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Theorem List for Metamath Proof Explorer - 25901-26000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfargshiftfv 25901* If a class is a function, then the values of the "shifted function" correspond to the function values of the class. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → (𝑋 ∈ (0..^𝑁) → (𝐺𝑋) = (𝐹‘(𝑋 + 1))))

Theoremfargshiftf 25902* If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(#‘𝐹))⟶dom 𝐸)

Theoremfargshiftf1 25903* If a function is 1-1, then also the shifted function is 1-1. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–1-1→dom 𝐸) → 𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸)

Theoremfargshiftfo 25904* If a function is onto, then also the shifted function is onto. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → 𝐺:(0..^(#‘𝐹))–onto→dom 𝐸)

Theoremfargshiftfva 25905* The values of a shifted function correspond to the value of the original function. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹𝑘)) = 𝑘 / 𝑥𝑃 → ∀𝑙 ∈ (0..^𝑁)(𝐸‘(𝐺𝑙)) = (𝑙 + 1) / 𝑥𝑃))

Theoremusgrcyclnl1 25906 In an undirected simple graph (with no loops!) there are no cycles with length 1 (consisting of one edge ). (Contributed by Alexander van der Vekens, 7-Nov-2017.)
((𝑉 USGrph 𝐸𝐹(𝑉 Cycles 𝐸)𝑃) → (#‘𝐹) ≠ 1)

Theoremusgrcyclnl2 25907 In an undirected simple graph (with no loops!) there are no cycles with length 2 (consisting of two edges ). (Contributed by Alexander van der Vekens, 9-Nov-2017.)
((𝑉 USGrph 𝐸𝐹(𝑉 Cycles 𝐸)𝑃) → (#‘𝐹) ≠ 2)

Theorem3cycl3dv 25908 In a simple graph, the vertices of a 3-cycle are mutually different. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (𝐴𝐵𝐵𝐶𝐶𝐴))

Theoremnvnencycllem 25909 Lemma for 3v3e3cycl1 25910 and 4cycl4v4e 25932. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
(((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (𝑋 ∈ ℕ0𝑋 < (#‘𝐹))) → ((𝐸‘(𝐹𝑋)) = {𝐴, 𝐵} → {𝐴, 𝐵} ∈ ran 𝐸))

Theorem3v3e3cycl1 25910* If there is a cycle of length 3 in a graph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
((Fun 𝐸𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))

Theoremconstr3lem1 25911 Lemma for constr3trl 25925 etc. (Contributed by Alexander van der Vekens, 10-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       (𝐹 ∈ V ∧ 𝑃 ∈ V)

Theoremconstr3lem2 25912 Lemma for constr3trl 25925 etc. (Contributed by Alexander van der Vekens, 10-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       (#‘𝐹) = 3

Theoremconstr3lem4 25913 Lemma for constr3trl 25925 etc. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝐴𝑉𝐵𝑉𝐶𝑉) → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)))

Theoremconstr3lem5 25914 Lemma for constr3trl 25925 etc. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴}))

Theoremconstr3lem6 25915 Lemma for constr3pthlem3 25923. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → ({(𝑃‘0), (𝑃‘3)} ∩ {(𝑃‘1), (𝑃‘2)}) = ∅)

Theoremconstr3trllem1 25916 Lemma for constr3trl 25925. (Contributed by Alexander van der Vekens, 10-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → 𝐹 ∈ Word dom 𝐸)

Theoremconstr3trllem2 25917 Lemma for constr3trl 25925. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → Fun 𝐹)

Theoremconstr3trllem3 25918 Lemma for constr3trl 25925. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝐴𝑉𝐵𝑉𝐶𝑉) → 𝑃:(0...(#‘𝐹))⟶𝑉)

Theoremconstr3trllem4 25919 Lemma for constr3trl 25925. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝐴𝑉𝐵𝑉𝐶𝑉) → 𝑃:(0...3)⟶𝑉)

Theoremconstr3trllem5 25920* Lemma for constr3trl 25925. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})

Theoremconstr3pthlem1 25921 Lemma for constr3pth 25926. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝐵𝑉𝐶𝑊) → (𝑃 ↾ (1..^(#‘𝐹))) = {⟨1, 𝐵⟩, ⟨2, 𝐶⟩})

Theoremconstr3pthlem2 25922 Lemma for constr3pth 25926. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ 𝐵𝐶) → Fun (𝑃 ↾ (1..^(#‘𝐹))))

Theoremconstr3pthlem3 25923 Lemma for constr3pth 25926. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)

Theoremconstr3cycllem1 25924 Lemma for constr3cycl 25927. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝑃‘0) = (𝑃‘(#‘𝐹)))

Theoremconstr3trl 25925 Construction of a trail from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → 𝐹(𝑉 Trails 𝐸)𝑃)

Theoremconstr3pth 25926 Construction of a path from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → 𝐹(𝑉 Paths 𝐸)𝑃)

Theoremconstr3cycl 25927 Construction of a 3-cycle from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3))

Theoremconstr3cyclp 25928 Construction of a 3-cycle from three given edges in a graph, containing an endpoint of one of these edges. (Contributed by Alexander van der Vekens, 17-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3 ∧ (𝑃‘0) = 𝐴))

Theoremconstr3cyclpe 25929* If there are three (different) vertices in a graph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.)
((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∃𝑓𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴))

Theorem3v3e3cycl2 25930* If there are three (different) vertices in a graph which are mutually connected by edges, there is a 3-cycle in the graph. (Contributed by Alexander van der Vekens, 14-Nov-2017.)
(𝑉 USGrph 𝐸 → (∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) → ∃𝑓𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3)))

Theorem3v3e3cycl 25931* If and only if there is a 3-cycle in a graph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 14-Nov-2017.)
(𝑉 USGrph 𝐸 → (∃𝑓𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))

Theorem4cycl4v4e 25932* If there is a cycle of length 4 in a graph, there are four (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
((Fun 𝐸𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 4) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))

Theorem4cycl4dv 25933 In a simple graph, the vertices of a 4-cycle are mutually different. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) → ((((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})) → ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) ∧ ((𝐴𝐵𝐴𝐶𝐴𝐷) ∧ (𝐵𝐶𝐵𝐷𝐶𝐷)))))

Theorem4cycl4dv4e 25934* If there is a cycle of length 4 in a graph, there are four (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
((𝑉 USGrph 𝐸𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 4) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑))))

16.1.5.4  Connected graphs

Syntaxcconngra 25935 Extend class notation with connected graphs.
class ConnGrph

Definitiondf-conngra 25936* Define the class of all connected graphs. A graph (or, more generally, any pair representing a structure consisting of "vertices" and "edges") is called connected if there is a path between every pair of (distinct) vertices. The distinctness of the vertices is not necessary for the definition, because there is always a path (of length 0) from a vertex to itself, see 0pthonv 25849 and dfconngra1 25937. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
ConnGrph = {⟨𝑣, 𝑒⟩ ∣ ∀𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(𝑣 PathOn 𝑒)𝑛)𝑝}

Theoremdfconngra1 25937* Alternative definition of the class of all connected graphs, requiring paths between distinct vertices. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
ConnGrph = {⟨𝑣, 𝑒⟩ ∣ ∀𝑘𝑣𝑛 ∈ (𝑣 ∖ {𝑘})∃𝑓𝑝 𝑓(𝑘(𝑣 PathOn 𝑒)𝑛)𝑝}

Theoremisconngra 25938* The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 ConnGrph 𝐸 ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))

Theoremisconngra1 25939* The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 ConnGrph 𝐸 ↔ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘})∃𝑓𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))

Theorem0conngra 25940 A class/graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
(𝐸𝑉 → ∅ ConnGrph 𝐸)

Theorem1conngra 25941 A class/graph with (at most) one vertex is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
(𝐸𝑉 → {𝐴} ConnGrph 𝐸)

Theoremcusconngra 25942 A complete (undirected simple) graph is connected. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
(𝑉 ComplUSGrph 𝐸𝑉 ConnGrph 𝐸)

16.1.5.5  Walks as words

Syntaxcwwlk 25943 Extend class notation with Walks (of a graph) as Word over the set of vertices.
class WWalks

Syntaxcwwlkn 25944 Extend class notation with Walks (of a graph) of a fixed length as Word over the set of vertices.
class WWalksN

Definitiondf-wwlk 25945* Define the set of all Walks (in an undirected graph) as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n-1) p(n) of the vertices in a walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) as defined in df-wlk 25774. 𝑤 = ∅ has to be excluded because a walk always consists of at least one vertex, see wlkn0 25793. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
WWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑤 ∈ Word 𝑣 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒)})

Definitiondf-wwlkn 25946* Define the set of all Walks (in an undirected graph) of a fixed length n as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n) of the vertices in a walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) as defined in df-wlk 25774. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
WWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)}))

Theoremwwlk 25947* The set of walks (in an undirected graph) as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
((𝑉𝑋𝐸𝑌) → (𝑉 WWalks 𝐸) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸)})

Theoremwwlkn 25948* The set of walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → ((𝑉 WWalksN 𝐸)‘𝑁) = {𝑤 ∈ (𝑉 WWalks 𝐸) ∣ (#‘𝑤) = (𝑁 + 1)})

Theoremiswwlk 25949* Properties of a word to represent a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.)
((𝑉𝑋𝐸𝑌) → (𝑊 ∈ (𝑉 WWalks 𝐸) ↔ (𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)))

Theoremiswwlkn 25950 Properties of a word to represent a walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.)
((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ (𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (𝑁 + 1))))

Theoremwwlkprop 25951 Properties of a walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
(𝑃 ∈ (𝑉 WWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉))

Theoremwwlknprop 25952 Properties of a walk of a fixed length (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 16-Jul-2018.)
(𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0𝑃 ∈ Word 𝑉)))

Theoremwwlknimp 25953* Implications for a set being a walk of length n (represented by a word). (Contributed by Alexander van der Vekens, 17-Jun-2018.)
(𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))

Theoremwwlksswrd 25954 Walks (represented by words) are words. (Contributed by Alexander van der Vekens, 17-Jul-2018.)
(𝑉 WWalks 𝐸) ⊆ Word 𝑉

Theoremwwlkn0 25955* A walk of length 0 is represented by a singleton word. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
(𝑊 ∈ ((𝑉 WWalksN 𝐸)‘0) → ∃𝑣𝑉 𝑊 = ⟨“𝑣”⟩)

Theoremwlkiswwlk1 25956 The sequence of vertices in a walk is a walk as word in an undirected simple graph. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
(𝑉 USGrph 𝐸 → (𝐹(𝑉 Walks 𝐸)𝑃𝑃 ∈ (𝑉 WWalks 𝐸)))

Theoremwlkiswwlk2lem1 25957* Lemma 1 for wlkiswwlk2 25963. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (#‘𝐹) = ((#‘𝑃) − 1))

Theoremwlkiswwlk2lem2 25958* Lemma 2 for wlkiswwlk2 25963. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       (((#‘𝑃) ∈ ℕ0𝐼 ∈ (0..^((#‘𝑃) − 1))) → (𝐹𝐼) = (𝐸‘{(𝑃𝐼), (𝑃‘(𝐼 + 1))}))

Theoremwlkiswwlk2lem3 25959* Lemma 3 for wlkiswwlk2 25963. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 𝑃:(0...(#‘𝐹))⟶𝑉)

Theoremwlkiswwlk2lem4 25960* Lemma 4 for wlkiswwlk2 25963. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       ((𝑉 USGrph 𝐸𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))

Theoremwlkiswwlk2lem5 25961* Lemma 5 for wlkiswwlk2 25963. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       ((𝑉 USGrph 𝐸𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸𝐹 ∈ Word dom 𝐸))

Theoremwlkiswwlk2lem6 25962* Lemma 6 for wlkiswwlk2 25963. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       ((𝑉 USGrph 𝐸𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))

Theoremwlkiswwlk2 25963* A walk as word corresponds to the sequence of vertices in a walk in an undirected simple graph. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
(𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 WWalks 𝐸) → ∃𝑓 𝑓(𝑉 Walks 𝐸)𝑃))

Theoremwlkiswwlk 25964* A walk as word corresponds to a walk in an undirected simple graph. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
(𝑉 USGrph 𝐸 → (∃𝑓 𝑓(𝑉 Walks 𝐸)𝑃𝑃 ∈ (𝑉 WWalks 𝐸)))

Theoremwlklniswwlkn1 25965 The sequence of vertices in a walk of length n is a walk as word of length n in an undirected simple graph. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
(𝑉 USGrph 𝐸 → ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁) → 𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))

Theoremwlklniswwlkn2 25966* A walk of length n as word corresponds to the sequence of vertices in a walk of length n in an undirected simple graph. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
(𝑉 USGrph 𝐸 → (𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ∃𝑓(𝑓(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝑓) = 𝑁)))

Theoremwlklniswwlkn 25967* A walk of length n as word corresponds to a walk with length n in an undirected simple graph. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
(𝑉 USGrph 𝐸 → (∃𝑓(𝑓(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝑓) = 𝑁) ↔ 𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))

Theoremwwlkiswwlkn 25968 A walk of a fixed length as word is a walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 17-Jul-2018.)
(𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑃 ∈ (𝑉 WWalks 𝐸))

Theoremwwlksswwlkn 25969 The walks of a fixed length as words are walks (in an undirected graph) as words. (Contributed by Alexander van der Vekens, 17-Jul-2018.)
((𝑉 WWalksN 𝐸)‘𝑁) ⊆ (𝑉 WWalks 𝐸)

Theoremwwlknimpb 25970 Basic implications for a set being a walk of length n (represented by a word). (Contributed by Alexander van der Vekens, 3-Oct-2018.)
(𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)))

Theoremwwlkn0s 25971* The set of all walks as words of length 0 is the set of all words of length 1 over the vertices. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
((𝑉𝑋𝐸𝑌) → ((𝑉 WWalksN 𝐸)‘0) = {𝑤 ∈ Word 𝑉 ∣ (#‘𝑤) = 1})

Theoremvfwlkniswwlkn 25972 If the edge function of a walk has length n, then the vertex function of the walk is a word representing the walk as word of length n. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑊)) = 𝑁)) → (2nd𝑊) ∈ ((𝑉 WWalksN 𝐸)‘𝑁))

Theorem2wlkeq 25973* Conditions for two walks (within the same graph) being the same. (Contributed by AV, 1-Jul-2018.) (Revised by AV, 16-May-2019.)
((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))

Theoremusg2wlkeq 25974* Conditions for two walks within the same undirected simple graph being the same. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 3-Jul-2018.)
((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦))))

Theoremusg2wlkeq2 25975 Conditions for which two walks within the same undirected simple graph are the same. It is sufficient that the vertices (in the same order) are identical. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
(((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) ∧ (𝑋 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑋)) = 𝑁) ∧ (𝑊 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑊)) = 𝑁)) → ((2nd𝑋) = (2nd𝑊) → 𝑋 = 𝑊))

Theoremwlknwwlknfun 25976* Lemma 1 for wlknwwlknbij2 25980. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁}    &   𝑊 = ((𝑉 WWalksN 𝐸)‘𝑁)    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       (𝑁 ∈ ℕ0𝐹:𝑇𝑊)

Theoremwlknwwlkninj 25977* Lemma 2 for wlknwwlknbij2 25980. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁}    &   𝑊 = ((𝑉 WWalksN 𝐸)‘𝑁)    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) → 𝐹:𝑇1-1𝑊)

Theoremwlknwwlknsur 25978* Lemma 3 for wlknwwlknbij2 25980. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁}    &   𝑊 = ((𝑉 WWalksN 𝐸)‘𝑁)    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)

Theoremwlknwwlknbij 25979* Lemma 4 for wlknwwlknbij2 25980. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁}    &   𝑊 = ((𝑉 WWalksN 𝐸)‘𝑁)    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) → 𝐹:𝑇1-1-onto𝑊)

Theoremwlknwwlknbij2 25980* There is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) → ∃𝑓 𝑓:{𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁}–1-1-onto→((𝑉 WWalksN 𝐸)‘𝑁))

Theoremwlknwwlknen 25981* The set of walks of a fixed length and the set of walks represented by words are equinumerous. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) → {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁} ≈ ((𝑉 WWalksN 𝐸)‘𝑁))

Theoremwlknwwlkneqs 25982* The set of walks of a fixed length and the set of walks represented by words have the same size. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) → (#‘{𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁}) = (#‘((𝑉 WWalksN 𝐸)‘𝑁)))

Theoremwlkiswwlkfun 25983* Lemma 1 for wlkiswwlkbij2 25987. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Proof shortened by Alexander van der Vekens, 25-Aug-2018.)
𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}    &   𝑊 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)

Theoremwlkiswwlkinj 25984* Lemma 2 for wlkiswwlkbij2 25987. (Contributed by Alexander van der Vekens, 23-Jul-2018.) (Proof shortened by Alexander van der Vekens, 25-Aug-2018.)
𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}    &   𝑊 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇1-1𝑊)

Theoremwlkiswwlksur 25985* Lemma 3 for wlkiswwlkbij2 25987. (Contributed by Alexander van der Vekens, 23-Jul-2018.)
𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}    &   𝑊 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)

Theoremwlkiswwlkbij 25986* Lemma 4 for wlkiswwlkbij2 25987. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}    &   𝑊 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇1-1-onto𝑊)

Theoremwlkiswwlkbij2 25987* There is a bijection between the set of walks of a fixed length, starting at a fixed vertex, and the set of walks represented as words of the same length, starting at the same vertex. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → ∃𝑓 𝑓:{𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃})

Theoremwwlkeq 25988* Equality of two walks (as words). (Contributed by Alexander van der Vekens, 4-Aug-2018.)
((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ 𝑇 ∈ (𝑉 WWalks 𝐸)) → (𝑊 = 𝑇 ↔ ((#‘𝑊) = (#‘𝑇) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊𝑖) = (𝑇𝑖))))

Theoremwwlknred 25989 Reduction of a walk (as word) by removing the trailing edge/vertex. (Contributed by Alexander van der Vekens, 4-Aug-2018.)
(𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))

Theoremwwlknext 25990 Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 4-Aug-2018.)
((𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ 𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))

Theoremwwlknextbi 25991 Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
(((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ↔ 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))

Theoremwwlknredwwlkn 25992* For each walk (as word) of length at least 1 there is a shorter walk (as word). (Contributed by Alexander van der Vekens, 22-Aug-2018.)
(𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)))

Theoremwwlknredwwlkn0 25993* For each walk (as word) of length at least 1 there is a shorter walk (as word) starting at the same vertex. (Contributed by Alexander van der Vekens, 22-Aug-2018.)
((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → ((𝑊‘0) = 𝑃 ↔ ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)))

Theoremwwlkextwrd 25994* Lemma 0 for wwlkextbij 25999. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}       (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)})

Theoremwwlkextfun 25995* Lemma 1 for wwlkextbij 25999. (Contributed by Alexander van der Vekens, 7-Aug-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}    &   𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸}    &   𝐹 = (𝑡𝐷 ↦ ( lastS ‘𝑡))       (𝑁 ∈ ℕ0𝐹:𝐷𝑅)

Theoremwwlkextinj 25996* Lemma 2 for wwlkextbij 25999. (Contributed by Alexander van der Vekens, 7-Aug-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}    &   𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸}    &   𝐹 = (𝑡𝐷 ↦ ( lastS ‘𝑡))       (𝑁 ∈ ℕ0𝐹:𝐷1-1𝑅)

Theoremwwlkextsur 25997* Lemma 3 for wwlkextbij 25999. (Contributed by Alexander van der Vekens, 7-Aug-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}    &   𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸}    &   𝐹 = (𝑡𝐷 ↦ ( lastS ‘𝑡))       (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝐹:𝐷onto𝑅)

Theoremwwlkextbij0 25998* Lemma 4 for wwlkextbij 25999. (Contributed by Alexander van der Vekens, 7-Aug-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}    &   𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸}    &   𝐹 = (𝑡𝐷 ↦ ( lastS ‘𝑡))       (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝐹:𝐷1-1-onto𝑅)

Theoremwwlkextbij 25999* There is a bijection between the extensions of a walk (as word) by an edge and the set of vertices being connected to the trailing vertex of the walk. (Contributed by Alexander van der Vekens, 21-Aug-2018.)
(𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ∃𝑓 𝑓:{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸})

Theoremwwlkexthasheq 26000* The number of the extensions of a walk (as word) by an edge equals the number of vertices being connected to the trailing vertex of the walk. (Contributed by Alexander van der Vekens, 23-Aug-2018.) (Proof shortened by AV, 4-May-2021.)
(𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}) = (#‘{𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸}))

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