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Theorem List for Metamath Proof Explorer - 16001-16100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmrcssvd 16001 The Moore closure of a set is a subset of the base. Deduction form of mrcssv 15992. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)       (𝜑 → (𝑁𝐵) ⊆ 𝑋)
 
Theoremmrcssd 16002 Moore closure preserves subset ordering. Deduction form of mrcss 15994. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑈𝑉)    &   (𝜑𝑉𝑋)       (𝜑 → (𝑁𝑈) ⊆ (𝑁𝑉))
 
Theoremmrcssidd 16003 A set is contained in its Moore closure. Deduction form of mrcssid 15995. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑈𝑋)       (𝜑𝑈 ⊆ (𝑁𝑈))
 
Theoremmrcidmd 16004 Moore closure is idempotent. Deduction form of mrcidm 15997. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑈𝑋)       (𝜑 → (𝑁‘(𝑁𝑈)) = (𝑁𝑈))
 
Theoremmressmrcd 16005 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 16006 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 16007 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 16016 and mrieqv2d 16017. 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 16008* 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 16009* 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 16010* 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 16011* 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 16012* 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 16013 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 16014 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 16015 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 16016* 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 16017* 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 16018 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 16005, and so are equal by mrieqv2d 16017.) (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝑆 ⊆ (𝑁𝑇))    &   (𝜑𝑇𝑆)    &   (𝜑𝑆𝐼)       (𝜑𝑆 = 𝑇)
 
Theoremmrissmrid 16019 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 16020* 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 16021* 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 16022* This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 16026. (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 16023* Used in mreexexlem4d 16025 to prove the induction step in mreexexd 16026. 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 16024* Base case of the induction in mreexexd 16026. (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 16025* Induction step of the induction in mreexexd 16026. (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 16026* 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 16024 for the base case and mreexexlem4d 16025 for the induction step. (Contributed by David Moews, 1-May-2017.) Removed dependencies on ax-rep 4597 and ax-ac2 9048. (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 16027* 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 16024 for the base case and mreexexlem4d 16025 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 16028* 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 16026. (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 16029* 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 16028 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 16030* 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 16031 Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
 
Theoremisacs2 16032* 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 16033* 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 16034* 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 16035 An algebraic closure system is also a Moore system. Deduction form of acsmre 16031. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (ACS‘𝑋))       (𝜑𝐴 ∈ (Moore‘𝑋))
 
Theoremisacs1i 16036* 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 16037 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 16038* 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 16039* Algebraicity of a point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((𝑋𝑉𝐾𝑋) → {𝑎 ∈ 𝒫 𝑋𝐾𝑎} ∈ (ACS‘𝑋))
 
Theoremacsfn1 16040* 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 16041* 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 16042* 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 16043 Extend class notation with the class of categories.
class Cat
 
Syntaxccid 16044 Extend class notation with the identity arrow of a category.
class Id
 
Syntaxchomf 16045 Extend class notation to include functionalized Hom-set extractor.
class Homf
 
Syntaxccomf 16046 Extend class notation to include functionalized composition operation.
class compf
 
Definitiondf-cat 16047* 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 16048. (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 16048* 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 16049* 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 16050* 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 16051* 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 16052* 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 16053* 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 16054* 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 16055* 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 16056* 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 16057 The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.)
Id Fn Cat
 
Theoremcidfn 16058 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 16059* 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 16060* Version of iscatd 16052 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 16061 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 16062 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 16063 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 16064 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 16065 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 16066 Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat)
 
Theorem0cat 16067 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 16068* 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 16069 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 16070 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 16071 The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐹 = (Homf𝐶)    &   𝐵 = (Base‘𝐶)       𝐹 Fn (𝐵 × 𝐵)
 
Theoremhomfeq 16072* Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   (𝜑𝐵 = (Base‘𝐶))    &   (𝜑𝐵 = (Base‘𝐷))       (𝜑 → ((Homf𝐶) = (Homf𝐷) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦)))
 
Theoremhomfeqd 16073 If two structures have the same Hom slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑 → (Base‘𝐶) = (Base‘𝐷))    &   (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷))       (𝜑 → (Homf𝐶) = (Homf𝐷))
 
Theoremhomfeqbas 16074 Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑 → (Homf𝐶) = (Homf𝐷))       (𝜑 → (Base‘𝐶) = (Base‘𝐷))
 
Theoremhomfeqval 16075 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   (𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐽𝑌))
 
Theoremcomfffval 16076* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)       𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
 
Theoremcomffval 16077* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)))
 
Theoremcomfval 16078 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
 
Theoremcomfffval2 16079* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Homf𝐶)    &    · = (comp‘𝐶)       𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
 
Theoremcomffval2 16080* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Homf𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)))
 
Theoremcomfval2 16081 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Homf𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
 
Theoremcomfffn 16082 The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)       𝑂 Fn ((𝐵 × 𝐵) × 𝐵)
 
Theoremcomffn 16083 The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)))
 
Theoremcomfeq 16084* Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
· = (comp‘𝐶)    &    = (comp‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐵 = (Base‘𝐶))    &   (𝜑𝐵 = (Base‘𝐷))    &   (𝜑 → (Homf𝐶) = (Homf𝐷))       (𝜑 → ((compf𝐶) = (compf𝐷) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓)))
 
Theoremcomfeqd 16085 Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑 → (comp‘𝐶) = (comp‘𝐷))    &   (𝜑 → (Homf𝐶) = (Homf𝐷))       (𝜑 → (compf𝐶) = (compf𝐷))
 
Theoremcomfeqval 16086 Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &    = (comp‘𝐷)    &   (𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌 𝑍)𝐹))
 
Theoremcatpropd 16087 Two structures with the same base, hom-sets and composition operation are either both categories or neither. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)       (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
 
Theoremcidpropd 16088 Two structures with the same base, hom-sets and composition operation have the same identity function. (Contributed by Mario Carneiro, 17-Jan-2017.)
(𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)       (𝜑 → (Id‘𝐶) = (Id‘𝐷))
 
8.1.2  Opposite category
 
Syntaxcoppc 16089 The opposite category operation.
class oppCat
 
Definitiondf-oppc 16090* Define an opposite category, which is the same as the original category but with the direction of arrows the other way around. Definition 3.5 of [Adamek] p. 25. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat = (𝑓 ∈ V ↦ ((𝑓 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑓)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝑓) × (Base‘𝑓)), 𝑧 ∈ (Base‘𝑓) ↦ tpos (⟨𝑧, (2nd𝑢)⟩(comp‘𝑓)(1st𝑢)))⟩))
 
Theoremoppcval 16091* Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   𝑂 = (oppCat‘𝐶)       (𝐶𝑉𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))⟩))
 
Theoremoppchomfval 16092 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐻 = (Hom ‘𝐶)    &   𝑂 = (oppCat‘𝐶)       tpos 𝐻 = (Hom ‘𝑂)
 
Theoremoppchom 16093 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐻 = (Hom ‘𝐶)    &   𝑂 = (oppCat‘𝐶)       (𝑋(Hom ‘𝑂)𝑌) = (𝑌𝐻𝑋)
 
Theoremoppccofval 16094 Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &    · = (comp‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝑂)𝑍) = tpos (⟨𝑍, 𝑌· 𝑋))
 
Theoremoppcco 16095 Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &    · = (comp‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝑂)𝑍)𝐹) = (𝐹(⟨𝑍, 𝑌· 𝑋)𝐺))
 
Theoremoppcbas 16096 Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝐵 = (Base‘𝐶)       𝐵 = (Base‘𝑂)
 
Theoremoppccatid 16097 Lemma for oppccat 16100. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝑂 = (oppCat‘𝐶)       (𝐶 ∈ Cat → (𝑂 ∈ Cat ∧ (Id‘𝑂) = (Id‘𝐶)))
 
Theoremoppchomf 16098 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝐻 = (Homf𝐶)       tpos 𝐻 = (Homf𝑂)
 
Theoremoppcid 16099 Identity function of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝐵 = (Id‘𝐶)       (𝐶 ∈ Cat → (Id‘𝑂) = 𝐵)
 
Theoremoppccat 16100 An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝑂 = (oppCat‘𝐶)       (𝐶 ∈ Cat → 𝑂 ∈ Cat)
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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