), (𝑃‘1)}))
95		preq12 4760	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) → {𝑥, 𝑦} = {(𝑃‘0), (𝑃‘1)})
96	95	difeq2d 4149	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) → (𝑉 ∖ {𝑥, 𝑦}) = (𝑉 ∖ {(𝑃‘0), (𝑃‘1)}))
97	96	eleq2d 2830	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) → ((𝑃‘2) ∈ (𝑉 ∖ {𝑥, 𝑦}) ↔ (𝑃‘2) ∈ (𝑉 ∖ {(𝑃‘0), (𝑃‘1)})))
98	97	adantll 713	. . . . . . . . . . . . . 14 ⊢ ((((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) ∧ 𝑥 = (𝑃‘0)) ∧ 𝑦 = (𝑃‘1)) → ((𝑃‘2) ∈ (𝑉 ∖ {𝑥, 𝑦}) ↔ (𝑃‘2) ∈ (𝑉 ∖ {(𝑃‘0), (𝑃‘1)})))
99	94, 98	mpbird 257	. . . . . . . . . . . . 13 ⊢ ((((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) ∧ 𝑥 = (𝑃‘0)) ∧ 𝑦 = (𝑃‘1)) → (𝑃‘2) ∈ (𝑉 ∖ {𝑥, 𝑦}))
100		eqcom 2747	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑃‘0) ↔ (𝑃‘0) = 𝑥)
101		eqcom 2747	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑃‘1) ↔ (𝑃‘1) = 𝑦)
102		eqcom 2747	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (𝑃‘2) ↔ (𝑃‘2) = 𝑧)
103	100, 101, 102	3anbi123i 1155	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1) ∧ 𝑧 = (𝑃‘2)) ↔ ((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧))
104	103	biimpi 216	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1) ∧ 𝑧 = (𝑃‘2)) → ((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧))
105	104	ad4ant123 1172	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) ∧ 𝑧 = (𝑃‘2)) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → ((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧))
106	100	biimpi 216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (𝑃‘0) → (𝑃‘0) = 𝑥)
107	106	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) ∧ 𝑧 = (𝑃‘2)) → (𝑃‘0) = 𝑥)
108	101	biimpi 216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (𝑃‘1) → (𝑃‘1) = 𝑦)
109	108	ad2antlr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) ∧ 𝑧 = (𝑃‘2)) → (𝑃‘1) = 𝑦)
110	107, 109	preq12d 4766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) ∧ 𝑧 = (𝑃‘2)) → {(𝑃‘0), (𝑃‘1)} = {𝑥, 𝑦})
111	110	eqeq2d 2751	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) ∧ 𝑧 = (𝑃‘2)) → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ↔ (𝐼‘(𝐹‘0)) = {𝑥, 𝑦}))
112	102	biimpi 216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = (𝑃‘2) → (𝑃‘2) = 𝑧)
113	112	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) ∧ 𝑧 = (𝑃‘2)) → (𝑃‘2) = 𝑧)
114	109, 113	preq12d 4766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) ∧ 𝑧 = (𝑃‘2)) → {(𝑃‘1), (𝑃‘2)} = {𝑦, 𝑧})
115	114	eqeq2d 2751	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) ∧ 𝑧 = (𝑃‘2)) → ((𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ↔ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))
116	111, 115	anbi12d 631	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) ∧ 𝑧 = (𝑃‘2)) → (((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ↔ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})))
117	116	biimpa 476	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) ∧ 𝑧 = (𝑃‘2)) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))
118	105, 117	jca 511	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) ∧ 𝑧 = (𝑃‘2)) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})))
119	118	exp41 434	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑃‘0) → (𝑦 = (𝑃‘1) → (𝑧 = (𝑃‘2) → (((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))))
120	119	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) ∧ 𝑥 = (𝑃‘0)) → (𝑦 = (𝑃‘1) → (𝑧 = (𝑃‘2) → (((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))))
121	120	imp31 417	. . . . . . . . . . . . 13 ⊢ (((((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) ∧ 𝑥 = (𝑃‘0)) ∧ 𝑦 = (𝑃‘1)) ∧ 𝑧 = (𝑃‘2)) → (((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))
122	99, 121	rspcimedv 3626	. . . . . . . . . . . 12 ⊢ ((((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) ∧ 𝑥 = (𝑃‘0)) ∧ 𝑦 = (𝑃‘1)) → (((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → ∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))
123	37, 122	rspcimedv 3626	. . . . . . . . . . 11 ⊢ (((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) ∧ 𝑥 = (𝑃‘0)) → (((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))
124	8, 123	rspcimedv 3626	. . . . . . . . . 10 ⊢ ((((𝐹:(0..^2)–1-1→dom 𝐼 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)⟶𝑉) → (((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))
125	124	exp41 434	. . . . . . . . 9 ⊢ (𝐹:(0..^2)–1-1→dom 𝐼 → (((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝐺 ∈ USGraph → (𝑃:(0...2)⟶𝑉 → (((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})))))))
126	125	com15 101	. . . . . . . 8 ⊢ (((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝐺 ∈ USGraph → (𝑃:(0...2)⟶𝑉 → (𝐹:(0..^2)–1-1→dom 𝐼 → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})))))))
127	126	pm2.43i 52	. . . . . . 7 ⊢ (((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝐺 ∈ USGraph → (𝑃:(0...2)⟶𝑉 → (𝐹:(0..^2)–1-1→dom 𝐼 → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))))
128	127	com12 32	. . . . . 6 ⊢ (𝐺 ∈ USGraph → (((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃:(0...2)⟶𝑉 → (𝐹:(0..^2)–1-1→dom 𝐼 → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))))
129	128	adantr 480	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃:(0...2)⟶𝑉 → (𝐹:(0..^2)–1-1→dom 𝐼 → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))))
130		oveq2 7456	. . . . . . . 8 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = (0..^2))
131	130	raleqdv 3334	. . . . . . 7 ⊢ ((♯‘𝐹) = 2 → (∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ∀𝑖 ∈ (0..^2)(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
132		fzo0to2pr 13801	. . . . . . . . 9 ⊢ (0..^2) = {0, 1}
133	132	raleqi 3332	. . . . . . . 8 ⊢ (∀𝑖 ∈ (0..^2)(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ∀𝑖 ∈ {0, 1} (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})
134		2wlklem 29703	. . . . . . . 8 ⊢ (∀𝑖 ∈ {0, 1} (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
135	133, 134	bitri 275	. . . . . . 7 ⊢ (∀𝑖 ∈ (0..^2)(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
136	131, 135	bitrdi 287	. . . . . 6 ⊢ ((♯‘𝐹) = 2 → (∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))
137	136	adantl 481	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))
138		oveq2 7456	. . . . . . . 8 ⊢ ((♯‘𝐹) = 2 → (0...(♯‘𝐹)) = (0...2))
139	138	feq2d 6733	. . . . . . 7 ⊢ ((♯‘𝐹) = 2 → (𝑃:(0...(♯‘𝐹))⟶𝑉 ↔ 𝑃:(0...2)⟶𝑉))
140	139	adantl 481	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝑃:(0...(♯‘𝐹))⟶𝑉 ↔ 𝑃:(0...2)⟶𝑉))
141		f1eq2 6813	. . . . . . . . 9 ⊢ ((0..^(♯‘𝐹)) = (0..^2) → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ↔ 𝐹:(0..^2)–1-1→dom 𝐼))
142	130, 141	syl 17	. . . . . . . 8 ⊢ ((♯‘𝐹) = 2 → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ↔ 𝐹:(0..^2)–1-1→dom 𝐼))
143	142	imbi1d 341	. . . . . . 7 ⊢ ((♯‘𝐹) = 2 → ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) ↔ (𝐹:(0..^2)–1-1→dom 𝐼 → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})))))
144	143	adantl 481	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))) ↔ (𝐹:(0..^2)–1-1→dom 𝐼 → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})))))
145	140, 144	imbi12d 344	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → ((𝑃:(0...(♯‘𝐹))⟶𝑉 → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})))) ↔ (𝑃:(0...2)⟶𝑉 → (𝐹:(0..^2)–1-1→dom 𝐼 → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))))
146	129, 137, 145	3imtr4d 294	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} → (𝑃:(0...(♯‘𝐹))⟶𝑉 → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))))
147	146	com14 96	. . 3 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → (∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} → (𝑃:(0...(♯‘𝐹))⟶𝑉 → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))))
148	147	com23 86	. 2 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → (𝑃:(0...(♯‘𝐹))⟶𝑉 → (∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))))
149	148	3imp 1111	1 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 Vcvv 3488 ∖ cdif 3973 {csn 4648 {cpr 4650 class class class wbr 5166 dom cdm 5700 ran crn 5701 ⟶wf 6569 –1-1→wf1 6570 ‘cfv 6573 (class class class)co 7448 0cc0 11184 1c1 11185 + caddc 11187 < clt 11324 ≤ cle 11325 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ...cfz 13567 ..^cfzo 13711 ♯chash 14379 Vtxcvtx 29031 iEdgciedg 29032 USGraphcusgr 29184

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-oadd 8526 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-fzo 13712 df-hash 14380 df-umgr 29118 df-usgr 29186

This theorem is referenced by: usgr2pth 29800

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