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Theorem usgr2pth 29289
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 = (Vtxβ€˜πΊ)
usgr2pthlem.i 𝐼 = (iEdgβ€˜πΊ)
Assertion
Ref Expression
usgr2pth (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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})))))
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 usgr2pth
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 usgr2pthspth 29287 . . . . 5 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (𝐹(Pathsβ€˜πΊ)𝑃 ↔ 𝐹(SPathsβ€˜πΊ)𝑃))
2 usgrupgr 28710 . . . . . . 7 (𝐺 ∈ USGraph β†’ 𝐺 ∈ UPGraph)
32adantr 480 . . . . . 6 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ 𝐺 ∈ UPGraph)
4 isspth 29249 . . . . . . . . 9 (𝐹(SPathsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃))
54a1i 11 . . . . . . . 8 (𝐺 ∈ UPGraph β†’ (𝐹(SPathsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃)))
6 usgr2pthlem.v . . . . . . . . . . 11 𝑉 = (Vtxβ€˜πΊ)
7 usgr2pthlem.i . . . . . . . . . . 11 𝐼 = (iEdgβ€˜πΊ)
86, 7upgrf1istrl 29228 . . . . . . . . . 10 (𝐺 ∈ UPGraph β†’ (𝐹(Trailsβ€˜πΊ)𝑃 ↔ (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
98anbi1d 629 . . . . . . . . 9 (𝐺 ∈ UPGraph β†’ ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃) ↔ ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ Fun ◑𝑃)))
10 oveq2 7420 . . . . . . . . . . . . . . . . 17 ((β™―β€˜πΉ) = 2 β†’ (0..^(β™―β€˜πΉ)) = (0..^2))
11 f1eq2 6783 . . . . . . . . . . . . . . . . 17 ((0..^(β™―β€˜πΉ)) = (0..^2) β†’ (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ↔ 𝐹:(0..^2)–1-1β†’dom 𝐼))
1210, 11syl 17 . . . . . . . . . . . . . . . 16 ((β™―β€˜πΉ) = 2 β†’ (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ↔ 𝐹:(0..^2)–1-1β†’dom 𝐼))
1312biimpd 228 . . . . . . . . . . . . . . 15 ((β™―β€˜πΉ) = 2 β†’ (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 β†’ 𝐹:(0..^2)–1-1β†’dom 𝐼))
1413adantl 481 . . . . . . . . . . . . . 14 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 β†’ 𝐹:(0..^2)–1-1β†’dom 𝐼))
1514com12 32 . . . . . . . . . . . . 13 (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ 𝐹:(0..^2)–1-1β†’dom 𝐼))
16153ad2ant1 1132 . . . . . . . . . . . 12 ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ 𝐹:(0..^2)–1-1β†’dom 𝐼))
1716ad2antrl 725 . . . . . . . . . . 11 ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ Fun ◑𝑃)) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ 𝐹:(0..^2)–1-1β†’dom 𝐼))
18 oveq2 7420 . . . . . . . . . . . . . . . . . 18 ((β™―β€˜πΉ) = 2 β†’ (0...(β™―β€˜πΉ)) = (0...2))
1918feq2d 6703 . . . . . . . . . . . . . . . . 17 ((β™―β€˜πΉ) = 2 β†’ (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ↔ 𝑃:(0...2)βŸΆπ‘‰))
20 df-f1 6548 . . . . . . . . . . . . . . . . . . 19 (𝑃:(0...2)–1-1→𝑉 ↔ (𝑃:(0...2)βŸΆπ‘‰ ∧ Fun ◑𝑃))
2120simplbi2 500 . . . . . . . . . . . . . . . . . 18 (𝑃:(0...2)βŸΆπ‘‰ β†’ (Fun ◑𝑃 β†’ 𝑃:(0...2)–1-1→𝑉))
2221a1i 11 . . . . . . . . . . . . . . . . 17 ((β™―β€˜πΉ) = 2 β†’ (𝑃:(0...2)βŸΆπ‘‰ β†’ (Fun ◑𝑃 β†’ 𝑃:(0...2)–1-1→𝑉)))
2319, 22sylbid 239 . . . . . . . . . . . . . . . 16 ((β™―β€˜πΉ) = 2 β†’ (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ β†’ (Fun ◑𝑃 β†’ 𝑃:(0...2)–1-1→𝑉)))
2423adantl 481 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ β†’ (Fun ◑𝑃 β†’ 𝑃:(0...2)–1-1→𝑉)))
2524com3l 89 . . . . . . . . . . . . . 14 (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ β†’ (Fun ◑𝑃 β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ 𝑃:(0...2)–1-1→𝑉)))
26253ad2ant2 1133 . . . . . . . . . . . . 13 ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) β†’ (Fun ◑𝑃 β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ 𝑃:(0...2)–1-1→𝑉)))
2726imp 406 . . . . . . . . . . . 12 (((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ Fun ◑𝑃) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ 𝑃:(0...2)–1-1→𝑉))
2827adantl 481 . . . . . . . . . . 11 ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ Fun ◑𝑃)) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ 𝑃:(0...2)–1-1→𝑉))
296, 7usgr2pthlem 29288 . . . . . . . . . . . 12 ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))
3029ad2antrl 725 . . . . . . . . . . 11 ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ Fun ◑𝑃)) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))
3117, 28, 303jcad 1128 . . . . . . . . . 10 ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ Fun ◑𝑃)) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})))))
3231ex 412 . . . . . . . . 9 (𝐺 ∈ UPGraph β†’ (((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ Fun ◑𝑃) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))))
339, 32sylbid 239 . . . . . . . 8 (𝐺 ∈ UPGraph β†’ ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))))
345, 33sylbid 239 . . . . . . 7 (𝐺 ∈ UPGraph β†’ (𝐹(SPathsβ€˜πΊ)𝑃 β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))))
3534com23 86 . . . . . 6 (𝐺 ∈ UPGraph β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (𝐹(SPathsβ€˜πΊ)𝑃 β†’ (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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))))
363, 35mpcom 38 . . . . 5 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (𝐹(SPathsβ€˜πΊ)𝑃 β†’ (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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})))))
371, 36sylbid 239 . . . 4 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})))))
3837ex 412 . . 3 (𝐺 ∈ USGraph β†’ ((β™―β€˜πΉ) = 2 β†’ (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))))
3938impcomd 411 . 2 (𝐺 ∈ USGraph β†’ ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 2) β†’ (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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})))))
40 2nn0 12494 . . . . . 6 2 ∈ β„•0
41 f1f 6787 . . . . . 6 (𝐹:(0..^2)–1-1β†’dom 𝐼 β†’ 𝐹:(0..^2)⟢dom 𝐼)
42 fnfzo0hash 14414 . . . . . 6 ((2 ∈ β„•0 ∧ 𝐹:(0..^2)⟢dom 𝐼) β†’ (β™―β€˜πΉ) = 2)
4340, 41, 42sylancr 586 . . . . 5 (𝐹:(0..^2)–1-1β†’dom 𝐼 β†’ (β™―β€˜πΉ) = 2)
44 oveq2 7420 . . . . . . . . . . . . . . . . . 18 (2 = (β™―β€˜πΉ) β†’ (0..^2) = (0..^(β™―β€˜πΉ)))
4544eqcoms 2739 . . . . . . . . . . . . . . . . 17 ((β™―β€˜πΉ) = 2 β†’ (0..^2) = (0..^(β™―β€˜πΉ)))
46 f1eq2 6783 . . . . . . . . . . . . . . . . 17 ((0..^2) = (0..^(β™―β€˜πΉ)) β†’ (𝐹:(0..^2)–1-1β†’dom 𝐼 ↔ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼))
4745, 46syl 17 . . . . . . . . . . . . . . . 16 ((β™―β€˜πΉ) = 2 β†’ (𝐹:(0..^2)–1-1β†’dom 𝐼 ↔ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼))
4847biimpd 228 . . . . . . . . . . . . . . 15 ((β™―β€˜πΉ) = 2 β†’ (𝐹:(0..^2)–1-1β†’dom 𝐼 β†’ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼))
4948imp 406 . . . . . . . . . . . . . 14 (((β™―β€˜πΉ) = 2 ∧ 𝐹:(0..^2)–1-1β†’dom 𝐼) β†’ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼)
5049adantr 480 . . . . . . . . . . . . 13 ((((β™―β€˜πΉ) = 2 ∧ 𝐹:(0..^2)–1-1β†’dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) β†’ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼)
5150ad2antrr 723 . . . . . . . . . . . 12 ((((((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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) β†’ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼)
52 f1f 6787 . . . . . . . . . . . . . . 15 (𝑃:(0...2)–1-1→𝑉 β†’ 𝑃:(0...2)βŸΆπ‘‰)
53 oveq2 7420 . . . . . . . . . . . . . . . . . 18 (2 = (β™―β€˜πΉ) β†’ (0...2) = (0...(β™―β€˜πΉ)))
5453eqcoms 2739 . . . . . . . . . . . . . . . . 17 ((β™―β€˜πΉ) = 2 β†’ (0...2) = (0...(β™―β€˜πΉ)))
5554adantr 480 . . . . . . . . . . . . . . . 16 (((β™―β€˜πΉ) = 2 ∧ 𝐹:(0..^2)–1-1β†’dom 𝐼) β†’ (0...2) = (0...(β™―β€˜πΉ)))
5655feq2d 6703 . . . . . . . . . . . . . . 15 (((β™―β€˜πΉ) = 2 ∧ 𝐹:(0..^2)–1-1β†’dom 𝐼) β†’ (𝑃:(0...2)βŸΆπ‘‰ ↔ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰))
5752, 56imbitrid 243 . . . . . . . . . . . . . 14 (((β™―β€˜πΉ) = 2 ∧ 𝐹:(0..^2)–1-1β†’dom 𝐼) β†’ (𝑃:(0...2)–1-1→𝑉 β†’ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰))
5857imp 406 . . . . . . . . . . . . 13 ((((β™―β€˜πΉ) = 2 ∧ 𝐹:(0..^2)–1-1β†’dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) β†’ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰)
5958ad2antrr 723 . . . . . . . . . . . 12 ((((((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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) β†’ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰)
60 eqcom 2738 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((π‘ƒβ€˜0) = π‘₯ ↔ π‘₯ = (π‘ƒβ€˜0))
6160biimpi 215 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((π‘ƒβ€˜0) = π‘₯ β†’ π‘₯ = (π‘ƒβ€˜0))
62613ad2ant1 1132 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) β†’ π‘₯ = (π‘ƒβ€˜0))
63 eqcom 2738 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((π‘ƒβ€˜1) = 𝑦 ↔ 𝑦 = (π‘ƒβ€˜1))
6463biimpi 215 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((π‘ƒβ€˜1) = 𝑦 β†’ 𝑦 = (π‘ƒβ€˜1))
65643ad2ant2 1133 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) β†’ 𝑦 = (π‘ƒβ€˜1))
6662, 65preq12d 4745 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) β†’ {π‘₯, 𝑦} = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)})
6766eqeq2d 2742 . . . . . . . . . . . . . . . . . . . . . . . 24 (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) β†’ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ↔ (πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)}))
6867biimpcd 248 . . . . . . . . . . . . . . . . . . . . . . 23 ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} β†’ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) β†’ (πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)}))
6968adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}) β†’ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) β†’ (πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)}))
7069impcom 407 . . . . . . . . . . . . . . . . . . . . 21 ((((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})) β†’ (πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)})
71 eqcom 2738 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((π‘ƒβ€˜2) = 𝑧 ↔ 𝑧 = (π‘ƒβ€˜2))
7271biimpi 215 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((π‘ƒβ€˜2) = 𝑧 β†’ 𝑧 = (π‘ƒβ€˜2))
73723ad2ant3 1134 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) β†’ 𝑧 = (π‘ƒβ€˜2))
7465, 73preq12d 4745 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) β†’ {𝑦, 𝑧} = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})
7574eqeq2d 2742 . . . . . . . . . . . . . . . . . . . . . . . 24 (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) β†’ ((πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧} ↔ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
7675biimpcd 248 . . . . . . . . . . . . . . . . . . . . . . 23 ((πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧} β†’ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) β†’ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
7776adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 (((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}) β†’ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) β†’ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
7877impcom 407 . . . . . . . . . . . . . . . . . . . . 21 ((((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})) β†’ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})
7970, 78jca 511 . . . . . . . . . . . . . . . . . . . 20 ((((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})) β†’ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
8079rexlimivw 3150 . . . . . . . . . . . . . . . . . . 19 (βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})) β†’ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
8180rexlimivw 3150 . . . . . . . . . . . . . . . . . 18 (βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})) β†’ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
8281rexlimivw 3150 . . . . . . . . . . . . . . . . 17 (βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})) β†’ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
8382a1i13 27 . . . . . . . . . . . . . . . 16 ((β™―β€˜πΉ) = 2 β†’ (βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})) β†’ (𝐺 ∈ USGraph β†’ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))))
84 fzo0to2pr 13722 . . . . . . . . . . . . . . . . . . . 20 (0..^2) = {0, 1}
8510, 84eqtrdi 2787 . . . . . . . . . . . . . . . . . . 19 ((β™―β€˜πΉ) = 2 β†’ (0..^(β™―β€˜πΉ)) = {0, 1})
8685raleqdv 3324 . . . . . . . . . . . . . . . . . 18 ((β™―β€˜πΉ) = 2 β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ βˆ€π‘– ∈ {0, 1} (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
87 2wlklem 29192 . . . . . . . . . . . . . . . . . 18 (βˆ€π‘– ∈ {0, 1} (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
8886, 87bitrdi 287 . . . . . . . . . . . . . . . . 17 ((β™―β€˜πΉ) = 2 β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})))
8988imbi2d 340 . . . . . . . . . . . . . . . 16 ((β™―β€˜πΉ) = 2 β†’ ((𝐺 ∈ USGraph β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ↔ (𝐺 ∈ USGraph β†’ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))))
9083, 89sylibrd 259 . . . . . . . . . . . . . . 15 ((β™―β€˜πΉ) = 2 β†’ (βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})) β†’ (𝐺 ∈ USGraph β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
9190ad2antrr 723 . . . . . . . . . . . . . 14 ((((β™―β€˜πΉ) = 2 ∧ 𝐹:(0..^2)–1-1β†’dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) β†’ (βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})) β†’ (𝐺 ∈ USGraph β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
9291imp 406 . . . . . . . . . . . . 13 (((((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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) β†’ (𝐺 ∈ USGraph β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
9392imp 406 . . . . . . . . . . . 12 ((((((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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})
9451, 59, 933jca 1127 . . . . . . . . . . 11 ((((((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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) β†’ (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
9520simprbi 496 . . . . . . . . . . . . 13 (𝑃:(0...2)–1-1→𝑉 β†’ Fun ◑𝑃)
9695adantl 481 . . . . . . . . . . . 12 ((((β™―β€˜πΉ) = 2 ∧ 𝐹:(0..^2)–1-1β†’dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) β†’ Fun ◑𝑃)
9796ad2antrr 723 . . . . . . . . . . 11 ((((((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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) β†’ Fun ◑𝑃)
9894, 97jca 511 . . . . . . . . . 10 ((((((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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) β†’ ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ Fun ◑𝑃))
995, 9bitrd 279 . . . . . . . . . . . 12 (𝐺 ∈ UPGraph β†’ (𝐹(SPathsβ€˜πΊ)𝑃 ↔ ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ Fun ◑𝑃)))
1002, 99syl 17 . . . . . . . . . . 11 (𝐺 ∈ USGraph β†’ (𝐹(SPathsβ€˜πΊ)𝑃 ↔ ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ Fun ◑𝑃)))
101100adantl 481 . . . . . . . . . 10 ((((((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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) β†’ (𝐹(SPathsβ€˜πΊ)𝑃 ↔ ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ∧ Fun ◑𝑃)))
10298, 101mpbird 257 . . . . . . . . 9 ((((((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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) β†’ 𝐹(SPathsβ€˜πΊ)𝑃)
103 simpr 484 . . . . . . . . . 10 ((((((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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) β†’ 𝐺 ∈ USGraph)
104 simp-4l 780 . . . . . . . . . 10 ((((((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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) β†’ (β™―β€˜πΉ) = 2)
105103, 104, 1syl2anc 583 . . . . . . . . 9 ((((((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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) β†’ (𝐹(Pathsβ€˜πΊ)𝑃 ↔ 𝐹(SPathsβ€˜πΊ)𝑃))
106102, 105mpbird 257 . . . . . . . 8 ((((((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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) β†’ 𝐹(Pathsβ€˜πΊ)𝑃)
107106, 104jca 511 . . . . . . 7 ((((((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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) ∧ 𝐺 ∈ USGraph) β†’ (𝐹(Pathsβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 2))
108107ex 412 . . . . . 6 (((((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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) β†’ (𝐺 ∈ USGraph β†’ (𝐹(Pathsβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 2)))
109108exp41 434 . . . . 5 ((β™―β€˜πΉ) = 2 β†’ (𝐹:(0..^2)–1-1β†’dom 𝐼 β†’ (𝑃:(0...2)–1-1→𝑉 β†’ (βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})) β†’ (𝐺 ∈ USGraph β†’ (𝐹(Pathsβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 2))))))
11043, 109mpcom 38 . . . 4 (𝐹:(0..^2)–1-1β†’dom 𝐼 β†’ (𝑃:(0...2)–1-1→𝑉 β†’ (βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})) β†’ (𝐺 ∈ USGraph β†’ (𝐹(Pathsβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 2)))))
1111103imp 1110 . . 3 ((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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) β†’ (𝐺 ∈ USGraph β†’ (𝐹(Pathsβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 2)))
112111com12 32 . 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‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) β†’ (𝐹(Pathsβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 2)))
11339, 112impbid 211 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) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  βˆƒwrex 3069   βˆ– cdif 3945  {csn 4628  {cpr 4630   class class class wbr 5148  β—‘ccnv 5675  dom cdm 5676  Fun wfun 6537  βŸΆwf 6539  β€“1-1β†’wf1 6540  β€˜cfv 6543  (class class class)co 7412  0cc0 11113  1c1 11114   + caddc 11116  2c2 12272  β„•0cn0 12477  ...cfz 13489  ..^cfzo 13632  β™―chash 14295  Vtxcvtx 28524  iEdgciedg 28525  UPGraphcupgr 28608  USGraphcusgr 28677  Trailsctrls 29215  Pathscpths 29237  SPathscspths 29238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1061  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-2o 8470  df-oadd 8473  df-er 8706  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-dju 9899  df-card 9937  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-3 12281  df-n0 12478  df-xnn0 12550  df-z 12564  df-uz 12828  df-fz 13490  df-fzo 13633  df-hash 14296  df-word 14470  df-concat 14526  df-s1 14551  df-s2 14804  df-s3 14805  df-edg 28576  df-uhgr 28586  df-upgr 28610  df-umgr 28611  df-uspgr 28678  df-usgr 28679  df-wlks 29124  df-wlkson 29125  df-trls 29217  df-trlson 29218  df-pths 29241  df-spths 29242  df-pthson 29243  df-spthson 29244
This theorem is referenced by:  usgr2pth0  29290
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