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Theorem List for Metamath Proof Explorer - 13501-13600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlsws4 13501 The last symbol of a 4 letter word is its fourth symbol. (Contributed by AV, 8-Feb-2021.)
(𝐷𝑉 → ( lastS ‘⟨“𝐴𝐵𝐶𝐷”⟩) = 𝐷)

Theorems2prop 13502 A length 2 word is an unordered pair of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
((𝐴𝑆𝐵𝑆) → ⟨“𝐴𝐵”⟩ = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩})

Theorems2dmALT 13503 Alternate version of s2dm 13485, having a shorter proof, but requiring that 𝐴 and 𝐵 are sets. (Contributed by AV, 9-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝑆𝐵𝑆) → dom ⟨“𝐴𝐵”⟩ = {0, 1})

Theorems3tpop 13504 A length 3 word is an unordered triple of ordered pairs. (Contributed by AV, 23-Jan-2021.)
((𝐴𝑆𝐵𝑆𝐶𝑆) → ⟨“𝐴𝐵𝐶”⟩ = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩})

Theorems4prop 13505 A length 4 word is a union of two unordered pairs of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
(((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ⟨“𝐴𝐵𝐶𝐷”⟩ = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐷⟩}))

Theorems3fn 13506 A length 3 word is a function with a triple as domain. (Contributed by Alexander van der Vekens, 5-Dec-2017.) (Revised by AV, 23-Jan-2021.)
((𝐴𝑉𝐵𝑉𝐶𝑉) → ⟨“𝐴𝐵𝐶”⟩ Fn {0, 1, 2})

Theoremfuncnvs1 13507 The converse of a singleton word is a function. (Contributed by AV, 22-Jan-2021.)
Fun ⟨“𝐴”⟩

Theoremfuncnvs2 13508 The converse of a length 2 word is a function if its symbols are different sets. (Contributed by AV, 23-Jan-2021.)
((𝐴𝑉𝐵𝑉𝐴𝐵) → Fun ⟨“𝐴𝐵”⟩)

Theoremfuncnvs3 13509 The converse of a length 3 word is a function if its symbols are different sets. (Contributed by Alexander van der Vekens, 31-Jan-2018.) (Revised by AV, 23-Jan-2021.)
(((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → Fun ⟨“𝐴𝐵𝐶”⟩)

Theoremfuncnvs4 13510 The converse of a length 4 word is a function if its symbols are different sets. (Contributed by AV, 10-Feb-2021.)
((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) ∧ ((𝐴𝐵𝐴𝐶𝐴𝐷) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷)) → Fun ⟨“𝐴𝐵𝐶𝐷”⟩)

Theorems2f1o 13511 A length 2 word with mutually different symbols is a one-to-one function onto the set of the symbols. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐸 = ⟨“𝐴𝐵”⟩ → 𝐸:{0, 1}–1-1-onto→{𝐴, 𝐵}))

Theoremf1oun2prg 13512 A union of unordered pairs of ordered pairs with different elements is a one-to-one onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → (((𝐴𝐵𝐴𝐶𝐴𝐷) ∧ (𝐵𝐶𝐵𝐷𝐶𝐷)) → ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐷⟩}):({0, 1} ∪ {2, 3})–1-1-onto→({𝐴, 𝐵} ∪ {𝐶, 𝐷})))

Theorems4f1o 13513 A length 4 word with mutually different symbols is a one-to-one function onto the set of the symbols. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
(((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (((𝐴𝐵𝐴𝐶𝐴𝐷) ∧ (𝐵𝐶𝐵𝐷𝐶𝐷)) → (𝐸 = ⟨“𝐴𝐵𝐶𝐷”⟩ → 𝐸:dom 𝐸1-1-onto→({𝐴, 𝐵} ∪ {𝐶, 𝐷}))))

Theorems4dom 13514 The domain of a length 4 word is the union of two (disjunct) pairs. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
(((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (𝐸 = ⟨“𝐴𝐵𝐶𝐷”⟩ → dom 𝐸 = ({0, 1} ∪ {2, 3})))

Theorems2co 13515 Mapping a doubleton word by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐹:𝑋𝑌)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)       (𝜑 → (𝐹 ∘ ⟨“𝐴𝐵”⟩) = ⟨“(𝐹𝐴)(𝐹𝐵)”⟩)

Theorems3co 13516 Mapping a length 3 string by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐹:𝑋𝑌)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)       (𝜑 → (𝐹 ∘ ⟨“𝐴𝐵𝐶”⟩) = ⟨“(𝐹𝐴)(𝐹𝐵)(𝐹𝐶)”⟩)

Theorems0s1 13517 Concatenation of fixed length strings. (This special case of ccatlid 13222 is provided to complete the pattern s0s1 13517, df-s2 13444, df-s3 13445, ...) (Contributed by Mario Carneiro, 28-Feb-2016.)
⟨“𝐴”⟩ = (∅ ++ ⟨“𝐴”⟩)

Theorems1s2 13518 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶”⟩)

Theorems1s3 13519 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷”⟩)

Theorems1s4 13520 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸”⟩)

Theorems1s5 13521 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸𝐹”⟩)

Theorems1s6 13522 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸𝐹𝐺”⟩)

Theorems1s7 13523 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩)

Theorems2s2 13524 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶𝐷”⟩)

Theorems4s2 13525 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸𝐹”⟩)

Theorems4s3 13526 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸𝐹𝐺”⟩)

Theorems4s4 13527 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸𝐹𝐺𝐻”⟩)

Theorems3s4 13528 Concatenation of fixed length strings. (Contributed by AV, 1-Mar-2021.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷𝐸𝐹𝐺”⟩)

Theorems2s5 13529 Concatenation of fixed length strings. (Contributed by AV, 1-Mar-2021.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶𝐷𝐸𝐹𝐺”⟩)

Theorems5s2 13530 Concatenation of fixed length strings. (Contributed by AV, 1-Mar-2021.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ++ ⟨“𝐹𝐺”⟩)

Theorems2eq2s1eq 13531 Two length 2 words are equal iff the corresponding singleton words consisting of their symbols are equal. (Contributed by Alexander van der Vekens, 24-Sep-2018.)
(((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (⟨“𝐴𝐵”⟩ = ⟨“𝐶𝐷”⟩ ↔ (⟨“𝐴”⟩ = ⟨“𝐶”⟩ ∧ ⟨“𝐵”⟩ = ⟨“𝐷”⟩)))

Theorems2eq2seq 13532 Two length 2 words are equal iff the corresponding symbols are equal. (Contributed by AV, 20-Oct-2018.)
(((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (⟨“𝐴𝐵”⟩ = ⟨“𝐶𝐷”⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremswrds2 13533 Extract two adjacent symbols from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
((𝑊 ∈ Word 𝐴𝐼 ∈ ℕ0 ∧ (𝐼 + 1) ∈ (0..^(#‘𝑊))) → (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩) = ⟨“(𝑊𝐼)(𝑊‘(𝐼 + 1))”⟩)

Theoremwrdlen2i 13534 Implications of a word of length 2. (Contributed by AV, 27-Jul-2018.) (Proof shortened by AV, 14-Oct-2018.)
((𝑆𝑉𝑇𝑉) → (𝑊 = {⟨0, 𝑆⟩, ⟨1, 𝑇⟩} → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 2) ∧ ((𝑊‘0) = 𝑆 ∧ (𝑊‘1) = 𝑇))))

Theoremwrd2pr2op 13535 A word of length 2 represented as unordered pair of ordered pairs. (Contributed by AV, 20-Oct-2018.) (Proof shortened by AV, 26-Jan-2021.)
((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 2) → 𝑊 = {⟨0, (𝑊‘0)⟩, ⟨1, (𝑊‘1)⟩})

Theoremwrdlen2 13536 A word of length 2. (Contributed by AV, 20-Oct-2018.)
((𝑆𝑉𝑇𝑉) → (𝑊 = {⟨0, 𝑆⟩, ⟨1, 𝑇⟩} ↔ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 2) ∧ ((𝑊‘0) = 𝑆 ∧ (𝑊‘1) = 𝑇))))

Theoremwrdlen2s2 13537 A word of length 2 as doubleton word. (Contributed by AV, 20-Oct-2018.)
((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 2) → 𝑊 = ⟨“(𝑊‘0)(𝑊‘1)”⟩)

Theoremwrdl2exs2 13538* A word of length 2 is a doubleton word. (Contributed by AV, 25-Jan-2021.)
((𝑊 ∈ Word 𝑆 ∧ (#‘𝑊) = 2) → ∃𝑠𝑆𝑡𝑆 𝑊 = ⟨“𝑠𝑡”⟩)

Theoremwrd3tpop 13539 A word of length 3 represented as triple of ordered pairs. (Contributed by AV, 26-Jan-2021.)
((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) → 𝑊 = {⟨0, (𝑊‘0)⟩, ⟨1, (𝑊‘1)⟩, ⟨2, (𝑊‘2)⟩})

Theoremwrdlen3s3 13540 A word of length 3 as length 3 string. (Contributed by AV, 26-Jan-2021.)
((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) → 𝑊 = ⟨“(𝑊‘0)(𝑊‘1)(𝑊‘2)”⟩)

Theoremrepsw2 13541 The "repeated symbol word" of length 2. (Contributed by AV, 6-Nov-2018.)
(𝑆𝑉 → (𝑆 repeatS 2) = ⟨“𝑆𝑆”⟩)

Theoremrepsw3 13542 The "repeated symbol word" of length 3. (Contributed by AV, 6-Nov-2018.)
(𝑆𝑉 → (𝑆 repeatS 3) = ⟨“𝑆𝑆𝑆”⟩)

Theoremswrd2lsw 13543 Extract the last two symbols from a word. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
((𝑊 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → (𝑊 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = ⟨“(𝑊‘((#‘𝑊) − 2))( lastS ‘𝑊)”⟩)

Theorem2swrd2eqwrdeq 13544 Two words of length at least 2 are equal if and only if they have the same prefix and the same two single symbols suffix. (Contributed by AV, 24-Sep-2018.) (Revised by Mario Carneiro/AV, 23-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 2)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 2)⟩) ∧ (𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑊) − 2)) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈)))))

Theoremccatw2s1ccatws2 13545 The concatenation of a word with two singleton words equals the concatenation of the word with the doubleton word consisting of the symbols of the two singletons. (Contributed by Mario Carneiro/AV, 21-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑋𝑉𝑌𝑉) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) = (𝑊 ++ ⟨“𝑋𝑌”⟩))

Theoremccat2s1fvwALT 13546 Alternate proof of ccat2s1fvw 13267 using words of length 2, see df-s2 13444. A symbol of the concatenation of a word with two single symbols corresponding to the symbol of the word. (Contributed by AV, 22-Sep-2018.) (Proof shortened by Mario Carneiro/AV, 21-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(((𝑊 ∈ Word 𝑉𝐼 ∈ ℕ0𝐼 < (#‘𝑊)) ∧ (𝑋𝑉𝑌𝑉)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = (𝑊𝐼))

Theoremwwlktovf 13547* Lemma 1 for wrd2f1tovbij 13551. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}    &   𝑅 = {𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}    &   𝐹 = (𝑡𝐷 ↦ (𝑡‘1))       𝐹:𝐷𝑅

Theoremwwlktovf1 13548* Lemma 2 for wrd2f1tovbij 13551. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}    &   𝑅 = {𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}    &   𝐹 = (𝑡𝐷 ↦ (𝑡‘1))       𝐹:𝐷1-1𝑅

Theoremwwlktovfo 13549* Lemma 3 for wrd2f1tovbij 13551. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}    &   𝑅 = {𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}    &   𝐹 = (𝑡𝐷 ↦ (𝑡‘1))       (𝑃𝑉𝐹:𝐷onto𝑅)

Theoremwwlktovf1o 13550* Lemma 4 for wrd2f1tovbij 13551. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}    &   𝑅 = {𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}    &   𝐹 = (𝑡𝐷 ↦ (𝑡‘1))       (𝑃𝑉𝐹:𝐷1-1-onto𝑅)

Theoremwrd2f1tovbij 13551* There is a bijection between words of length two with a fixed first symbol contained in a pair and the symbols contained in a pair together with the fixed symbol. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
((𝑉𝑌𝑃𝑉) → ∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}–1-1-onto→{𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋})

Theoremeqwrds3 13552 A word is equal with a length 3 string iff it has length 3 and the same symbol at each position. (Contributed by AV, 12-May-2021.)
((𝑊 ∈ Word 𝑉 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝑊 = ⟨“𝐴𝐵𝐶”⟩ ↔ ((#‘𝑊) = 3 ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = 𝐵 ∧ (𝑊‘2) = 𝐶))))

Theoremwrdl3s3 13553* A word of length 3 is a length 3 string. (Contributed by AV, 18-May-2021.)
((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩)

Theorems3sndisj 13554* The singletons consisting of length 3 strings which have distinct third symbols are disjunct. (Contributed by AV, 17-May-2021.)
((𝐴𝑋𝐵𝑌) → Disj 𝑐𝑍 {⟨“𝐴𝐵𝑐”⟩})

Theorems3iunsndisj 13555* The union of singletons consisting of length 3 strings which have distinct first and third symbols are disjunct. (Contributed by AV, 17-May-2021.)
(𝐵𝑋Disj 𝑎𝑌 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩})

Theoremofccat 13556 Letterwise operations on word concatenations. (Contributed by Thierry Arnoux, 28-Sep-2018.)
(𝜑𝐸 ∈ Word 𝑆)    &   (𝜑𝐹 ∈ Word 𝑆)    &   (𝜑𝐺 ∈ Word 𝑇)    &   (𝜑𝐻 ∈ Word 𝑇)    &   (𝜑 → (#‘𝐸) = (#‘𝐺))    &   (𝜑 → (#‘𝐹) = (#‘𝐻))       (𝜑 → ((𝐸 ++ 𝐹) ∘𝑓 𝑅(𝐺 ++ 𝐻)) = ((𝐸𝑓 𝑅𝐺) ++ (𝐹𝑓 𝑅𝐻)))

Theoremofs1 13557 Letterwise operations on a single letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.)
((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩ ∘𝑓 𝑅⟨“𝐵”⟩) = ⟨“(𝐴𝑅𝐵)”⟩)

Theoremofs2 13558 Letterwise operations on a double letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.)
(((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑇𝐷𝑇)) → (⟨“𝐴𝐵”⟩ ∘𝑓 𝑅⟨“𝐶𝐷”⟩) = ⟨“(𝐴𝑅𝐶)(𝐵𝑅𝐷)”⟩)

5.8  Reflexive and transitive closures of relations

A relation, 𝑅, has the reflexive property if 𝐴𝑅𝐴 holds whenever 𝐴 is an element which could be related by the the relation, namely elements of its domain and range. Eliminating dummy variables we see that a segment of the identity relation must be a subset of the relation or ( I ↾ (ran 𝑅 ∪ dom 𝑅)) ⊆ 𝑅. See issref 5428.

A relation, 𝑅, has the transitive property if 𝐴𝑅𝐶 holds whenever there exists an intermediate value 𝐵 such that both 𝐴𝑅𝐵 and 𝐵𝑅𝐶 hold. This can be expressed without dummy variables as (𝑅𝑅) ⊆ 𝑅. See cotr 5427.

The transitive closure of a relation, (t+‘𝑅), is the smallest superset of the relation which has the transitive property. Likewise the reflexive-transitive closure, (t*‘𝑅), is the smallest superset which has both the reflexive and transitive properties.

Not to be confused with the transitive closure of a set, trcl 8487, which is a closure relative to a different transitive property, df-tr 4681.

5.8.1  The reflexive and transitive properties of relations

Theoremcoss12d 13559 Subset deduction for composition of two classes. (Contributed by RP, 24-Dec-2019.)
(𝜑𝐴𝐵)    &   (𝜑𝐶𝐷)       (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐷))

Theoremtrrelssd 13560 The composition of subclasses of a transitive relation is a subclass of that relation. (Contributed by RP, 24-Dec-2019.)
(𝜑 → (𝑅𝑅) ⊆ 𝑅)    &   (𝜑𝑆𝑅)    &   (𝜑𝑇𝑅)       (𝜑 → (𝑆𝑇) ⊆ 𝑅)

Theoremxpcogend 13561 The most interesting case of the composition of two cross products. (Contributed by RP, 24-Dec-2019.)
(𝜑 → (𝐵𝐶) ≠ ∅)       (𝜑 → ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐷))

Theoremxpcoidgend 13562 If two classes are not disjoint, then the composition of their cross-product with itself is idempotent. (Contributed by RP, 24-Dec-2019.)
(𝜑 → (𝐴𝐵) ≠ ∅)       (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))

Theoremcotr2g 13563* Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr2 13564 for the main application. (Contributed by RP, 22-Mar-2020.)
dom 𝐵𝐷    &   (ran 𝐵 ∩ dom 𝐴) ⊆ 𝐸    &   ran 𝐴𝐹       ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝐷𝑦𝐸𝑧𝐹 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))

Theoremcotr2 13564* Two ways of saying a relation is transitive. Special instance of cotr2g 13563. (Contributed by RP, 22-Mar-2020.)
dom 𝑅𝐴    &   (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵    &   ran 𝑅𝐶       ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Theoremcotr3 13565* Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020.)
𝐴 = dom 𝑅    &   𝐵 = (𝐴𝐶)    &   𝐶 = ran 𝑅       ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Theoremcoemptyd 13566 Deduction about composition of classes with no relational content in common. (Contributed by RP, 24-Dec-2019.)
(𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅)       (𝜑 → (𝐴𝐵) = ∅)

Theoremxptrrel 13567 The cross product is always a transitive relation. (Contributed by RP, 24-Dec-2019.)
((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)

Theorem0trrel 13568 The empty class is a transitive relation. (Contributed by RP, 24-Dec-2019.)
(∅ ∘ ∅) ⊆ ∅

5.8.2  Basic properties of closures

Theoremcleq1lem 13569 Equality implies bijection. (Contributed by RP, 9-May-2020.)
(𝐴 = 𝐵 → ((𝐴𝐶𝜑) ↔ (𝐵𝐶𝜑)))

Theoremcleq1 13570* Equality of relations implies equality of closures. (Contributed by RP, 9-May-2020.)
(𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} = {𝑟 ∣ (𝑆𝑟𝜑)})

Theoremclsslem 13571* The closure of a subclass is a subclass of the closure. (Contributed by RP, 16-May-2020.)
(𝑅𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} ⊆ {𝑟 ∣ (𝑆𝑟𝜑)})

5.8.3  Definitions and basic properties of transitive closures

Syntaxctcl 13572 Extend class notation to include the transitive closure symbol.
class t+

Syntaxcrtcl 13573 Extend class notation with reflexive-transitive closure.
class t*

Definitiondf-trcl 13574* Transitive closure of a relation. This is the smallest superset which has the transitive property. (Contributed by FL, 27-Jun-2011.)
t+ = (𝑥 ∈ V ↦ {𝑧 ∣ (𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})

Definitiondf-rtrcl 13575* Reflexive-transitive closure of a relation. This is the smallest superset which is reflexive property over all elements of its domain and range and has the transitive property. (Contributed by FL, 27-Jun-2011.)
t* = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})

Theoremtrcleq1 13576* Equality of relations implies equality of transitive closures. (Contributed by RP, 9-May-2020.)
(𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} = {𝑟 ∣ (𝑆𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})

Theoremtrclsslem 13577* The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020.)
(𝑅𝑆 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ⊆ {𝑟 ∣ (𝑆𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})

Theoremtrcleq2lem 13578 Equality implies bijection. (Contributed by RP, 5-May-2020.)
(𝐴 = 𝐵 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑅𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))

Theoremcvbtrcl 13579* Change of bound variable in class of all transitive relations which are supersets of a relation. (Contributed by RP, 5-May-2020.)
{𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑦 ∣ (𝑅𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)}

Theoremtrcleq12lem 13580 Equality implies bijection. (Contributed by RP, 9-May-2020.)
((𝑅 = 𝑆𝐴 = 𝐵) → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑆𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))

Theoremtrclexlem 13581 Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 5-May-2020.)
(𝑅𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)

Theoremtrclublem 13582* If a relation exists then the class of transitive relations which are supersets of that relation is not empty. (Contributed by RP, 28-Apr-2020.)
(𝑅𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})

Theoremtrclubi 13583* The Cartesian product of the domain and range of a relation is an upper bound for its transitive closure. (Contributed by RP, 2-Jan-2020.) (Revised by RP, 28-Apr-2020.) (Revised by AV, 26-Mar-2021.)
Rel 𝑅    &   𝑅 ∈ V        {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ⊆ (dom 𝑅 × ran 𝑅)

TheoremtrclubiOLD 13584* Obsolete version of trclubi 13583 as of 26-Mar-2021. (Contributed by RP, 2-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Rel 𝑅    &   𝑅𝑉        {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ⊆ (dom 𝑅 × ran 𝑅)

Theoremtrclubgi 13585* The union with the Cartesian product of its domain and range is an upper bound for a set's transitive closure. (Contributed by RP, 3-Jan-2020.) (Revised by RP, 28-Apr-2020.) (Revised by AV, 26-Mar-2021.)
𝑅 ∈ V        {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))

TheoremtrclubgiOLD 13586* Obsolete version of trclubgi 13585 as of 26-Mar-2021. (Contributed by RP, 3-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑅𝑉        {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))

Theoremtrclub 13587* The Cartesian product of the domain and range of a relation is an upper bound for its transitive closure. (Contributed by RP, 17-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅) → {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ⊆ (dom 𝑅 × ran 𝑅))

Theoremtrclubg 13588* The union with the Cartesian product of its domain and range is an upper bound for a set's transitive closure (as a relation). (Contributed by RP, 17-May-2020.)
(𝑅𝑉 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))

Theoremtrclfv 13589* The transitive closure of a relation. (Contributed by RP, 28-Apr-2020.)
(𝑅𝑉 → (t+‘𝑅) = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})

Theorembrintclab 13590* Two ways to express a binary relation which is the intersection of a class. (Contributed by RP, 4-Apr-2020.)
(𝐴 {𝑥𝜑}𝐵 ↔ ∀𝑥(𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑥))

Theorembrtrclfv 13591* Two ways of expressing the transitive closure of a binary relation. (Contributed by RP, 9-May-2020.)
(𝑅𝑉 → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵)))

Theorembrcnvtrclfv 13592* Two ways of expressing the transitive closure of the converse of a binary relation. (Contributed by RP, 9-May-2020.)
((𝑅𝑈𝐴𝑉𝐵𝑊) → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐵𝑟𝐴)))

Theorembrtrclfvcnv 13593* Two ways of expressing the transitive closure of the converse of a binary relation. (Contributed by RP, 10-May-2020.)
(𝑅𝑉 → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵)))

Theorembrcnvtrclfvcnv 13594* Two ways of expressing the transitive closure of the converse of the converse of a binary relation. (Contributed by RP, 10-May-2020.)
((𝑅𝑈𝐴𝑉𝐵𝑊) → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐵𝑟𝐴)))

Theoremtrclfvss 13595 The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020.)
((𝑅𝑉𝑆𝑊𝑅𝑆) → (t+‘𝑅) ⊆ (t+‘𝑆))

Theoremtrclfvub 13596 The transitive closure of a relation has an upper bound. (Contributed by RP, 28-Apr-2020.)
(𝑅𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))

Theoremtrclfvlb 13597 The transitive closure of a relation has a lower bound. (Contributed by RP, 28-Apr-2020.)
(𝑅𝑉𝑅 ⊆ (t+‘𝑅))

Theoremtrclfvcotr 13598 The transitive closure of a relation is a transitive relation. (Contributed by RP, 29-Apr-2020.)
(𝑅𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))

Theoremtrclfvlb2 13599 The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.)
(𝑅𝑉 → (𝑅𝑅) ⊆ (t+‘𝑅))

Theoremtrclfvlb3 13600 The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.)
(𝑅𝑉 → (𝑅 ∪ (𝑅𝑅)) ⊆ (t+‘𝑅))

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