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Theorem usgr2pth 27551
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 27549 . . . . 5 ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Paths‘𝐺)𝑃𝐹(SPaths‘𝐺)𝑃))
2 usgrupgr 26973 . . . . . . 7 (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)
32adantr 484 . . . . . 6 ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → 𝐺 ∈ UPGraph)
4 isspth 27511 . . . . . . . . 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 27491 . . . . . . . . . 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 632 . . . . . . . . 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 7148 . . . . . . . . . . . . . . . . 17 ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = (0..^2))
11 f1eq2 6552 . . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . . 15 ((♯‘𝐹) = 2 → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝐹:(0..^2)–1-1→dom 𝐼))
1413adantl 485 . . . . . . . . . . . . . 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 1130 . . . . . . . . . . . 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 727 . . . . . . . . . . 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 7148 . . . . . . . . . . . . . . . . . 18 ((♯‘𝐹) = 2 → (0...(♯‘𝐹)) = (0...2))
1918feq2d 6480 . . . . . . . . . . . . . . . . 17 ((♯‘𝐹) = 2 → (𝑃:(0...(♯‘𝐹))⟶𝑉𝑃:(0...2)⟶𝑉))
20 df-f1 6339 . . . . . . . . . . . . . . . . . . 19 (𝑃:(0...2)–1-1𝑉 ↔ (𝑃:(0...2)⟶𝑉 ∧ Fun 𝑃))
2120simplbi2 504 . . . . . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . . . . 16 ((♯‘𝐹) = 2 → (𝑃:(0...(♯‘𝐹))⟶𝑉 → (Fun 𝑃𝑃:(0...2)–1-1𝑉)))
2423adantl 485 . . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . 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 410 . . . . . . . . . . . 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 485 . . . . . . . . . . 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 27550 . . . . . . . . . . . 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 727 . . . . . . . . . . 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 1126 . . . . . . . . . 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 416 . . . . . . . . 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 243 . . . . . . . 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 243 . . . . . . 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 243 . . . 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 416 . . 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 415 . 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 11902 . . . . . 6 2 ∈ ℕ0
41 f1f 6556 . . . . . 6 (𝐹:(0..^2)–1-1→dom 𝐼𝐹:(0..^2)⟶dom 𝐼)
42 fnfzo0hash 13804 . . . . . 6 ((2 ∈ ℕ0𝐹:(0..^2)⟶dom 𝐼) → (♯‘𝐹) = 2)
4340, 41, 42sylancr 590 . . . . 5 (𝐹:(0..^2)–1-1→dom 𝐼 → (♯‘𝐹) = 2)
44 oveq2 7148 . . . . . . . . . . . . . . . . . 18 (2 = (♯‘𝐹) → (0..^2) = (0..^(♯‘𝐹)))
4544eqcoms 2830 . . . . . . . . . . . . . . . . 17 ((♯‘𝐹) = 2 → (0..^2) = (0..^(♯‘𝐹)))
46 f1eq2 6552 . . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . . 15 ((♯‘𝐹) = 2 → (𝐹:(0..^2)–1-1→dom 𝐼𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼))
4948imp 410 . . . . . . . . . . . . . 14 (((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
5049adantr 484 . . . . . . . . . . . . 13 ((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
5150ad2antrr 725 . . . . . . . . . . . 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 6556 . . . . . . . . . . . . . . 15 (𝑃:(0...2)–1-1𝑉𝑃:(0...2)⟶𝑉)
53 oveq2 7148 . . . . . . . . . . . . . . . . . 18 (2 = (♯‘𝐹) → (0...2) = (0...(♯‘𝐹)))
5453eqcoms 2830 . . . . . . . . . . . . . . . . 17 ((♯‘𝐹) = 2 → (0...2) = (0...(♯‘𝐹)))
5554adantr 484 . . . . . . . . . . . . . . . 16 (((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) → (0...2) = (0...(♯‘𝐹)))
5655feq2d 6480 . . . . . . . . . . . . . . 15 (((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) → (𝑃:(0...2)⟶𝑉𝑃:(0...(♯‘𝐹))⟶𝑉))
5752, 56syl5ib 247 . . . . . . . . . . . . . 14 (((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) → (𝑃:(0...2)–1-1𝑉𝑃:(0...(♯‘𝐹))⟶𝑉))
5857imp 410 . . . . . . . . . . . . 13 ((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) → 𝑃:(0...(♯‘𝐹))⟶𝑉)
5958ad2antrr 725 . . . . . . . . . . . 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 2829 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃‘0) = 𝑥𝑥 = (𝑃‘0))
6160biimpi 219 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃‘0) = 𝑥𝑥 = (𝑃‘0))
62613ad2ant1 1130 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → 𝑥 = (𝑃‘0))
63 eqcom 2829 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃‘1) = 𝑦𝑦 = (𝑃‘1))
6463biimpi 219 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃‘1) = 𝑦𝑦 = (𝑃‘1))
65643ad2ant2 1131 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → 𝑦 = (𝑃‘1))
6662, 65preq12d 4651 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → {𝑥, 𝑦} = {(𝑃‘0), (𝑃‘1)})
6766eqeq2d 2833 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ↔ (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}))
6867biimpcd 252 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} → (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}))
6968adantr 484 . . . . . . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . . . . . . 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 2829 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃‘2) = 𝑧𝑧 = (𝑃‘2))
7271biimpi 219 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃‘2) = 𝑧𝑧 = (𝑃‘2))
73723ad2ant3 1132 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → 𝑧 = (𝑃‘2))
7465, 73preq12d 4651 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → {𝑦, 𝑧} = {(𝑃‘1), (𝑃‘2)})
7574eqeq2d 2833 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → ((𝐼‘(𝐹‘1)) = {𝑦, 𝑧} ↔ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
7675biimpcd 252 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐼‘(𝐹‘1)) = {𝑦, 𝑧} → (((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) → (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
7776adantl 485 . . . . . . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . . . . . . 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 515 . . . . . . . . . . . . . . . . . . . 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 3268 . . . . . . . . . . . . . . . . . . 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 3268 . . . . . . . . . . . . . . . . . 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 3268 . . . . . . . . . . . . . . . . 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 13117 . . . . . . . . . . . . . . . . . . . 20 (0..^2) = {0, 1}
8510, 84syl6eq 2873 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = {0, 1})
8685raleqdv 3392 . . . . . . . . . . . . . . . . . 18 ((♯‘𝐹) = 2 → (∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ↔ ∀𝑖 ∈ {0, 1} (𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))
87 2wlklem 27455 . . . . . . . . . . . . . . . . . 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, 87syl6bb 290 . . . . . . . . . . . . . . . . 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 344 . . . . . . . . . . . . . . . 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 262 . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . 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 410 . . . . . . . . . . . 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 1125 . . . . . . . . . . 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 500 . . . . . . . . . . . . 13 (𝑃:(0...2)–1-1𝑉 → Fun 𝑃)
9695adantl 485 . . . . . . . . . . . 12 ((((♯‘𝐹) = 2 ∧ 𝐹:(0..^2)–1-1→dom 𝐼) ∧ 𝑃:(0...2)–1-1𝑉) → Fun 𝑃)
9796ad2antrr 725 . . . . . . . . . . 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 515 . . . . . . . . . 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 282 . . . . . . . . . . . 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 485 . . . . . . . . . 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 260 . . . . . . . . 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 488 . . . . . . . . . 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 782 . . . . . . . . . 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 587 . . . . . . . . 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 260 . . . . . . . 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 515 . . . . . . 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 416 . . . . . 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 438 . . . . 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 1108 . . 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 215 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wral 3130  wrex 3131  cdif 3905  {csn 4539  {cpr 4541   class class class wbr 5042  ccnv 5531  dom cdm 5532  Fun wfun 6328  wf 6330  1-1wf1 6331  cfv 6334  (class class class)co 7140  0cc0 10526  1c1 10527   + caddc 10529  2c2 11680  0cn0 11885  ...cfz 12885  ..^cfzo 13028  chash 13686  Vtxcvtx 26787  iEdgciedg 26788  UPGraphcupgr 26871  USGraphcusgr 26940  Trailsctrls 27478  Pathscpths 27499  SPathscspths 27500
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-dju 9318  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-hash 13687  df-word 13858  df-concat 13914  df-s1 13941  df-s2 14201  df-s3 14202  df-edg 26839  df-uhgr 26849  df-upgr 26873  df-umgr 26874  df-uspgr 26941  df-usgr 26942  df-wlks 27387  df-wlkson 27388  df-trls 27480  df-trlson 27481  df-pths 27503  df-spths 27504  df-pthson 27505  df-spthson 27506
This theorem is referenced by:  usgr2pth0  27552
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