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Theorem usgr2pth 29059
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 29057 . . . . 5 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (𝐹(Pathsβ€˜πΊ)𝑃 ↔ 𝐹(SPathsβ€˜πΊ)𝑃))
2 usgrupgr 28480 . . . . . . 7 (𝐺 ∈ USGraph β†’ 𝐺 ∈ UPGraph)
32adantr 481 . . . . . 6 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ 𝐺 ∈ UPGraph)
4 isspth 29019 . . . . . . . . 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 28998 . . . . . . . . . 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 630 . . . . . . . . 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 7419 . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . 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 1133 . . . . . . . . . . . 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 726 . . . . . . . . . . 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 7419 . . . . . . . . . . . . . . . . . 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 501 . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . 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 407 . . . . . . . . . . . 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 482 . . . . . . . . . . 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 29058 . . . . . . . . . . . 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 726 . . . . . . . . . . 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 1129 . . . . . . . . . 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 413 . . . . . . . . 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 413 . . 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 412 . 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 12491 . . . . . 6 2 ∈ β„•0
41 f1f 6787 . . . . . 6 (𝐹:(0..^2)–1-1β†’dom 𝐼 β†’ 𝐹:(0..^2)⟢dom 𝐼)
42 fnfzo0hash 14411 . . . . . 6 ((2 ∈ β„•0 ∧ 𝐹:(0..^2)⟢dom 𝐼) β†’ (β™―β€˜πΉ) = 2)
4340, 41, 42sylancr 587 . . . . 5 (𝐹:(0..^2)–1-1β†’dom 𝐼 β†’ (β™―β€˜πΉ) = 2)
44 oveq2 7419 . . . . . . . . . . . . . . . . . 18 (2 = (β™―β€˜πΉ) β†’ (0..^2) = (0..^(β™―β€˜πΉ)))
4544eqcoms 2740 . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 724 . . . . . . . . . . . 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 7419 . . . . . . . . . . . . . . . . . 18 (2 = (β™―β€˜πΉ) β†’ (0...2) = (0...(β™―β€˜πΉ)))
5453eqcoms 2740 . . . . . . . . . . . . . . . . 17 ((β™―β€˜πΉ) = 2 β†’ (0...2) = (0...(β™―β€˜πΉ)))
5554adantr 481 . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . 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 724 . . . . . . . . . . . 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 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((π‘ƒβ€˜0) = π‘₯ ↔ π‘₯ = (π‘ƒβ€˜0))
6160biimpi 215 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((π‘ƒβ€˜0) = π‘₯ β†’ π‘₯ = (π‘ƒβ€˜0))
62613ad2ant1 1133 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) β†’ π‘₯ = (π‘ƒβ€˜0))
63 eqcom 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((π‘ƒβ€˜1) = 𝑦 ↔ 𝑦 = (π‘ƒβ€˜1))
6463biimpi 215 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((π‘ƒβ€˜1) = 𝑦 β†’ 𝑦 = (π‘ƒβ€˜1))
65643ad2ant2 1134 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2743 . . . . . . . . . . . . . . . . . . . . . . . 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 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€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)}))
7069impcom 408 . . . . . . . . . . . . . . . . . . . . 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 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((π‘ƒβ€˜2) = 𝑧 ↔ 𝑧 = (π‘ƒβ€˜2))
7271biimpi 215 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((π‘ƒβ€˜2) = 𝑧 β†’ 𝑧 = (π‘ƒβ€˜2))
73723ad2ant3 1135 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2743 . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . . . . . 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 512 . . . . . . . . . . . . . . . . . . . 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 3151 . . . . . . . . . . . . . . . . . . 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 3151 . . . . . . . . . . . . . . . . . 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 3151 . . . . . . . . . . . . . . . . 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 13719 . . . . . . . . . . . . . . . . . . . 20 (0..^2) = {0, 1}
8510, 84eqtrdi 2788 . . . . . . . . . . . . . . . . . . 19 ((β™―β€˜πΉ) = 2 β†’ (0..^(β™―β€˜πΉ)) = {0, 1})
8685raleqdv 3325 . . . . . . . . . . . . . . . . . 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 28962 . . . . . . . . . . . . . . . . . 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 286 . . . . . . . . . . . . . . . . 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 258 . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . 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 407 . . . . . . . . . . . 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 1128 . . . . . . . . . . 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 497 . . . . . . . . . . . . 13 (𝑃:(0...2)–1-1→𝑉 β†’ Fun ◑𝑃)
9695adantl 482 . . . . . . . . . . . 12 ((((β™―β€˜πΉ) = 2 ∧ 𝐹:(0..^2)–1-1β†’dom 𝐼) ∧ 𝑃:(0...2)–1-1→𝑉) β†’ Fun ◑𝑃)
9796ad2antrr 724 . . . . . . . . . . 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 512 . . . . . . . . . 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 278 . . . . . . . . . . . 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 482 . . . . . . . . . 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 256 . . . . . . . . 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 485 . . . . . . . . . 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 781 . . . . . . . . . 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 584 . . . . . . . . 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 256 . . . . . . . 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 512 . . . . . . 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 413 . . . . . 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 435 . . . . 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 1111 . . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   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 7411  0cc0 11112  1c1 11113   + caddc 11115  2c2 12269  β„•0cn0 12474  ...cfz 13486  ..^cfzo 13629  β™―chash 14292  Vtxcvtx 28294  iEdgciedg 28295  UPGraphcupgr 28378  USGraphcusgr 28447  Trailsctrls 28985  Pathscpths 29007  SPathscspths 29008
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  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 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-3 12278  df-n0 12475  df-xnn0 12547  df-z 12561  df-uz 12825  df-fz 13487  df-fzo 13630  df-hash 14293  df-word 14467  df-concat 14523  df-s1 14548  df-s2 14801  df-s3 14802  df-edg 28346  df-uhgr 28356  df-upgr 28380  df-umgr 28381  df-uspgr 28448  df-usgr 28449  df-wlks 28894  df-wlkson 28895  df-trls 28987  df-trlson 28988  df-pths 29011  df-spths 29012  df-pthson 29013  df-spthson 29014
This theorem is referenced by:  usgr2pth0  29060
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