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Theorem usgr2pth0 28755
Description: In a simply 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.)
Hypotheses
Ref Expression
usgr2pthlem.v 𝑉 = (Vtxβ€˜πΊ)
usgr2pthlem.i 𝐼 = (iEdgβ€˜πΊ)
Assertion
Ref Expression
usgr2pth0 (𝐺 ∈ USGraph β†’ ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 2) ↔ (𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))))
Distinct variable groups:   π‘₯,𝐹,𝑦,𝑧   π‘₯,𝐺,𝑦,𝑧   π‘₯,𝐼,𝑦,𝑧   π‘₯,𝑃,𝑦,𝑧   π‘₯,𝑉,𝑦,𝑧

Proof of Theorem usgr2pth0
StepHypRef Expression
1 usgr2pthlem.v . . 3 𝑉 = (Vtxβ€˜πΊ)
2 usgr2pthlem.i . . 3 𝐼 = (iEdgβ€˜πΊ)
31, 2usgr2pth 28754 . 2 (𝐺 ∈ USGraph β†’ ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 2) ↔ (𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯, 𝑧})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))))
4 r19.42v 3188 . . . . . . . . 9 (βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯, 𝑧})(𝑧 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))) ↔ (𝑧 β‰  π‘₯ ∧ βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯, 𝑧})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))))
5 rexdifpr 4624 . . . . . . . . 9 (βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯, 𝑧})(𝑧 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))) ↔ βˆƒπ‘¦ ∈ 𝑉 (𝑦 β‰  π‘₯ ∧ 𝑦 β‰  𝑧 ∧ (𝑧 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))))
64, 5bitr3i 277 . . . . . . . 8 ((𝑧 β‰  π‘₯ ∧ βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯, 𝑧})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))) ↔ βˆƒπ‘¦ ∈ 𝑉 (𝑦 β‰  π‘₯ ∧ 𝑦 β‰  𝑧 ∧ (𝑧 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))))
76rexbii 3098 . . . . . . 7 (βˆƒπ‘§ ∈ 𝑉 (𝑧 β‰  π‘₯ ∧ βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯, 𝑧})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))) ↔ βˆƒπ‘§ ∈ 𝑉 βˆƒπ‘¦ ∈ 𝑉 (𝑦 β‰  π‘₯ ∧ 𝑦 β‰  𝑧 ∧ (𝑧 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))))
8 rexcom 3276 . . . . . . 7 (βˆƒπ‘§ ∈ 𝑉 βˆƒπ‘¦ ∈ 𝑉 (𝑦 β‰  π‘₯ ∧ 𝑦 β‰  𝑧 ∧ (𝑧 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))) ↔ βˆƒπ‘¦ ∈ 𝑉 βˆƒπ‘§ ∈ 𝑉 (𝑦 β‰  π‘₯ ∧ 𝑦 β‰  𝑧 ∧ (𝑧 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))))
9 df-3an 1090 . . . . . . . . . . 11 ((𝑦 β‰  π‘₯ ∧ 𝑦 β‰  𝑧 ∧ (𝑧 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))) ↔ ((𝑦 β‰  π‘₯ ∧ 𝑦 β‰  𝑧) ∧ (𝑧 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))))
10 anass 470 . . . . . . . . . . 11 ((((𝑦 β‰  π‘₯ ∧ 𝑦 β‰  𝑧) ∧ 𝑧 β‰  π‘₯) ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))) ↔ ((𝑦 β‰  π‘₯ ∧ 𝑦 β‰  𝑧) ∧ (𝑧 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))))
11 anass 470 . . . . . . . . . . . 12 ((((𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦) ∧ 𝑦 β‰  π‘₯) ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))) ↔ ((𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦) ∧ (𝑦 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))))
12 anass 470 . . . . . . . . . . . . . 14 (((𝑦 β‰  π‘₯ ∧ 𝑦 β‰  𝑧) ∧ 𝑧 β‰  π‘₯) ↔ (𝑦 β‰  π‘₯ ∧ (𝑦 β‰  𝑧 ∧ 𝑧 β‰  π‘₯)))
13 ancom 462 . . . . . . . . . . . . . 14 ((𝑦 β‰  π‘₯ ∧ (𝑦 β‰  𝑧 ∧ 𝑧 β‰  π‘₯)) ↔ ((𝑦 β‰  𝑧 ∧ 𝑧 β‰  π‘₯) ∧ 𝑦 β‰  π‘₯))
14 necom 2998 . . . . . . . . . . . . . . . 16 (𝑦 β‰  𝑧 ↔ 𝑧 β‰  𝑦)
1514anbi2ci 626 . . . . . . . . . . . . . . 15 ((𝑦 β‰  𝑧 ∧ 𝑧 β‰  π‘₯) ↔ (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦))
1615anbi1i 625 . . . . . . . . . . . . . 14 (((𝑦 β‰  𝑧 ∧ 𝑧 β‰  π‘₯) ∧ 𝑦 β‰  π‘₯) ↔ ((𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦) ∧ 𝑦 β‰  π‘₯))
1712, 13, 163bitri 297 . . . . . . . . . . . . 13 (((𝑦 β‰  π‘₯ ∧ 𝑦 β‰  𝑧) ∧ 𝑧 β‰  π‘₯) ↔ ((𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦) ∧ 𝑦 β‰  π‘₯))
1817anbi1i 625 . . . . . . . . . . . 12 ((((𝑦 β‰  π‘₯ ∧ 𝑦 β‰  𝑧) ∧ 𝑧 β‰  π‘₯) ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))) ↔ (((𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦) ∧ 𝑦 β‰  π‘₯) ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))))
19 df-3an 1090 . . . . . . . . . . . 12 ((𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ (𝑦 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))) ↔ ((𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦) ∧ (𝑦 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))))
2011, 18, 193bitr4i 303 . . . . . . . . . . 11 ((((𝑦 β‰  π‘₯ ∧ 𝑦 β‰  𝑧) ∧ 𝑧 β‰  π‘₯) ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))) ↔ (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ (𝑦 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))))
219, 10, 203bitr2i 299 . . . . . . . . . 10 ((𝑦 β‰  π‘₯ ∧ 𝑦 β‰  𝑧 ∧ (𝑧 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))) ↔ (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ (𝑦 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))))
2221rexbii 3098 . . . . . . . . 9 (βˆƒπ‘§ ∈ 𝑉 (𝑦 β‰  π‘₯ ∧ 𝑦 β‰  𝑧 ∧ (𝑧 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))) ↔ βˆƒπ‘§ ∈ 𝑉 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ (𝑦 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))))
23 rexdifpr 4624 . . . . . . . . 9 (βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(𝑦 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))) ↔ βˆƒπ‘§ ∈ 𝑉 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ (𝑦 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))))
24 r19.42v 3188 . . . . . . . . 9 (βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(𝑦 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))) ↔ (𝑦 β‰  π‘₯ ∧ βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))))
2522, 23, 243bitr2i 299 . . . . . . . 8 (βˆƒπ‘§ ∈ 𝑉 (𝑦 β‰  π‘₯ ∧ 𝑦 β‰  𝑧 ∧ (𝑧 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))) ↔ (𝑦 β‰  π‘₯ ∧ βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))))
2625rexbii 3098 . . . . . . 7 (βˆƒπ‘¦ ∈ 𝑉 βˆƒπ‘§ ∈ 𝑉 (𝑦 β‰  π‘₯ ∧ 𝑦 β‰  𝑧 ∧ (𝑧 β‰  π‘₯ ∧ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))) ↔ βˆƒπ‘¦ ∈ 𝑉 (𝑦 β‰  π‘₯ ∧ βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))))
277, 8, 263bitri 297 . . . . . 6 (βˆƒπ‘§ ∈ 𝑉 (𝑧 β‰  π‘₯ ∧ βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯, 𝑧})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))) ↔ βˆƒπ‘¦ ∈ 𝑉 (𝑦 β‰  π‘₯ ∧ βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))))
28 rexdifsn 4759 . . . . . 6 (βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯, 𝑧})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})) ↔ βˆƒπ‘§ ∈ 𝑉 (𝑧 β‰  π‘₯ ∧ βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯, 𝑧})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))))
29 rexdifsn 4759 . . . . . 6 (βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})) ↔ βˆƒπ‘¦ ∈ 𝑉 (𝑦 β‰  π‘₯ ∧ βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))))
3027, 28, 293bitr4i 303 . . . . 5 (βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯, 𝑧})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})) ↔ βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))
3130a1i 11 . . . 4 ((𝐺 ∈ USGraph ∧ π‘₯ ∈ 𝑉) β†’ (βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯, 𝑧})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})) ↔ βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))))
3231rexbidva 3174 . . 3 (𝐺 ∈ USGraph β†’ (βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯, 𝑧})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})) ↔ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))))
33323anbi3d 1443 . 2 (𝐺 ∈ USGraph β†’ ((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯, 𝑧})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦}))) ↔ (𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))))
343, 33bitrd 279 1 (𝐺 ∈ USGraph β†’ ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 2) ↔ (𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆƒwrex 3074   βˆ– cdif 3912  {csn 4591  {cpr 4593   class class class wbr 5110  dom cdm 5638  β€“1-1β†’wf1 6498  β€˜cfv 6501  (class class class)co 7362  0cc0 11058  1c1 11059  2c2 12215  ...cfz 13431  ..^cfzo 13574  β™―chash 14237  Vtxcvtx 27989  iEdgciedg 27990  USGraphcusgr 28142  Pathscpths 28702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-oadd 8421  df-er 8655  df-map 8774  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-fz 13432  df-fzo 13575  df-hash 14238  df-word 14410  df-concat 14466  df-s1 14491  df-s2 14744  df-s3 14745  df-edg 28041  df-uhgr 28051  df-upgr 28075  df-umgr 28076  df-uspgr 28143  df-usgr 28144  df-wlks 28589  df-wlkson 28590  df-trls 28682  df-trlson 28683  df-pths 28706  df-spths 28707  df-pthson 28708  df-spthson 28709
This theorem is referenced by: (None)
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