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Theorem usgr2pth 29851

Description: In a simple graph, there is a path of length 2 iff there are three distinct vertices so that one of them is connected to each of the two others by an edge. (Contributed by Alexander van der Vekens, 27-Jan-2018.) (Revised by AV, 5-Jun-2021.) (Proof shortened by AV, 31-Oct-2021.)

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

**Assertion**
Ref	Expression
usgr2pth	⊢ (𝐺 ∈ USGraph → ((𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃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: 𝑥,𝐹,𝑦,𝑧 𝑥,𝐺,𝑦,𝑧 𝑥,𝐼,𝑦,𝑧 𝑥,𝑃,𝑦,𝑧 𝑥,𝑉,𝑦,𝑧

Proof of Theorem usgr2pth Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgr2pthspth 29849	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Paths‘𝐺)𝑃 ↔ 𝐹(SPaths‘𝐺)𝑃))
2		usgrupgr 29272	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)
3	2	adantr 480	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝐺 ∈ UPGraph)
4		isspth 29809	. . . . . . . . 9 ⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃))
5	4	a1i 11	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃)))
6		usgr2pthlem.v	. . . . . . . . . . 11 ⊢ 𝑉 = (Vtx‘𝐺)
7		usgr2pthlem.i	. . . . . . . . . . 11 ⊢ 𝐼 = (iEdg‘𝐺)
8	6, 7	upgrf1istrl 29789	. . . . . . . . . 10 ⊢ (𝐺 ∈ UPGraph → (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
9	8	anbi1d 632	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃) ↔ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃)))
10		oveq2 7370	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = (0..^2))
11		f1eq2 6728	. . . . . . . . . . . . . . . . 17 ⊢ ((0..^(♯‘𝐹)) = (0..^2) → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ↔ 𝐹:(0..^2)–1-1→dom 𝐼))
12	10, 11	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) = 2 → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ↔ 𝐹:(0..^2)–1-1→dom 𝐼))
13	12	biimpd 229	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) = 2 → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → 𝐹:(0..^2)–1-1→dom 𝐼))
14	13	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → 𝐹:(0..^2)–1-1→dom 𝐼))
15	14	com12 32	. . . . . . . . . . . . 13 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝐹:(0..^2)–1-1→dom 𝐼))
16	15	3ad2ant1 1134	. . . . . . . . . . . 12 ⊢ ((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) → 𝐹:(0..^2)–1-1→dom 𝐼))
17	16	ad2antrl 729	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃)) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝐹:(0..^2)–1-1→dom 𝐼))
18		oveq2 7370	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) = 2 → (0...(♯‘𝐹)) = (0...2))
19	18	feq2d 6648	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐹) = 2 → (𝑃:(0...(♯‘𝐹))⟶𝑉 ↔ 𝑃:(0...2)⟶𝑉))
20		df-f1 6499	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃:(0...2)–1-1→𝑉 ↔ (𝑃:(0...2)⟶𝑉 ∧ Fun ^◡𝑃))
21	20	simplbi2 500	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃:(0...2)⟶𝑉 → (Fun ^◡𝑃 → 𝑃:(0...2)–1-1→𝑉))
22	21	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐹) = 2 → (𝑃:(0...2)⟶𝑉 → (Fun ^◡𝑃 → 𝑃:(0...2)–1-1→𝑉)))
23	19, 22	sylbid 240	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) = 2 → (𝑃:(0...(♯‘𝐹))⟶𝑉 → (Fun ^◡𝑃 → 𝑃:(0...2)–1-1→𝑉)))
24	23	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝑃:(0...(♯‘𝐹))⟶𝑉 → (Fun ^◡𝑃 → 𝑃:(0...2)–1-1→𝑉)))
25	24	com3l 89	. . . . . . . . . . . . . 14 ⊢ (𝑃:(0...(♯‘𝐹))⟶𝑉 → (Fun ^◡𝑃 → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝑃:(0...2)–1-1→𝑉)))
26	25	3ad2ant2 1135	. . . . . . . . . . . . 13 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (Fun ^◡𝑃 → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝑃:(0...2)–1-1→𝑉)))
27	26	imp 406	. . . . . . . . . . . 12 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝑃:(0...2)–1-1→𝑉))
28	27	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃)) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝑃:(0...2)–1-1→𝑉))
29	6, 7	usgr2pthlem 29850	. . . . . . . . . . . 12 ⊢ ((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}))))
30	29	ad2antrl 729	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃)) → ((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}))))
31	17, 28, 30	3jcad 1130	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃)) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃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})))))
32	31	ex 412	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → (((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃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}))))))
33	9, 32	sylbid 240	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃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}))))))
34	5, 33	sylbid 240	. . . . . . 7 ⊢ (𝐺 ∈ UPGraph → (𝐹(SPaths‘𝐺)𝑃 → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃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}))))))
35	34	com23 86	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(SPaths‘𝐺)𝑃 → (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃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}))))))
36	3, 35	mpcom 38	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(SPaths‘𝐺)𝑃 → (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃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})))))
37	1, 36	sylbid 240	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Paths‘𝐺)𝑃 → (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃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})))))
38	37	ex 412	. . 3 ⊢ (𝐺 ∈ USGraph → ((♯‘𝐹) = 2 → (𝐹(Paths‘𝐺)𝑃 → (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃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}))))))
39	38	impcomd 411	. 2 ⊢ (𝐺 ∈ USGraph → ((𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) → (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃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})))))
40		2nn0 12449	. . . . . 6 ⊢ 2 ∈ ℕ₀
41		f1f 6732	. . . . . 6 ⊢ (𝐹:(0..^2)–1-1→dom 𝐼 → 𝐹:(0..^2)⟶dom 𝐼)
42		fnfzo0hash 14407	. . . . . 6 ⊢ ((2 ∈ ℕ₀ ∧ 𝐹:(0..^2)⟶dom 𝐼) → (♯‘𝐹) = 2)
43	40, 41, 42	sylancr 588	. . . . 5 ⊢ (𝐹:(0..^2)–1-1→dom 𝐼 → (♯‘𝐹) = 2)
44		oveq2 7370	. . . . . . . . . . . . . . . . . 18 ⊢ (2 = (♯‘𝐹) → (0..^2) = (0..^(♯‘𝐹)))
45	44	eqcoms 2745	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐹) = 2 → (0..^2) = (0..^(♯‘𝐹)))
46		f1eq2 6728	. . . . . . . . . . . . . . . . 17 ⊢ ((0..^2) = (0..^(♯‘𝐹)) → (𝐹:(0..^2)–1-1→dom 𝐼 ↔ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼))
47	45, 46	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) = 2 → (𝐹:(0..^2)–1-1→dom 𝐼 ↔ 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼))
48	47	biimpd 229	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) = 2 → (𝐹:(0..^2)–1-1→dom 𝐼 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼))
49	48	imp 406	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
50	49	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
51	50	ad2antrr 727	. . . . . . . . . . . 12 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃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}))) ∧ 𝐺 ∈ USGraph) → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
52		f1f 6732	. . . . . . . . . . . . . . 15 ⊢ (𝑃:(0...2)–1-1→𝑉 → 𝑃:(0...2)⟶𝑉)
53		oveq2 7370	. . . . . . . . . . . . . . . . . 18 ⊢ (2 = (♯‘𝐹) → (0...2) = (0...(♯‘𝐹)))
54	53	eqcoms 2745	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐹) = 2 → (0...2) = (0...(♯‘𝐹)))
55	54	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) → (0...2) = (0...(♯‘𝐹)))
56	55	feq2d 6648	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) → (𝑃:(0...2)⟶𝑉 ↔ 𝑃:(0...(♯‘𝐹))⟶𝑉))
57	52, 56	imbitrid 244	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) → (𝑃:(0...2)–1-1→𝑉 → 𝑃:(0...(♯‘𝐹))⟶𝑉))
58	57	imp 406	. . . . . . . . . . . . 13 ⊢ ((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) → 𝑃:(0...(♯‘𝐹))⟶𝑉)
59	58	ad2antrr 727	. . . . . . . . . . . 12 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃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}))) ∧ 𝐺 ∈ USGraph) → 𝑃:(0...(♯‘𝐹))⟶𝑉)
60		eqcom 2744	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃‘0) = 𝑥 ↔ 𝑥 = (𝑃‘0))
61	60	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃‘0) = 𝑥 → 𝑥 = (𝑃‘0))
62	61	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → 𝑥 = (𝑃‘0))
63		eqcom 2744	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃‘1) = 𝑦 ↔ 𝑦 = (𝑃‘1))
64	63	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃‘1) = 𝑦 → 𝑦 = (𝑃‘1))
65	64	3ad2ant2 1135	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → 𝑦 = (𝑃‘1))
66	62, 65	preq12d 4686	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → {𝑥, 𝑦} = {(𝑃‘0), (𝑃‘1)})
67	66	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ↔ (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}))
68	67	biimpcd 249	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} → (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}))
69	68	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}) → (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}))
70	69	impcom 407	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})) → (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)})
71		eqcom 2744	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃‘2) = 𝑧 ↔ 𝑧 = (𝑃‘2))
72	71	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃‘2) = 𝑧 → 𝑧 = (𝑃‘2))
73	72	3ad2ant3 1136	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → 𝑧 = (𝑃‘2))
74	65, 73	preq12d 4686	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → {𝑦, 𝑧} = {(𝑃‘1), (𝑃‘2)})
75	74	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → ((𝐼‘(𝐹‘1)) = {𝑦, 𝑧} ↔ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
76	75	biimpcd 249	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐼‘(𝐹‘1)) = {𝑦, 𝑧} → (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
77	76	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}) → (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
78	77	impcom 407	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})) → (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})
79	70, 78	jca 511	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})) → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
80	79	rexlimivw 3135	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})) → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
81	80	rexlimivw 3135	. . . . . . . . . . . . . . . . . 18 ⊢ (∃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})) → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
82	81	rexlimivw 3135	. . . . . . . . . . . . . . . . 17 ⊢ (∃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})) → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
83	82	a1i13 27	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘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})) → (𝐺 ∈ USGraph → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
84		fzo0to2pr 13700	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0..^2) = {0, 1}
85	10, 84	eqtrdi 2788	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = {0, 1})
86	85	raleqdv 3296	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) = 2 → (∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ∀𝑖 ∈ {0, 1} (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
87		2wlklem 29753	. . . . . . . . . . . . . . . . . 18 ⊢ (∀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)}))
88	86, 87	bitrdi 287	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘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)})))
89	88	imbi2d 340	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) = 2 → ((𝐺 ∈ USGraph → ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ↔ (𝐺 ∈ USGraph → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
90	83, 89	sylibrd 259	. . . . . . . . . . . . . . 15 ⊢ ((♯‘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})) → (𝐺 ∈ USGraph → ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
91	90	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ ((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) → (∃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})) → (𝐺 ∈ USGraph → ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
92	91	imp 406	. . . . . . . . . . . . 13 ⊢ (((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃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}))) → (𝐺 ∈ USGraph → ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
93	92	imp 406	. . . . . . . . . . . 12 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃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}))) ∧ 𝐺 ∈ USGraph) → ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})
94	51, 59, 93	3jca 1129	. . . . . . . . . . 11 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃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}))) ∧ 𝐺 ∈ USGraph) → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
95	20	simprbi 497	. . . . . . . . . . . . 13 ⊢ (𝑃:(0...2)–1-1→𝑉 → Fun ^◡𝑃)
96	95	adantl 481	. . . . . . . . . . . 12 ⊢ ((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) → Fun ^◡𝑃)
97	96	ad2antrr 727	. . . . . . . . . . 11 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃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}))) ∧ 𝐺 ∈ USGraph) → Fun ^◡𝑃)
98	94, 97	jca 511	. . . . . . . . . 10 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃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}))) ∧ 𝐺 ∈ USGraph) → ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃))
99	5, 9	bitrd 279	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ UPGraph → (𝐹(SPaths‘𝐺)𝑃 ↔ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃)))
100	2, 99	syl 17	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → (𝐹(SPaths‘𝐺)𝑃 ↔ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃)))
101	100	adantl 481	. . . . . . . . . 10 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃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}))) ∧ 𝐺 ∈ USGraph) → (𝐹(SPaths‘𝐺)𝑃 ↔ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun ^◡𝑃)))
102	98, 101	mpbird 257	. . . . . . . . 9 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃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}))) ∧ 𝐺 ∈ USGraph) → 𝐹(SPaths‘𝐺)𝑃)
103		simpr 484	. . . . . . . . . 10 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃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}))) ∧ 𝐺 ∈ USGraph) → 𝐺 ∈ USGraph)
104		simp-4l 783	. . . . . . . . . 10 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃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}))) ∧ 𝐺 ∈ USGraph) → (♯‘𝐹) = 2)
105	103, 104, 1	syl2anc 585	. . . . . . . . 9 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃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}))) ∧ 𝐺 ∈ USGraph) → (𝐹(Paths‘𝐺)𝑃 ↔ 𝐹(SPaths‘𝐺)𝑃))
106	102, 105	mpbird 257	. . . . . . . 8 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃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}))) ∧ 𝐺 ∈ USGraph) → 𝐹(Paths‘𝐺)𝑃)
107	106, 104	jca 511	. . . . . . 7 ⊢ ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃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}))) ∧ 𝐺 ∈ USGraph) → (𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2))
108	107	ex 412	. . . . . 6 ⊢ (((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) ∧ ∃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}))) → (𝐺 ∈ USGraph → (𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2)))
109	108	exp41 434	. . . . 5 ⊢ ((♯‘𝐹) = 2 → (𝐹:(0..^2)–1-1→dom 𝐼 → (𝑃:(0...2)–1-1→𝑉 → (∃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})) → (𝐺 ∈ USGraph → (𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2))))))
110	43, 109	mpcom 38	. . . 4 ⊢ (𝐹:(0..^2)–1-1→dom 𝐼 → (𝑃:(0...2)–1-1→𝑉 → (∃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})) → (𝐺 ∈ USGraph → (𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2)))))
111	110	3imp 1111	. . 3 ⊢ ((𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃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}))) → (𝐺 ∈ USGraph → (𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2)))
112	111	com12 32	. 2 ⊢ (𝐺 ∈ USGraph → ((𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃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(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2)))
113	39, 112	impbid 212	1 ⊢ (𝐺 ∈ USGraph → ((𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃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: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 ∖ cdif 3887 {csn 4568 {cpr 4570 class class class wbr 5086 ^◡ccnv 5625 dom cdm 5626 Fun wfun 6488 ⟶wf 6490 –1-1→wf1 6491 ‘cfv 6494 (class class class)co 7362 0cc0 11033 1c1 11034 + caddc 11036 2c2 12231 ℕ₀cn0 12432 ...cfz 13456 ..^cfzo 13603 ♯chash 14287 Vtxcvtx 29083 iEdgciedg 29084 UPGraphcupgr 29167 USGraphcusgr 29236 Trailsctrls 29776 Pathscpths 29797 SPathscspths 29798

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-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 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 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-oadd 8404 df-er 8638 df-map 8770 df-pm 8771 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-dju 9820 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-n0 12433 df-xnn0 12506 df-z 12520 df-uz 12784 df-fz 13457 df-fzo 13604 df-hash 14288 df-word 14471 df-concat 14528 df-s1 14554 df-s2 14805 df-s3 14806 df-edg 29135 df-uhgr 29145 df-upgr 29169 df-umgr 29170 df-uspgr 29237 df-usgr 29238 df-wlks 29687 df-wlkson 29688 df-trls 29778 df-trlson 29779 df-pths 29801 df-spths 29802 df-pthson 29803 df-spthson 29804

This theorem is referenced by: usgr2pth0 29852

W3C validator