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Theorem usgr2pthlem 28127

Description: Lemma for usgr2pth 28128. (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 12248	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ₀
2		2nn0 12250	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ₀
3		0le2 12075	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 2
4		elfz2nn0 13346	. . . . . . . . . . . . . 14 ⊢ (0 ∈ (0...2) ↔ (0 ∈ ℕ₀ ∧ 2 ∈ ℕ₀ ∧ 0 ≤ 2))
5	1, 2, 3, 4	mpbir3an 1340	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0...2)
6		ffvelrn 6956	. . . . . . . . . . . . 13 ⊢ ((𝑃:(0...2)⟶𝑉 ∧ 0 ∈ (0...2)) → (𝑃‘0) ∈ 𝑉)
7	5, 6	mpan2 688	. . . . . . . . . . . 12 ⊢ (𝑃:(0...2)⟶𝑉 → (𝑃‘0) ∈ 𝑉)
8	7	adantl 482	. . . . . . . . . . 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 12249	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ₀
10		1le2 12182	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ≤ 2
11		elfz2nn0 13346	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ (0...2) ↔ (1 ∈ ℕ₀ ∧ 2 ∈ ℕ₀ ∧ 1 ≤ 2))
12	9, 2, 10, 11	mpbir3an 1340	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ (0...2)
13		ffvelrn 6956	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...2)⟶𝑉 ∧ 1 ∈ (0...2)) → (𝑃‘1) ∈ 𝑉)
14	12, 13	mpan2 688	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...2)⟶𝑉 → (𝑃‘1) ∈ 𝑉)
15	14	adantl 482	. . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . 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 6784	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃‘1) ∈ V
18	16, 17	jctir 521	. . . . . . . . . . . . . . . . . 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 4674	. . . . . . . . . . . . . . . . . . . . . 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 215	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} → (𝐼‘(𝐹‘0)) = {(𝑃‘1), (𝑃‘0)})
22	21	adantr 481	. . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . 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 27565	. . . . . . . . . . . . . . . . . 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 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‘1) ≠ (𝑃‘0))
28	17	elsn 4582	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃‘1) ∈ {(𝑃‘0)} ↔ (𝑃‘1) = (𝑃‘0))
29	28	necon3bbii 2993	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑃‘1) ∈ {(𝑃‘0)} ↔ (𝑃‘1) ≠ (𝑃‘0))
30	27, 29	sylibr 233	. . . . . . . . . . . . . . 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 3903	. . . . . . . . . . . . . 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 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 ∖ {(𝑃‘0)}))
33		sneq 4577	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑃‘0) → {𝑥} = {(𝑃‘0)})
34	33	difeq2d 4062	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑃‘0) → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {(𝑃‘0)}))
35	34	eleq2d 2826	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑃‘0) → ((𝑃‘1) ∈ (𝑉 ∖ {𝑥}) ↔ (𝑃‘1) ∈ (𝑉 ∖ {(𝑃‘0)})))
36	35	adantl 482	. . . . . . . . . . . . 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 256	. . . . . . . . . . . 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 12047	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ
39	38	leidi 11509	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ≤ 2
40		elfz2nn0 13346	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 ∈ (0...2) ↔ (2 ∈ ℕ₀ ∧ 2 ∈ ℕ₀ ∧ 2 ≤ 2))
41	2, 2, 39, 40	mpbir3an 1340	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ (0...2)
42		ffvelrn 6956	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...2)⟶𝑉 ∧ 2 ∈ (0...2)) → (𝑃‘2) ∈ 𝑉)
43	41, 42	mpan2 688	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃:(0...2)⟶𝑉 → (𝑃‘2) ∈ 𝑉)
44	43	adantl 482	. . . . . . . . . . . . . . . 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 27540	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1→ran 𝐼)
46	45	ad2antlr 724	. . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . . 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 512	. . . . . . . . . . . . . . . . . . . 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 12046	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 2 ∈ ℕ
51		lbfzo0 13425	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
52	50, 51	mpbir 230	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ (0..^2)
53		1lt2 12144	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 < 2
54		elfzo0 13426	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 ∈ (0..^2) ↔ (1 ∈ ℕ₀ ∧ 2 ∈ ℕ ∧ 1 < 2))
55	9, 50, 53, 54	mpbir3an 1340	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ (0..^2)
56	52, 55	pm3.2i 471	. . . . . . . . . . . . . . . . . . . . 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 12044	. . . . . . . . . . . . . . . . . . . . 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 1127	. . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . 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 7130	. . . . . . . . . . . . . . . . . . 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 2999	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃‘2) ≠ (𝑃‘0) ↔ (𝑃‘0) ≠ (𝑃‘2))
66		fvex 6784	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃‘0) ∈ V
67		fvex 6784	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃‘2) ∈ V
68	66, 17, 67	3pm3.2i 1338	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃‘0) ∈ V ∧ (𝑃‘1) ∈ V ∧ (𝑃‘2) ∈ V)
69		fvexd 6786	. . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . . . . 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 515	. . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . 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 27565	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ USGraph ∧ (𝑃‘0) ∈ V) → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} → (𝑃‘0) ≠ (𝑃‘1)))
75	74	imp 407	. . . . . . . . . . . . . . . . . . . . 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 4794	. . . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . . 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	syl5bb 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 256	. . . . . . . . . . . . . . . . 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 6786	. . . . . . . . . . . . . . . . . . . 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 4674	. . . . . . . . . . . . . . . . . . . . . . . 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 215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} → (𝐼‘(𝐹‘1)) = {(𝑃‘2), (𝑃‘1)})
85	84	adantl 482	. . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . 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 515	. . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 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 27565	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USGraph ∧ (𝑃‘2) ∈ V) → ((𝐼‘(𝐹‘1)) = {(𝑃‘2), (𝑃‘1)} → (𝑃‘2) ≠ (𝑃‘1)))
90	89	imp 407	. . . . . . . . . . . . . . . . . 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 4598	. . . . . . . . . . . . . . . 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 3903	. . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . 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 4677	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) → {𝑥, 𝑦} = {(𝑃‘0), (𝑃‘1)})
96	95	difeq2d 4062	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) → (𝑉 ∖ {𝑥, 𝑦}) = (𝑉 ∖ {(𝑃‘0), (𝑃‘1)}))
97	96	eleq2d 2826	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) → ((𝑃‘2) ∈ (𝑉 ∖ {𝑥, 𝑦}) ↔ (𝑃‘2) ∈ (𝑉 ∖ {(𝑃‘0), (𝑃‘1)})))
98	97	adantll 711	. . . . . . . . . . . . . 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 256	. . . . . . . . . . . . 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 1154	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1) ∧ 𝑧 = (𝑃‘2)) ↔ ((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧))
104	103	biimpi 215	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1) ∧ 𝑧 = (𝑃‘2)) → ((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧))
105	104	ad4ant123 1171	. . . . . . . . . . . . . . . . 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 215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (𝑃‘0) → (𝑃‘0) = 𝑥)
107	106	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) ∧ 𝑧 = (𝑃‘2)) → (𝑃‘0) = 𝑥)
108	101	biimpi 215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (𝑃‘1) → (𝑃‘1) = 𝑦)
109	108	ad2antlr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) ∧ 𝑧 = (𝑃‘2)) → (𝑃‘1) = 𝑦)
110	107, 109	preq12d 4683	. . . . . . . . . . . . . . . . . . . 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 215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = (𝑃‘2) → (𝑃‘2) = 𝑧)
113	112	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 = (𝑃‘0) ∧ 𝑦 = (𝑃‘1)) ∧ 𝑧 = (𝑃‘2)) → (𝑃‘2) = 𝑧)
114	109, 113	preq12d 4683	. . . . . . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . . . 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 512	. . . . . . . . . . . . . . . 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 435	. . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . 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 418	. . . . . . . . . . . . 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 3551	. . . . . . . . . . . 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 3551	. . . . . . . . . . 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 3551	. . . . . . . . . 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 435	. . . . . . . . 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 481	. . . . 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 7279	. . . . . . . 8 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = (0..^2))
131	130	raleqdv 3347	. . . . . . 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 13470	. . . . . . . . 9 ⊢ (0..^2) = {0, 1}
133	132	raleqi 3345	. . . . . . . 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 28032	. . . . . . . 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 274	. . . . . . 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 482	. . . . 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 7279	. . . . . . . 8 ⊢ ((♯‘𝐹) = 2 → (0...(♯‘𝐹)) = (0...2))
139	138	feq2d 6584	. . . . . . 7 ⊢ ((♯‘𝐹) = 2 → (𝑃:(0...(♯‘𝐹))⟶𝑉 ↔ 𝑃:(0...2)⟶𝑉))
140	139	adantl 482	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝑃:(0...(♯‘𝐹))⟶𝑉 ↔ 𝑃:(0...2)⟶𝑉))
141		f1eq2 6664	. . . . . . . . 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 342	. . . . . . 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 482	. . . . . 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 345	. . . . 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 1110	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ∀wral 3066 ∃wrex 3067 Vcvv 3431 ∖ cdif 3889 {csn 4567 {cpr 4569 class class class wbr 5079 dom cdm 5590 ran crn 5591 ⟶wf 6428 –1-1→wf1 6429 ‘cfv 6432 (class class class)co 7271 0cc0 10872 1c1 10873 + caddc 10875 < clt 11010 ≤ cle 11011 ℕcn 11973 2c2 12028 ℕ₀cn0 12233 ...cfz 13238 ..^cfzo 13381 ♯chash 14042 Vtxcvtx 27364 iEdgciedg 27365 USGraphcusgr 27517

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-oadd 8292 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-dju 9660 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-fzo 13382 df-hash 14043 df-umgr 27451 df-usgr 27519

This theorem is referenced by: usgr2pth 28128

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