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Theorem usgra2wlkspthlem2 26148

Description: Lemma 2 for usgra2wlkspth 26149. (Contributed by Alexander van der Vekens, 2-Mar-2018.)

**Assertion**
Ref	Expression
usgra2wlkspthlem2	⊢ (((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ (𝑉 USGrph 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉)) → ((((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → Fun ^◡𝑃))

Distinct variable groups: 𝑖,𝐸 𝑖,𝐹 𝑃,𝑖 Allowed substitution hints: 𝐴(𝑖) 𝐵(𝑖) 𝑉(𝑖)

**Proof of Theorem usgra2wlkspthlem2**
Step	Hyp	Ref	Expression
1		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 2 → (0...(#‘𝐹)) = (0...2))
2	1	feq2d 5944	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 2 → (𝑃:(0...(#‘𝐹))⟶𝑉 ↔ 𝑃:(0...2)⟶𝑉))
3		fz0tp 12309	. . . . . . . . . . . . . . 15 ⊢ (0...2) = {0, 1, 2}
4	3	feq2i 5950	. . . . . . . . . . . . . 14 ⊢ (𝑃:(0...2)⟶𝑉 ↔ 𝑃:{0, 1, 2}⟶𝑉)
5	4	biimpi 205	. . . . . . . . . . . . 13 ⊢ (𝑃:(0...2)⟶𝑉 → 𝑃:{0, 1, 2}⟶𝑉)
6	2, 5	syl6bi 242	. . . . . . . . . . . 12 ⊢ ((#‘𝐹) = 2 → (𝑃:(0...(#‘𝐹))⟶𝑉 → 𝑃:{0, 1, 2}⟶𝑉))
7	6	ad2antll 761	. . . . . . . . . . 11 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2)) → (𝑃:(0...(#‘𝐹))⟶𝑉 → 𝑃:{0, 1, 2}⟶𝑉))
8	7	adantr 480	. . . . . . . . . 10 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2)) ∧ (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) → (𝑃:(0...(#‘𝐹))⟶𝑉 → 𝑃:{0, 1, 2}⟶𝑉))
9	8	impcom 445	. . . . . . . . 9 ⊢ ((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2)) ∧ (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))) → 𝑃:{0, 1, 2}⟶𝑉)
10		eqeq12 2623	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘2) = 𝐵) → ((𝑃‘0) = (𝑃‘2) ↔ 𝐴 = 𝐵))
11	10	biimpd 218	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘2) = 𝐵) → ((𝑃‘0) = (𝑃‘2) → 𝐴 = 𝐵))
12	11	necon3d 2803	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘2) = 𝐵) → (𝐴 ≠ 𝐵 → (𝑃‘0) ≠ (𝑃‘2)))
13	12	3impia 1253	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘2) = 𝐵 ∧ 𝐴 ≠ 𝐵) → (𝑃‘0) ≠ (𝑃‘2))
14	13	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑃‘0) = 𝐴 ∧ (𝑃‘2) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃‘0) ≠ (𝑃‘2))
15	14	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) ∧ (((𝑃‘0) = 𝐴 ∧ (𝑃‘2) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → (𝑃‘0) ≠ (𝑃‘2))
16	15	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) ∧ (((𝑃‘0) = 𝐴 ∧ (𝑃‘2) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ∧ 𝑃:(0...2)⟶𝑉) → (𝑃‘0) ≠ (𝑃‘2))
17		simplr 788	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) ∧ (𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}) → 𝑉 USGrph 𝐸)
18		usgrafun 25878	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑉 USGrph 𝐸 → Fun 𝐸)
19		wrdf 13165	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐹 ∈ Word dom 𝐸 → 𝐹:(0..^(#‘𝐹))⟶dom 𝐸)
20		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = (0..^2))
21	20	feq2d 5944	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((#‘𝐹) = 2 → (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 ↔ 𝐹:(0..^2)⟶dom 𝐸))
22		c0ex 9913	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 0 ∈ V
23	22	prid1 4241	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 0 ∈ {0, 1}
24		fzo0to2pr 12420	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (0..^2) = {0, 1}
25	23, 24	eleqtrri 2687	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ (0..^2)
26		ffvelrn 6265	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐹:(0..^2)⟶dom 𝐸 ∧ 0 ∈ (0..^2)) → (𝐹‘0) ∈ dom 𝐸)
27	25, 26	mpan2 703	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹:(0..^2)⟶dom 𝐸 → (𝐹‘0) ∈ dom 𝐸)
28	21, 27	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((#‘𝐹) = 2 → (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 → (𝐹‘0) ∈ dom 𝐸))
29	19, 28	mpan9 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) → (𝐹‘0) ∈ dom 𝐸)
30		fvelrn 6260	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Fun 𝐸 ∧ (𝐹‘0) ∈ dom 𝐸) → (𝐸‘(𝐹‘0)) ∈ ran 𝐸)
31	18, 29, 30	syl2anr 494	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) → (𝐸‘(𝐹‘0)) ∈ ran 𝐸)
32	31	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) ∧ (𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}) → (𝐸‘(𝐹‘0)) ∈ ran 𝐸)
33		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} → ((𝐸‘(𝐹‘0)) ∈ ran 𝐸 ↔ {(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸))
34	33	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) ∧ (𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}) → ((𝐸‘(𝐹‘0)) ∈ ran 𝐸 ↔ {(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸))
35	32, 34	mpbid 221	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) ∧ (𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}) → {(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸)
36		usgraedgrn 25910	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑉 USGrph 𝐸 ∧ {(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸) → (𝑃‘0) ≠ (𝑃‘1))
37	17, 35, 36	syl2anc 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) ∧ (𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}) → (𝑃‘0) ≠ (𝑃‘1))
38	37	ex 449	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) → ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} → (𝑃‘0) ≠ (𝑃‘1)))
39		simplr 788	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → 𝑉 USGrph 𝐸)
40		1ex 9914	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 1 ∈ V
41	40	prid2 4242	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ {0, 1}
42	41, 24	eleqtrri 2687	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 1 ∈ (0..^2)
43		ffvelrn 6265	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐹:(0..^2)⟶dom 𝐸 ∧ 1 ∈ (0..^2)) → (𝐹‘1) ∈ dom 𝐸)
44	42, 43	mpan2 703	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹:(0..^2)⟶dom 𝐸 → (𝐹‘1) ∈ dom 𝐸)
45	21, 44	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((#‘𝐹) = 2 → (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 → (𝐹‘1) ∈ dom 𝐸))
46	19, 45	mpan9 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) → (𝐹‘1) ∈ dom 𝐸)
47		fvelrn 6260	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Fun 𝐸 ∧ (𝐹‘1) ∈ dom 𝐸) → (𝐸‘(𝐹‘1)) ∈ ran 𝐸)
48	18, 46, 47	syl2anr 494	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) → (𝐸‘(𝐹‘1)) ∈ ran 𝐸)
49	48	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝐸‘(𝐹‘1)) ∈ ran 𝐸)
50		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} → ((𝐸‘(𝐹‘1)) ∈ ran 𝐸 ↔ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸))
51	50	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → ((𝐸‘(𝐹‘1)) ∈ ran 𝐸 ↔ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸))
52	49, 51	mpbid 221	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸)
53		usgraedgrn 25910	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑉 USGrph 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) → (𝑃‘1) ≠ (𝑃‘2))
54	39, 52, 53	syl2anc 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘1) ≠ (𝑃‘2))
55	54	ex 449	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) → ((𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} → (𝑃‘1) ≠ (𝑃‘2)))
56	38, 55	anim12d 584	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) → (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2))))
57	56	adantld 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) → ((((𝑃‘0) = 𝐴 ∧ (𝑃‘2) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2))))
58	57	imp 444	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) ∧ (((𝑃‘0) = 𝐴 ∧ (𝑃‘2) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2)))
59	58	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) ∧ (((𝑃‘0) = 𝐴 ∧ (𝑃‘2) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ∧ 𝑃:(0...2)⟶𝑉) → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2)))
60		3anan12 1044	. . . . . . . . . . . . . . 15 ⊢ (((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2)) ↔ ((𝑃‘0) ≠ (𝑃‘2) ∧ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2))))
61	16, 59, 60	sylanbrc 695	. . . . . . . . . . . . . 14 ⊢ (((((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑉 USGrph 𝐸) ∧ (((𝑃‘0) = 𝐴 ∧ (𝑃‘2) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ∧ 𝑃:(0...2)⟶𝑉) → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2)))
62	61	exp41 636	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) → (𝑉 USGrph 𝐸 → ((((𝑃‘0) = 𝐴 ∧ (𝑃‘2) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃:(0...2)⟶𝑉 → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2))))))
63	62	impcom 445	. . . . . . . . . . . 12 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2)) → ((((𝑃‘0) = 𝐴 ∧ (𝑃‘2) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃:(0...2)⟶𝑉 → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2)))))
64		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝐹) = 2 → (𝑃‘(#‘𝐹)) = (𝑃‘2))
65	64	eqeq1d 2612	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐹) = 2 → ((𝑃‘(#‘𝐹)) = 𝐵 ↔ (𝑃‘2) = 𝐵))
66	65	3anbi2d 1396	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐹) = 2 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ↔ ((𝑃‘0) = 𝐴 ∧ (𝑃‘2) = 𝐵 ∧ 𝐴 ≠ 𝐵)))
67	20, 24	syl6eq 2660	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = {0, 1})
68	67	raleqdv 3121	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐹) = 2 → (∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ∀𝑖 ∈ {0, 1} (𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
69		2wlklem 26094	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑖 ∈ {0, 1} (𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
70	68, 69	syl6bb 275	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐹) = 2 → (∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))
71	66, 70	anbi12d 743	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 2 → ((((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ↔ (((𝑃‘0) = 𝐴 ∧ (𝑃‘2) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
72	2	imbi1d 330	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 2 → ((𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2))) ↔ (𝑃:(0...2)⟶𝑉 → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2)))))
73	71, 72	imbi12d 333	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 2 → (((((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2)))) ↔ ((((𝑃‘0) = 𝐴 ∧ (𝑃‘2) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃:(0...2)⟶𝑉 → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2))))))
74	73	ad2antll 761	. . . . . . . . . . . 12 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2)) → (((((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2)))) ↔ ((((𝑃‘0) = 𝐴 ∧ (𝑃‘2) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃:(0...2)⟶𝑉 → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2))))))
75	63, 74	mpbird 246	. . . . . . . . . . 11 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2)) → ((((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2)))))
76	75	imp 444	. . . . . . . . . 10 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2)) ∧ (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2))))
77	76	impcom 445	. . . . . . . . 9 ⊢ ((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2)) ∧ (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))) → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2)))
78		2z 11286	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
79	22, 40, 78	3pm3.2i 1232	. . . . . . . . . . 11 ⊢ (0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ)
80		0ne1 10965	. . . . . . . . . . . 12 ⊢ 0 ≠ 1
81		0ne2 11116	. . . . . . . . . . . 12 ⊢ 0 ≠ 2
82		1ne2 11117	. . . . . . . . . . . 12 ⊢ 1 ≠ 2
83	80, 81, 82	3pm3.2i 1232	. . . . . . . . . . 11 ⊢ (0 ≠ 1 ∧ 0 ≠ 2 ∧ 1 ≠ 2)
84	79, 83	pm3.2i 470	. . . . . . . . . 10 ⊢ ((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (0 ≠ 1 ∧ 0 ≠ 2 ∧ 1 ≠ 2))
85		eqid 2610	. . . . . . . . . . 11 ⊢ {0, 1, 2} = {0, 1, 2}
86	85	f13dfv 6430	. . . . . . . . . 10 ⊢ (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (0 ≠ 1 ∧ 0 ≠ 2 ∧ 1 ≠ 2)) → (𝑃:{0, 1, 2}–1-1→𝑉 ↔ (𝑃:{0, 1, 2}⟶𝑉 ∧ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2)))))
87	84, 86	mp1i 13	. . . . . . . . 9 ⊢ ((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2)) ∧ (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))) → (𝑃:{0, 1, 2}–1-1→𝑉 ↔ (𝑃:{0, 1, 2}⟶𝑉 ∧ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2)))))
88	9, 77, 87	mpbir2and 959	. . . . . . . 8 ⊢ ((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2)) ∧ (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))) → 𝑃:{0, 1, 2}–1-1→𝑉)
89		df-f1 5809	. . . . . . . 8 ⊢ (𝑃:{0, 1, 2}–1-1→𝑉 ↔ (𝑃:{0, 1, 2}⟶𝑉 ∧ Fun ^◡𝑃))
90	88, 89	sylib 207	. . . . . . 7 ⊢ ((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2)) ∧ (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))) → (𝑃:{0, 1, 2}⟶𝑉 ∧ Fun ^◡𝑃))
91	90	simprd 478	. . . . . 6 ⊢ ((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2)) ∧ (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))) → Fun ^◡𝑃)
92	91	expcom 450	. . . . 5 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2)) ∧ (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) → (𝑃:(0...(#‘𝐹))⟶𝑉 → Fun ^◡𝑃))
93	92	ex 449	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2)) → ((((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (𝑃:(0...(#‘𝐹))⟶𝑉 → Fun ^◡𝑃)))
94	93	com23 84	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2)) → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → Fun ^◡𝑃)))
95	94	impancom 455	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → ((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) → ((((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → Fun ^◡𝑃)))
96	95	impcom 445	1 ⊢ (((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ (𝑉 USGrph 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉)) → ((((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵 ∧ 𝐴 ≠ 𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → Fun ^◡𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 Vcvv 3173 {cpr 4127 {ctp 4129 class class class wbr 4583 ^◡ccnv 5037 dom cdm 5038 ran crn 5039 Fun wfun 5798 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 2c2 10947 ℤcz 11254 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 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-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-fzo 12335 df-hash 12980 df-word 13154 df-usgra 25862

This theorem is referenced by: usgra2wlkspth 26149

W3C validator