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Theorem usgr2pth 29784
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 𝐼𝑃:(0...2)–1-1𝑉 ∧ ∃𝑥𝑉𝑦 ∈ (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.
StepHypRef Expression
1 usgr2pthspth 29782 . . . . 5 ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Paths‘𝐺)𝑃𝐹(SPaths‘𝐺)𝑃))
2 usgrupgr 29202 . . . . . . 7 (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)
32adantr 480 . . . . . 6 ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝐺 ∈ UPGraph)
4 isspth 29742 . . . . . . . . 9 (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun 𝑃))
54a1i 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)
86, 7upgrf1istrl 29721 . . . . . . . . . 10 (𝐺 ∈ UPGraph → (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))
98anbi1d 631 . . . . . . . . 9 (𝐺 ∈ UPGraph → ((𝐹(Trails‘𝐺)𝑃 ∧ Fun 𝑃) ↔ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun 𝑃)))
10 oveq2 7439 . . . . . . . . . . . . . . . . 17 ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = (0..^2))
11 f1eq2 6800 . . . . . . . . . . . . . . . . 17 ((0..^(♯‘𝐹)) = (0..^2) → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝐹:(0..^2)–1-1→dom 𝐼))
1210, 11syl 17 . . . . . . . . . . . . . . . 16 ((♯‘𝐹) = 2 → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝐹:(0..^2)–1-1→dom 𝐼))
1312biimpd 229 . . . . . . . . . . . . . . 15 ((♯‘𝐹) = 2 → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝐹:(0..^2)–1-1→dom 𝐼))
1413adantl 481 . . . . . . . . . . . . . 14 ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝐹:(0..^2)–1-1→dom 𝐼))
1514com12 32 . . . . . . . . . . . . 13 (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝐹:(0..^2)–1-1→dom 𝐼))
16153ad2ant1 1134 . . . . . . . . . . . 12 ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝐹:(0..^2)–1-1→dom 𝐼))
1716ad2antrl 728 . . . . . . . . . . 11 ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun 𝑃)) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝐹:(0..^2)–1-1→dom 𝐼))
18 oveq2 7439 . . . . . . . . . . . . . . . . . 18 ((♯‘𝐹) = 2 → (0...(♯‘𝐹)) = (0...2))
1918feq2d 6722 . . . . . . . . . . . . . . . . 17 ((♯‘𝐹) = 2 → (𝑃:(0...(♯‘𝐹))⟶𝑉𝑃:(0...2)⟶𝑉))
20 df-f1 6566 . . . . . . . . . . . . . . . . . . 19 (𝑃:(0...2)–1-1𝑉 ↔ (𝑃:(0...2)⟶𝑉 ∧ Fun 𝑃))
2120simplbi2 500 . . . . . . . . . . . . . . . . . 18 (𝑃:(0...2)⟶𝑉 → (Fun 𝑃𝑃:(0...2)–1-1𝑉))
2221a1i 11 . . . . . . . . . . . . . . . . 17 ((♯‘𝐹) = 2 → (𝑃:(0...2)⟶𝑉 → (Fun 𝑃𝑃:(0...2)–1-1𝑉)))
2319, 22sylbid 240 . . . . . . . . . . . . . . . 16 ((♯‘𝐹) = 2 → (𝑃:(0...(♯‘𝐹))⟶𝑉 → (Fun 𝑃𝑃:(0...2)–1-1𝑉)))
2423adantl 481 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝑃:(0...(♯‘𝐹))⟶𝑉 → (Fun 𝑃𝑃:(0...2)–1-1𝑉)))
2524com3l 89 . . . . . . . . . . . . . 14 (𝑃:(0...(♯‘𝐹))⟶𝑉 → (Fun 𝑃 → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝑃:(0...2)–1-1𝑉)))
26253ad2ant2 1135 . . . . . . . . . . . . 13 ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → (Fun 𝑃 → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝑃:(0...2)–1-1𝑉)))
2726imp 406 . . . . . . . . . . . 12 (((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun 𝑃) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝑃:(0...2)–1-1𝑉))
2827adantl 481 . . . . . . . . . . 11 ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun 𝑃)) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝑃:(0...2)–1-1𝑉))
296, 7usgr2pthlem 29783 . . . . . . . . . . . 12 ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → ∃𝑥𝑉𝑦 ∈ (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}))))
3029ad2antrl 728 . . . . . . . . . . 11 ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun 𝑃)) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → ∃𝑥𝑉𝑦 ∈ (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}))))
3117, 28, 303jcad 1130 . . . . . . . . . 10 ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun 𝑃)) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹:(0..^2)–1-1→dom 𝐼𝑃:(0...2)–1-1𝑉 ∧ ∃𝑥𝑉𝑦 ∈ (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})))))
3231ex 412 . . . . . . . . 9 (𝐺 ∈ UPGraph → (((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun 𝑃) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹:(0..^2)–1-1→dom 𝐼𝑃:(0...2)–1-1𝑉 ∧ ∃𝑥𝑉𝑦 ∈ (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}))))))
339, 32sylbid 240 . . . . . . . 8 (𝐺 ∈ UPGraph → ((𝐹(Trails‘𝐺)𝑃 ∧ Fun 𝑃) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹:(0..^2)–1-1→dom 𝐼𝑃:(0...2)–1-1𝑉 ∧ ∃𝑥𝑉𝑦 ∈ (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}))))))
345, 33sylbid 240 . . . . . . 7 (𝐺 ∈ UPGraph → (𝐹(SPaths‘𝐺)𝑃 → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹:(0..^2)–1-1→dom 𝐼𝑃:(0...2)–1-1𝑉 ∧ ∃𝑥𝑉𝑦 ∈ (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}))))))
3534com23 86 . . . . . 6 (𝐺 ∈ UPGraph → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(SPaths‘𝐺)𝑃 → (𝐹:(0..^2)–1-1→dom 𝐼𝑃:(0...2)–1-1𝑉 ∧ ∃𝑥𝑉𝑦 ∈ (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}))))))
363, 35mpcom 38 . . . . 5 ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(SPaths‘𝐺)𝑃 → (𝐹:(0..^2)–1-1→dom 𝐼𝑃:(0...2)–1-1𝑉 ∧ ∃𝑥𝑉𝑦 ∈ (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})))))
371, 36sylbid 240 . . . 4 ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Paths‘𝐺)𝑃 → (𝐹:(0..^2)–1-1→dom 𝐼𝑃:(0...2)–1-1𝑉 ∧ ∃𝑥𝑉𝑦 ∈ (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})))))
3837ex 412 . . 3 (𝐺 ∈ USGraph → ((♯‘𝐹) = 2 → (𝐹(Paths‘𝐺)𝑃 → (𝐹:(0..^2)–1-1→dom 𝐼𝑃:(0...2)–1-1𝑉 ∧ ∃𝑥𝑉𝑦 ∈ (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}))))))
3938impcomd 411 . 2 (𝐺 ∈ USGraph → ((𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) → (𝐹:(0..^2)–1-1→dom 𝐼𝑃:(0...2)–1-1𝑉 ∧ ∃𝑥𝑉𝑦 ∈ (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 12543 . . . . . 6 2 ∈ ℕ0
41 f1f 6804 . . . . . 6 (𝐹:(0..^2)–1-1→dom 𝐼𝐹:(0..^2)⟶dom 𝐼)
42 fnfzo0hash 14489 . . . . . 6 ((2 ∈ ℕ0𝐹:(0..^2)⟶dom 𝐼) → (♯‘𝐹) = 2)
4340, 41, 42sylancr 587 . . . . 5 (𝐹:(0..^2)–1-1→dom 𝐼 → (♯‘𝐹) = 2)
44 oveq2 7439 . . . . . . . . . . . . . . . . . 18 (2 = (♯‘𝐹) → (0..^2) = (0..^(♯‘𝐹)))
4544eqcoms 2745 . . . . . . . . . . . . . . . . 17 ((♯‘𝐹) = 2 → (0..^2) = (0..^(♯‘𝐹)))
46 f1eq2 6800 . . . . . . . . . . . . . . . . 17 ((0..^2) = (0..^(♯‘𝐹)) → (𝐹:(0..^2)–1-1→dom 𝐼𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼))
4745, 46syl 17 . . . . . . . . . . . . . . . 16 ((♯‘𝐹) = 2 → (𝐹:(0..^2)–1-1→dom 𝐼𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼))
4847biimpd 229 . . . . . . . . . . . . . . 15 ((♯‘𝐹) = 2 → (𝐹:(0..^2)–1-1→dom 𝐼𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼))
4948imp 406 . . . . . . . . . . . . . 14 (((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
5049adantr 480 . . . . . . . . . . . . 13 ((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
5150ad2antrr 726 . . . . . . . . . . . 12 ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) ∧ ∃𝑥𝑉𝑦 ∈ (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 6804 . . . . . . . . . . . . . . 15 (𝑃:(0...2)–1-1𝑉𝑃:(0...2)⟶𝑉)
53 oveq2 7439 . . . . . . . . . . . . . . . . . 18 (2 = (♯‘𝐹) → (0...2) = (0...(♯‘𝐹)))
5453eqcoms 2745 . . . . . . . . . . . . . . . . 17 ((♯‘𝐹) = 2 → (0...2) = (0...(♯‘𝐹)))
5554adantr 480 . . . . . . . . . . . . . . . 16 (((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) → (0...2) = (0...(♯‘𝐹)))
5655feq2d 6722 . . . . . . . . . . . . . . 15 (((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) → (𝑃:(0...2)⟶𝑉𝑃:(0...(♯‘𝐹))⟶𝑉))
5752, 56imbitrid 244 . . . . . . . . . . . . . 14 (((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) → (𝑃:(0...2)–1-1𝑉𝑃:(0...(♯‘𝐹))⟶𝑉))
5857imp 406 . . . . . . . . . . . . 13 ((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) → 𝑃:(0...(♯‘𝐹))⟶𝑉)
5958ad2antrr 726 . . . . . . . . . . . 12 ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) ∧ ∃𝑥𝑉𝑦 ∈ (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) = 𝑥𝑥 = (𝑃‘0))
6160biimpi 216 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃‘0) = 𝑥𝑥 = (𝑃‘0))
62613ad2ant1 1134 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → 𝑥 = (𝑃‘0))
63 eqcom 2744 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃‘1) = 𝑦𝑦 = (𝑃‘1))
6463biimpi 216 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃‘1) = 𝑦𝑦 = (𝑃‘1))
65643ad2ant2 1135 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → 𝑦 = (𝑃‘1))
6662, 65preq12d 4741 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → {𝑥, 𝑦} = {(𝑃‘0), (𝑃‘1)})
6766eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ↔ (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}))
6867biimpcd 249 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} → (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}))
6968adantr 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)}))
7069impcom 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) = 𝑧𝑧 = (𝑃‘2))
7271biimpi 216 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃‘2) = 𝑧𝑧 = (𝑃‘2))
73723ad2ant3 1136 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → 𝑧 = (𝑃‘2))
7465, 73preq12d 4741 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → {𝑦, 𝑧} = {(𝑃‘1), (𝑃‘2)})
7574eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → ((𝐼‘(𝐹‘1)) = {𝑦, 𝑧} ↔ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
7675biimpcd 249 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐼‘(𝐹‘1)) = {𝑦, 𝑧} → (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
7776adantl 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)}))
7877impcom 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)})
7970, 78jca 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)}))
8079rexlimivw 3151 . . . . . . . . . . . . . . . . . . 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)}))
8180rexlimivw 3151 . . . . . . . . . . . . . . . . . 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)}))
8281rexlimivw 3151 . . . . . . . . . . . . . . . . 17 (∃𝑥𝑉𝑦 ∈ (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)}))
8382a1i13 27 . . . . . . . . . . . . . . . 16 ((♯‘𝐹) = 2 → (∃𝑥𝑉𝑦 ∈ (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 13789 . . . . . . . . . . . . . . . . . . . 20 (0..^2) = {0, 1}
8510, 84eqtrdi 2793 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = {0, 1})
8685raleqdv 3326 . . . . . . . . . . . . . . . . . 18 ((♯‘𝐹) = 2 → (∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ↔ ∀𝑖 ∈ {0, 1} (𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))
87 2wlklem 29685 . . . . . . . . . . . . . . . . . 18 (∀𝑖 ∈ {0, 1} (𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ↔ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
8886, 87bitrdi 287 . . . . . . . . . . . . . . . . 17 ((♯‘𝐹) = 2 → (∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ↔ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))
8988imbi2d 340 . . . . . . . . . . . . . . . 16 ((♯‘𝐹) = 2 → ((𝐺 ∈ USGraph → ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) ↔ (𝐺 ∈ USGraph → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
9083, 89sylibrd 259 . . . . . . . . . . . . . . 15 ((♯‘𝐹) = 2 → (∃𝑥𝑉𝑦 ∈ (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‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))
9190ad2antrr 726 . . . . . . . . . . . . . 14 ((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) → (∃𝑥𝑉𝑦 ∈ (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‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))
9291imp 406 . . . . . . . . . . . . 13 (((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) ∧ ∃𝑥𝑉𝑦 ∈ (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‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))
9392imp 406 . . . . . . . . . . . 12 ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) ∧ ∃𝑥𝑉𝑦 ∈ (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‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})
9451, 59, 933jca 1129 . . . . . . . . . . 11 ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) ∧ ∃𝑥𝑉𝑦 ∈ (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 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))
9520simprbi 496 . . . . . . . . . . . . 13 (𝑃:(0...2)–1-1𝑉 → Fun 𝑃)
9695adantl 481 . . . . . . . . . . . 12 ((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) → Fun 𝑃)
9796ad2antrr 726 . . . . . . . . . . 11 ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) ∧ ∃𝑥𝑉𝑦 ∈ (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)
9894, 97jca 511 . . . . . . . . . 10 ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) ∧ ∃𝑥𝑉𝑦 ∈ (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 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun 𝑃))
995, 9bitrd 279 . . . . . . . . . . . 12 (𝐺 ∈ UPGraph → (𝐹(SPaths‘𝐺)𝑃 ↔ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun 𝑃)))
1002, 99syl 17 . . . . . . . . . . 11 (𝐺 ∈ USGraph → (𝐹(SPaths‘𝐺)𝑃 ↔ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun 𝑃)))
101100adantl 481 . . . . . . . . . 10 ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) ∧ ∃𝑥𝑉𝑦 ∈ (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 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) ∧ Fun 𝑃)))
10298, 101mpbird 257 . . . . . . . . 9 ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) ∧ ∃𝑥𝑉𝑦 ∈ (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)–1-1𝑉) ∧ ∃𝑥𝑉𝑦 ∈ (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)–1-1𝑉) ∧ ∃𝑥𝑉𝑦 ∈ (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)
105103, 104, 1syl2anc 584 . . . . . . . . 9 ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) ∧ ∃𝑥𝑉𝑦 ∈ (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)𝑃𝐹(SPaths‘𝐺)𝑃))
106102, 105mpbird 257 . . . . . . . 8 ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) ∧ ∃𝑥𝑉𝑦 ∈ (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)
107106, 104jca 511 . . . . . . 7 ((((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) ∧ ∃𝑥𝑉𝑦 ∈ (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))
108107ex 412 . . . . . 6 (((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) ∧ ∃𝑥𝑉𝑦 ∈ (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)))
109108exp41 434 . . . . 5 ((♯‘𝐹) = 2 → (𝐹:(0..^2)–1-1→dom 𝐼 → (𝑃:(0...2)–1-1𝑉 → (∃𝑥𝑉𝑦 ∈ (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))))))
11043, 109mpcom 38 . . . 4 (𝐹:(0..^2)–1-1→dom 𝐼 → (𝑃:(0...2)–1-1𝑉 → (∃𝑥𝑉𝑦 ∈ (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)))))
1111103imp 1111 . . 3 ((𝐹:(0..^2)–1-1→dom 𝐼𝑃:(0...2)–1-1𝑉 ∧ ∃𝑥𝑉𝑦 ∈ (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)))
112111com12 32 . 2 (𝐺 ∈ USGraph → ((𝐹:(0..^2)–1-1→dom 𝐼𝑃:(0...2)–1-1𝑉 ∧ ∃𝑥𝑉𝑦 ∈ (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)))
11339, 112impbid 212 1 (𝐺 ∈ USGraph → ((𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ (𝐹:(0..^2)–1-1→dom 𝐼𝑃:(0...2)–1-1𝑉 ∧ ∃𝑥𝑉𝑦 ∈ (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 1540  wcel 2108  wral 3061  wrex 3070  cdif 3948  {csn 4626  {cpr 4628   class class class wbr 5143  ccnv 5684  dom cdm 5685  Fun wfun 6555  wf 6557  1-1wf1 6558  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156   + caddc 11158  2c2 12321  0cn0 12526  ...cfz 13547  ..^cfzo 13694  chash 14369  Vtxcvtx 29013  iEdgciedg 29014  UPGraphcupgr 29097  USGraphcusgr 29166  Trailsctrls 29708  Pathscpths 29730  SPathscspths 29731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
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 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-concat 14609  df-s1 14634  df-s2 14887  df-s3 14888  df-edg 29065  df-uhgr 29075  df-upgr 29099  df-umgr 29100  df-uspgr 29167  df-usgr 29168  df-wlks 29617  df-wlkson 29618  df-trls 29710  df-trlson 29711  df-pths 29734  df-spths 29735  df-pthson 29736  df-spthson 29737
This theorem is referenced by:  usgr2pth0  29785
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