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Theorem List for Metamath Proof Explorer - 16301-16400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmre1cl 16301 In any Moore collection the base set is closed. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)

Theoremmreintcl 16302 A nonempty collection of closed sets has a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝑆𝐶)

Theoremmreiincl 16303* A nonempty indexed intersection of closed sets is closed. (Contributed by Stefan O'Rear, 1-Feb-2015.)
((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → 𝑦𝐼 𝑆𝐶)

Theoremmrerintcl 16304 The relative intersection of a set of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → (𝑋 𝑆) ∈ 𝐶)

Theoremmreriincl 16305* The relative intersection of a family of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) → (𝑋 𝑦𝐼 𝑆) ∈ 𝐶)

Theoremmreincl 16306 Two closed sets have a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶𝐵𝐶) → (𝐴𝐵) ∈ 𝐶)

Theoremmreuni 16307 Since the entire base set of a Moore collection is the greatest element of it, the base set can be recovered from a Moore collection by set union. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝐶 ∈ (Moore‘𝑋) → 𝐶 = 𝑋)

Theoremmreunirn 16308 Two ways to express the notion of being a Moore collection on an unspecified base. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝐶 ran Moore ↔ 𝐶 ∈ (Moore‘ 𝐶))

Theoremismred 16309* Properties that determine a Moore collection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝜑𝐶 ⊆ 𝒫 𝑋)    &   (𝜑𝑋𝐶)    &   ((𝜑𝑠𝐶𝑠 ≠ ∅) → 𝑠𝐶)       (𝜑𝐶 ∈ (Moore‘𝑋))

Theoremismred2 16310* Properties that determine a Moore collection, using restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝜑𝐶 ⊆ 𝒫 𝑋)    &   ((𝜑𝑠𝐶) → (𝑋 𝑠) ∈ 𝐶)       (𝜑𝐶 ∈ (Moore‘𝑋))

Theoremmremre 16311 The Moore collections of subsets of a space, viewed as a kind of subset of the power set, form a Moore collection in their own right on the power set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝑋𝑉 → (Moore‘𝑋) ∈ (Moore‘𝒫 𝑋))

Theoremsubmre 16312 The subcollection of a closed set system below a given closed set is itself a closed set system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → (𝐶 ∩ 𝒫 𝐴) ∈ (Moore‘𝐴))

7.2.1  Moore closures

Theoremmrcflem 16313* The domain and range of the function expression for Moore closures. (Contributed by Stefan O'Rear, 31-Jan-2015.)
(𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}):𝒫 𝑋𝐶)

Theoremfnmrc 16314 Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
mrCls Fn ran Moore

Theoremmrcfval 16315* Value of the function expression for the Moore closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))

Theoremmrcf 16316 The Moore closure is a function mapping arbitrary subsets to closed sets. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)

Theoremmrcval 16317* Evaluation of the Moore closure of a set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})

Theoremmrccl 16318 The Moore closure of a set is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) ∈ 𝐶)

Theoremmrcsncl 16319 The Moore closure of a singleton is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹‘{𝑈}) ∈ 𝐶)

Theoremmrcid 16320 The closure of a closed set is itself. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐶) → (𝐹𝑈) = 𝑈)

Theoremmrcssv 16321 The closure of a set is a subset of the base. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (Moore‘𝑋) → (𝐹𝑈) ⊆ 𝑋)

Theoremmrcidb 16322 A set is closed iff it is equal to its closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (Moore‘𝑋) → (𝑈𝐶 ↔ (𝐹𝑈) = 𝑈))

Theoremmrcss 16323 Closure preserves subset ordering. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → (𝐹𝑈) ⊆ (𝐹𝑉))

Theoremmrcssid 16324 The closure of a set is a superset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈 ⊆ (𝐹𝑈))

Theoremmrcidb2 16325 A set is closed iff it contains its closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝑈𝐶 ↔ (𝐹𝑈) ⊆ 𝑈))

Theoremmrcidm 16326 The closure operation is idempotent. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹‘(𝐹𝑈)) = (𝐹𝑈))

Theoremmrcsscl 16327 The closure is the minimal closed set; any closed set which contains the generators is a superset of the closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝐶) → (𝐹𝑈) ⊆ 𝑉)

Theoremmrcuni 16328 Idempotence of closure under a general union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 𝑈) = (𝐹 (𝐹𝑈)))

Theoremmrcun 16329 Idempotence of closure under a pair union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹‘(𝑈𝑉)) = (𝐹‘((𝐹𝑈) ∪ (𝐹𝑉))))

Theoremmrcssvd 16330 The Moore closure of a set is a subset of the base. Deduction form of mrcssv 16321. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)       (𝜑 → (𝑁𝐵) ⊆ 𝑋)

Theoremmrcssd 16331 Moore closure preserves subset ordering. Deduction form of mrcss 16323. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑈𝑉)    &   (𝜑𝑉𝑋)       (𝜑 → (𝑁𝑈) ⊆ (𝑁𝑉))

Theoremmrcssidd 16332 A set is contained in its Moore closure. Deduction form of mrcssid 16324. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑈𝑋)       (𝜑𝑈 ⊆ (𝑁𝑈))

Theoremmrcidmd 16333 Moore closure is idempotent. Deduction form of mrcidm 16326. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑈𝑋)       (𝜑 → (𝑁‘(𝑁𝑈)) = (𝑁𝑈))

Theoremmressmrcd 16334 In a Moore system, if a set is between another set and its closure, the two sets have the same closure. Deduction form. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑆 ⊆ (𝑁𝑇))    &   (𝜑𝑇𝑆)       (𝜑 → (𝑁𝑆) = (𝑁𝑇))

Theoremsubmrc 16335 In a closure system which is cut off above some level, closures below that level act as normal. (Contributed by Stefan O'Rear, 9-Mar-2015.)
𝐹 = (mrCls‘𝐶)    &   𝐺 = (mrCls‘(𝐶 ∩ 𝒫 𝐷))       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) = (𝐹𝑈))

Theoremmrieqvlemd 16336 In a Moore system, if 𝑌 is a member of 𝑆, (𝑆 ∖ {𝑌}) and 𝑆 have the same closure if and only if 𝑌 is in the closure of (𝑆 ∖ {𝑌}). Used in the proof of mrieqvd 16345 and mrieqv2d 16346. Deduction form. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑆𝑋)    &   (𝜑𝑌𝑆)       (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)))

7.2.2  Independent sets in a Moore system

Theoremmrisval 16337* Value of the set of independent sets of a Moore system. (Contributed by David Moews, 1-May-2017.)
𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)       (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})

Theoremismri 16338* Criterion for a set to be an independent set of a Moore system. (Contributed by David Moews, 1-May-2017.)
𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)       (𝐴 ∈ (Moore‘𝑋) → (𝑆𝐼 ↔ (𝑆𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))

Theoremismri2 16339* Criterion for a subset of the base set in a Moore system to be independent. (Contributed by David Moews, 1-May-2017.)
𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)       ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))

Theoremismri2d 16340* Criterion for a subset of the base set in a Moore system to be independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝐴 ∈ (Moore‘𝑋))    &   (𝜑𝑆𝑋)       (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))

Theoremismri2dd 16341* Definition of independence of a subset of the base set in a Moore system. One-way deduction form. (Contributed by David Moews, 1-May-2017.)
𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝐴 ∈ (Moore‘𝑋))    &   (𝜑𝑆𝑋)    &   (𝜑 → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))       (𝜑𝑆𝐼)

Theoremmriss 16342 An independent set of a Moore system is a subset of the base set. (Contributed by David Moews, 1-May-2017.)
𝐼 = (mrInd‘𝐴)       ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆𝐼) → 𝑆𝑋)

Theoremmrissd 16343 An independent set of a Moore system is a subset of the base set. Deduction form. (Contributed by David Moews, 1-May-2017.)
𝐼 = (mrInd‘𝐴)    &   (𝜑𝐴 ∈ (Moore‘𝑋))    &   (𝜑𝑆𝐼)       (𝜑𝑆𝑋)

Theoremismri2dad 16344 Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝐴 ∈ (Moore‘𝑋))    &   (𝜑𝑆𝐼)    &   (𝜑𝑌𝑆)       (𝜑 → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))

Theoremmrieqvd 16345* In a Moore system, a set is independent if and only if, for all elements of the set, the closure of the set with the element removed is unequal to the closure of the original set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝑆𝑋)       (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁𝑆)))

Theoremmrieqv2d 16346* In a Moore system, a set is independent if and only if all its proper subsets have closure properly contained in the closure of the set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝑆𝑋)       (𝜑 → (𝑆𝐼 ↔ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))))

Theoremmrissmrcd 16347 In a Moore system, if an independent set is between a set and its closure, the two sets are equal (since the two sets must have equal closures by mressmrcd 16334, and so are equal by mrieqv2d 16346.) (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝑆 ⊆ (𝑁𝑇))    &   (𝜑𝑇𝑆)    &   (𝜑𝑆𝐼)       (𝜑𝑆 = 𝑇)

Theoremmrissmrid 16348 In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝑆𝐼)    &   (𝜑𝑇𝑆)       (𝜑𝑇𝐼)

Theoremmreexd 16349* In a Moore system, the closure operator is said to have the exchange property if, for all elements 𝑦 and 𝑧 of the base set and subsets 𝑆 of the base set such that 𝑧 is in the closure of (𝑆 ∪ {𝑦}) but not in the closure of 𝑆, 𝑦 is in the closure of (𝑆 ∪ {𝑧}) (Definition 3.1.9 in [FaureFrolicher] p. 57 to 58.) This theorem allows us to construct substitution instances of this definition. (Contributed by David Moews, 1-May-2017.)
(𝜑𝑋𝑉)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝑆𝑋)    &   (𝜑𝑌𝑋)    &   (𝜑𝑍 ∈ (𝑁‘(𝑆 ∪ {𝑌})))    &   (𝜑 → ¬ 𝑍 ∈ (𝑁𝑆))       (𝜑𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍})))

Theoremmreexmrid 16350* In a Moore system whose closure operator has the exchange property, if a set is independent and an element is not in its closure, then adding the element to the set gives another independent set. Lemma 4.1.5 in [FaureFrolicher] p. 84. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝑆𝐼)    &   (𝜑𝑌𝑋)    &   (𝜑 → ¬ 𝑌 ∈ (𝑁𝑆))       (𝜑 → (𝑆 ∪ {𝑌}) ∈ 𝐼)

Theoremmreexexlemd 16351* This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 16355. (Contributed by David Moews, 1-May-2017.)
(𝜑𝑋𝐽)    &   (𝜑𝐹 ⊆ (𝑋𝐻))    &   (𝜑𝐺 ⊆ (𝑋𝐻))    &   (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))    &   (𝜑 → (𝐹𝐻) ∈ 𝐼)    &   (𝜑 → (𝐹𝐾𝐺𝐾))    &   (𝜑 → ∀𝑡𝑢 ∈ 𝒫 (𝑋𝑡)∀𝑣 ∈ 𝒫 (𝑋𝑡)(((𝑢𝐾𝑣𝐾) ∧ 𝑢 ⊆ (𝑁‘(𝑣𝑡)) ∧ (𝑢𝑡) ∈ 𝐼) → ∃𝑖 ∈ 𝒫 𝑣(𝑢𝑖 ∧ (𝑖𝑡) ∈ 𝐼)))       (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))

Theoremmreexexlem2d 16352* Used in mreexexlem4d 16354 to prove the induction step in mreexexd 16355. See the proof of Proposition 4.2.1 in [FaureFrolicher] p. 86 to 87. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝐹 ⊆ (𝑋𝐻))    &   (𝜑𝐺 ⊆ (𝑋𝐻))    &   (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))    &   (𝜑 → (𝐹𝐻) ∈ 𝐼)    &   (𝜑𝑌𝐹)       (𝜑 → ∃𝑔𝐺𝑔 ∈ (𝐹 ∖ {𝑌}) ∧ ((𝐹 ∖ {𝑌}) ∪ (𝐻 ∪ {𝑔})) ∈ 𝐼))

Theoremmreexexlem3d 16353* Base case of the induction in mreexexd 16355. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝐹 ⊆ (𝑋𝐻))    &   (𝜑𝐺 ⊆ (𝑋𝐻))    &   (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))    &   (𝜑 → (𝐹𝐻) ∈ 𝐼)    &   (𝜑 → (𝐹 = ∅ ∨ 𝐺 = ∅))       (𝜑 → ∃𝑖 ∈ 𝒫 𝐺(𝐹𝑖 ∧ (𝑖𝐻) ∈ 𝐼))

Theoremmreexexlem4d 16354* Induction step of the induction in mreexexd 16355. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝐹 ⊆ (𝑋𝐻))    &   (𝜑𝐺 ⊆ (𝑋𝐻))    &   (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))    &   (𝜑 → (𝐹𝐻) ∈ 𝐼)    &   (𝜑𝐿 ∈ ω)    &   (𝜑 → ∀𝑓 ∈ 𝒫 (𝑋)∀𝑔 ∈ 𝒫 (𝑋)(((𝑓𝐿𝑔𝐿) ∧ 𝑓 ⊆ (𝑁‘(𝑔)) ∧ (𝑓) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓𝑗 ∧ (𝑗) ∈ 𝐼)))    &   (𝜑 → (𝐹 ≈ suc 𝐿𝐺 ≈ suc 𝐿))       (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))

Theoremmreexexd 16355* Exchange-type theorem. In a Moore system whose closure operator has the exchange property, if 𝐹 and 𝐺 are disjoint from 𝐻, (𝐹𝐻) is independent, 𝐹 is contained in the closure of (𝐺𝐻), and either 𝐹 or 𝐺 is finite, then there is a subset 𝑞 of 𝐺 equinumerous to 𝐹 such that (𝑞𝐻) is independent. This implies the case of Proposition 4.2.1 in [FaureFrolicher] p. 86 where either (𝐴𝐵) or (𝐵𝐴) is finite. The theorem is proven by induction using mreexexlem3d 16353 for the base case and mreexexlem4d 16354 for the induction step. (Contributed by David Moews, 1-May-2017.) Removed dependencies on ax-rep 4804 and ax-ac2 9323. (Revised by Brendan Leahy, 2-Jun-2021.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝐹 ⊆ (𝑋𝐻))    &   (𝜑𝐺 ⊆ (𝑋𝐻))    &   (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))    &   (𝜑 → (𝐹𝐻) ∈ 𝐼)    &   (𝜑 → (𝐹 ∈ Fin ∨ 𝐺 ∈ Fin))       (𝜑 → ∃𝑞 ∈ 𝒫 𝐺(𝐹𝑞 ∧ (𝑞𝐻) ∈ 𝐼))

TheoremmreexexdOLD 16356* Obsolete proof of mreexexd 16355 as of 2-Jun-2021. Exchange-type theorem. In a Moore system whose closure operator has the exchange property, if 𝐹 and 𝐺 are disjoint from 𝐻, (𝐹𝐻) is independent, 𝐹 is contained in the closure of (𝐺𝐻), and either 𝐹 or 𝐺 is finite, then there is a subset 𝑞 of 𝐺 equinumerous to 𝐹 such that (𝑞𝐻) is independent. This implies the case of Proposition 4.2.1 in [FaureFrolicher] p. 86 where either (𝐴𝐵) or (𝐵𝐴) is finite. The theorem is proven by induction using mreexexlem3d 16353 for the base case and mreexexlem4d 16354 for the induction step. (Contributed by David Moews, 1-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝐹 ⊆ (𝑋𝐻))    &   (𝜑𝐺 ⊆ (𝑋𝐻))    &   (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))    &   (𝜑 → (𝐹𝐻) ∈ 𝐼)    &   (𝜑 → (𝐹 ∈ Fin ∨ 𝐺 ∈ Fin))       (𝜑 → ∃𝑞 ∈ 𝒫 𝐺(𝐹𝑞 ∧ (𝑞𝐻) ∈ 𝐼))

Theoremmreexdomd 16357* In a Moore system whose closure operator has the exchange property, if 𝑆 is independent and contained in the closure of 𝑇, and either 𝑆 or 𝑇 is finite, then 𝑇 dominates 𝑆. This is an immediate consequence of mreexexd 16355. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝑆 ⊆ (𝑁𝑇))    &   (𝜑𝑇𝑋)    &   (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin))    &   (𝜑𝑆𝐼)       (𝜑𝑆𝑇)

Theoremmreexfidimd 16358* In a Moore system whose closure operator has the exchange property, if two independent sets have equal closure and one is finite, then they are equinumerous. Proven by using mreexdomd 16357 twice. This implies a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝑆𝐼)    &   (𝜑𝑇𝐼)    &   (𝜑𝑆 ∈ Fin)    &   (𝜑 → (𝑁𝑆) = (𝑁𝑇))       (𝜑𝑆𝑇)

7.2.3  Algebraic closure systems

Theoremisacs 16359* A set is an algebraic closure system iff it is specified by some function of the finite subsets, such that a set is closed iff it does not expand under the operation. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))))

Theoremacsmre 16360 Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))

Theoremisacs2 16361* In the definition of an algebraic closure system, we may always take the operation being closed over as the Moore closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)))

Theoremacsfiel 16362* A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (ACS‘𝑋) → (𝑆𝐶 ↔ (𝑆𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆)))

Theoremacsfiel2 16363* A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 3-Apr-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆))

Theoremacsmred 16364 An algebraic closure system is also a Moore system. Deduction form of acsmre 16360. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (ACS‘𝑋))       (𝜑𝐴 ∈ (Moore‘𝑋))

Theoremisacs1i 16365* A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (ACS‘𝑋))

Theoremmreacs 16366 Algebraicity is a composable property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋𝑉 → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋))

Theoremacsfn 16367* Algebraicity of a conditional point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ (𝑇𝑎𝐾𝑎)} ∈ (ACS‘𝑋))

Theoremacsfn0 16368* Algebraicity of a point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((𝑋𝑉𝐾𝑋) → {𝑎 ∈ 𝒫 𝑋𝐾𝑎} ∈ (ACS‘𝑋))

Theoremacsfn1 16369* Algebraicity of a one-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((𝑋𝑉 ∧ ∀𝑏𝑋 𝐸𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏𝑎 𝐸𝑎} ∈ (ACS‘𝑋))

Theoremacsfn1c 16370* Algebraicity of a one-argument closure condition with additional constant. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((𝑋𝑉 ∧ ∀𝑏𝐾𝑐𝑋 𝐸𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏𝐾𝑐𝑎 𝐸𝑎} ∈ (ACS‘𝑋))

Theoremacsfn2 16371* Algebraicity of a two-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((𝑋𝑉 ∧ ∀𝑏𝑋𝑐𝑋 𝐸𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏𝑎𝑐𝑎 𝐸𝑎} ∈ (ACS‘𝑋))

PART 8  BASIC CATEGORY THEORY

8.1  Categories

8.1.1  Categories

Syntaxccat 16372 Extend class notation with the class of categories.
class Cat

Syntaxccid 16373 Extend class notation with the identity arrow of a category.
class Id

Syntaxchomf 16374 Extend class notation to include functionalized Hom-set extractor.
class Homf

Syntaxccomf 16375 Extend class notation to include functionalized composition operation.
class compf

Definitiondf-cat 16376* A category is an abstraction of a structure (a group, a topology, an order...) Category theory consists in finding new formulation of the concepts associated with those structures (product, substructure...) using morphisms instead of the belonging relation. That trick has the interesting property that heterogeneous structures like topologies or groups for instance become comparable. Definition in [Lang] p. 53. In contrast to definition 3.1 of [Adamek] p. 21, where "A category is a quadruple A = (O, hom, id, o)", a category is defined as an extensible structure consisting of three slots: the objects "O" ((Base‘𝑐)), the morphisms "hom" ((Hom ‘𝑐)) and the composition law "o" ((comp‘𝑐)). The identities "id" are defined by their properties related to morphisms and their composition, see condition 3.1(b) in [Adamek] p. 21 and df-cid 16377. (Note: in category theory morphisms are also called arrows.) (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 2-Jan-2017.)
Cat = {𝑐[(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ][(comp‘𝑐) / 𝑜]𝑥𝑏 (∃𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝑏𝑧𝑏𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)((𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓) ∈ (𝑥𝑧) ∧ ∀𝑤𝑏𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))))}

Definitiondf-cid 16377* Define the category identity arrow. Since it is uniquely defined when it exists, we do not need to add it to the data of the category, and instead extract it by uniqueness. (Contributed by Mario Carneiro, 3-Jan-2017.)
Id = (𝑐 ∈ Cat ↦ (Base‘𝑐) / 𝑏(Hom ‘𝑐) / (comp‘𝑐) / 𝑜(𝑥𝑏 ↦ (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓))))

Definitiondf-homf 16378* Define the functionalized Hom-set operator, which is exactly like Hom but is guaranteed to be a function on the base. (Contributed by Mario Carneiro, 4-Jan-2017.)
Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)))

Definitiondf-comf 16379* Define the functionalized composition operator, which is exactly like comp but is guaranteed to be a function of the proper type. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf = (𝑐 ∈ V ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓))))

Theoremiscat 16380* The predicate "is a category". (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)       (𝐶𝑉 → (𝐶 ∈ Cat ↔ ∀𝑥𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))))))

Theoremiscatd 16381* Properties that determine a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝐵 = (Base‘𝐶))    &   (𝜑𝐻 = (Hom ‘𝐶))    &   (𝜑· = (comp‘𝐶))    &   (𝜑𝐶𝑉)    &   ((𝜑𝑥𝐵) → 1 ∈ (𝑥𝐻𝑥))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))    &   ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))       (𝜑𝐶 ∈ Cat)

Theoremcatidex 16382* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → ∃𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))

Theoremcatideu 16383* Each object in a category has a unique identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓))

Theoremcidfval 16384* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &    1 = (Id‘𝐶)       (𝜑1 = (𝑥𝐵 ↦ (𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))))

Theoremcidval 16385* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &    1 = (Id‘𝐶)    &   (𝜑𝑋𝐵)       (𝜑 → ( 1𝑋) = (𝑔 ∈ (𝑋𝐻𝑋)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋· 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋· 𝑦)𝑔) = 𝑓)))

Theoremcidffn 16386 The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.)
Id Fn Cat

Theoremcidfn 16387 The identity arrow operator is a function from objects to arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)       (𝐶 ∈ Cat → 1 Fn 𝐵)

Theoremcatidd 16388* Deduce the identity arrow in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
(𝜑𝐵 = (Base‘𝐶))    &   (𝜑𝐻 = (Hom ‘𝐶))    &   (𝜑· = (comp‘𝐶))    &   (𝜑𝐶 ∈ Cat)    &   ((𝜑𝑥𝐵) → 1 ∈ (𝑥𝐻𝑥))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓)       (𝜑 → (Id‘𝐶) = (𝑥𝐵1 ))

Theoremiscatd2 16389* Version of iscatd 16381 with a uniform assumption list, for increased proof sharing capabilities. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐵 = (Base‘𝐶))    &   (𝜑𝐻 = (Hom ‘𝐶))    &   (𝜑· = (comp‘𝐶))    &   (𝜑𝐶𝑉)    &   (𝜓 ↔ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))    &   ((𝜑𝑦𝐵) → 1 ∈ (𝑦𝐻𝑦))    &   ((𝜑𝜓) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = 𝑓)    &   ((𝜑𝜓) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = 𝑔)    &   ((𝜑𝜓) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))    &   ((𝜑𝜓) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))       (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦𝐵1 )))

Theoremcatidcl 16390 Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    1 = (Id‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → ( 1𝑋) ∈ (𝑋𝐻𝑋))

Theoremcatlid 16391 Left identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    1 = (Id‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &    · = (comp‘𝐶)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))       (𝜑 → (( 1𝑌)(⟨𝑋, 𝑌· 𝑌)𝐹) = 𝐹)

Theoremcatrid 16392 Right identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    1 = (Id‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &    · = (comp‘𝐶)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))       (𝜑 → (𝐹(⟨𝑋, 𝑋· 𝑌)( 1𝑋)) = 𝐹)

Theoremcatcocl 16393 Closure of a composition arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐻𝑍))

Theoremcatass 16394 Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))    &   (𝜑𝑊𝐵)    &   (𝜑𝐾 ∈ (𝑍𝐻𝑊))       (𝜑 → ((𝐾(⟨𝑌, 𝑍· 𝑊)𝐺)(⟨𝑋, 𝑌· 𝑊)𝐹) = (𝐾(⟨𝑋, 𝑍· 𝑊)(𝐺(⟨𝑋, 𝑌· 𝑍)𝐹)))

Theorem0catg 16395 Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat)

Theorem0cat 16396 The empty set is a category, the empty category, see example 3.3(4.c) in [Adamek] p. 24. (Contributed by Mario Carneiro, 3-Jan-2017.)
∅ ∈ Cat

Theoremhomffval 16397* Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐹 = (Homf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)       𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))

Theoremfnhomeqhomf 16398 If the Hom-set operation is a function it is equal to the corresponding functionalized Hom-set operation. (Contributed by AV, 1-Mar-2020.)
𝐹 = (Homf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)       (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻)

Theoremhomfval 16399 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐹 = (Homf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌))

Theoremhomffn 16400 The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐹 = (Homf𝐶)    &   𝐵 = (Base‘𝐶)       𝐹 Fn (𝐵 × 𝐵)

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