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Type | Label | Description |
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Statement | ||
Theorem | mrissmrcd 16901 | 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 16888, and so are equal by mrieqv2d 16900.) (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) & ⊢ (𝜑 → 𝑇 ⊆ 𝑆) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) ⇒ ⊢ (𝜑 → 𝑆 = 𝑇) | ||
Theorem | mrissmrid 16902 | In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) & ⊢ (𝜑 → 𝑇 ⊆ 𝑆) ⇒ ⊢ (𝜑 → 𝑇 ∈ 𝐼) | ||
Theorem | mreexd 16903* | 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.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) & ⊢ (𝜑 → 𝑌 ∈ 𝑋) & ⊢ (𝜑 → 𝑍 ∈ (𝑁‘(𝑆 ∪ {𝑌}))) & ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘𝑆)) ⇒ ⊢ (𝜑 → 𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍}))) | ||
Theorem | mreexmrid 16904* | 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.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) & ⊢ (𝜑 → 𝑌 ∈ 𝑋) & ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘𝑆)) ⇒ ⊢ (𝜑 → (𝑆 ∪ {𝑌}) ∈ 𝐼) | ||
Theorem | mreexexlemd 16905* | This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 16909. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝑋 ∈ 𝐽) & ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻))) & ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼) & ⊢ (𝜑 → (𝐹 ≈ 𝐾 ∨ 𝐺 ≈ 𝐾)) & ⊢ (𝜑 → ∀𝑡∀𝑢 ∈ 𝒫 (𝑋 ∖ 𝑡)∀𝑣 ∈ 𝒫 (𝑋 ∖ 𝑡)(((𝑢 ≈ 𝐾 ∨ 𝑣 ≈ 𝐾) ∧ 𝑢 ⊆ (𝑁‘(𝑣 ∪ 𝑡)) ∧ (𝑢 ∪ 𝑡) ∈ 𝐼) → ∃𝑖 ∈ 𝒫 𝑣(𝑢 ≈ 𝑖 ∧ (𝑖 ∪ 𝑡) ∈ 𝐼))) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)) | ||
Theorem | mreexexlem2d 16906* | Used in mreexexlem4d 16908 to prove the induction step in mreexexd 16909. See the proof of Proposition 4.2.1 in [FaureFrolicher] p. 86 to 87. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻))) & ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼) & ⊢ (𝜑 → 𝑌 ∈ 𝐹) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 (¬ 𝑔 ∈ (𝐹 ∖ {𝑌}) ∧ ((𝐹 ∖ {𝑌}) ∪ (𝐻 ∪ {𝑔})) ∈ 𝐼)) | ||
Theorem | mreexexlem3d 16907* | Base case of the induction in mreexexd 16909. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻))) & ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼) & ⊢ (𝜑 → (𝐹 = ∅ ∨ 𝐺 = ∅)) ⇒ ⊢ (𝜑 → ∃𝑖 ∈ 𝒫 𝐺(𝐹 ≈ 𝑖 ∧ (𝑖 ∪ 𝐻) ∈ 𝐼)) | ||
Theorem | mreexexlem4d 16908* | Induction step of the induction in mreexexd 16909. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻))) & ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼) & ⊢ (𝜑 → 𝐿 ∈ ω) & ⊢ (𝜑 → ∀ℎ∀𝑓 ∈ 𝒫 (𝑋 ∖ ℎ)∀𝑔 ∈ 𝒫 (𝑋 ∖ ℎ)(((𝑓 ≈ 𝐿 ∨ 𝑔 ≈ 𝐿) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼))) & ⊢ (𝜑 → (𝐹 ≈ suc 𝐿 ∨ 𝐺 ≈ suc 𝐿)) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)) | ||
Theorem | mreexexd 16909* | Exchange-type theorem. In a Moore system whose closure operator has the exchange property, if 𝐹 and 𝐺 are disjoint from 𝐻, (𝐹 ∪ 𝐻) is independent, 𝐹 is contained in the closure of (𝐺 ∪ 𝐻), and either 𝐹 or 𝐺 is finite, then there is a subset 𝑞 of 𝐺 equinumerous to 𝐹 such that (𝑞 ∪ 𝐻) is independent. This implies the case of Proposition 4.2.1 in [FaureFrolicher] p. 86 where either (𝐴 ∖ 𝐵) or (𝐵 ∖ 𝐴) is finite. The theorem is proven by induction using mreexexlem3d 16907 for the base case and mreexexlem4d 16908 for the induction step. (Contributed by David Moews, 1-May-2017.) Remove dependencies on ax-rep 5182 and ax-ac2 9874. (Revised by Brendan Leahy, 2-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻)) & ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻))) & ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼) & ⊢ (𝜑 → (𝐹 ∈ Fin ∨ 𝐺 ∈ Fin)) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝒫 𝐺(𝐹 ≈ 𝑞 ∧ (𝑞 ∪ 𝐻) ∈ 𝐼)) | ||
Theorem | mreexdomd 16910* | 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 16909. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) & ⊢ (𝜑 → 𝑇 ⊆ 𝑋) & ⊢ (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin)) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) ⇒ ⊢ (𝜑 → 𝑆 ≼ 𝑇) | ||
Theorem | mreexfidimd 16911* | 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 16910 twice. This implies a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) & ⊢ (𝜑 → 𝑆 ∈ 𝐼) & ⊢ (𝜑 → 𝑇 ∈ 𝐼) & ⊢ (𝜑 → 𝑆 ∈ Fin) & ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇)) ⇒ ⊢ (𝜑 → 𝑆 ≈ 𝑇) | ||
Theorem | isacs 16912* | 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(𝑠 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)))) | ||
Theorem | acsmre 16913 | Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋)) | ||
Theorem | isacs2 16914* | 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(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠))) | ||
Theorem | acsfiel 16915* | 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) → (𝑆 ∈ 𝐶 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆))) | ||
Theorem | acsfiel2 16916* | 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) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆)) | ||
Theorem | acsmred 16917 | An algebraic closure system is also a Moore system. Deduction form of acsmre 16913. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋)) ⇒ ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) | ||
Theorem | isacs1i 16918* | A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (ACS‘𝑋)) | ||
Theorem | mreacs 16919 | Algebraicity is a composable property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ (𝑋 ∈ 𝑉 → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋)) | ||
Theorem | acsfn 16920* | Algebraicity of a conditional point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ (𝑇 ⊆ 𝑎 → 𝐾 ∈ 𝑎)} ∈ (ACS‘𝑋)) | ||
Theorem | acsfn0 16921* | Algebraicity of a point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ 𝐾 ∈ 𝑎} ∈ (ACS‘𝑋)) | ||
Theorem | acsfn1 16922* | Algebraicity of a one-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑋 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝑎 𝐸 ∈ 𝑎} ∈ (ACS‘𝑋)) | ||
Theorem | acsfn1c 16923* | Algebraicity of a one-argument closure condition with additional constant. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝐾 ∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝐾 ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎} ∈ (ACS‘𝑋)) | ||
Theorem | acsfn2 16924* | Algebraicity of a two-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎} ∈ (ACS‘𝑋)) | ||
Syntax | ccat 16925 | Extend class notation with the class of categories. |
class Cat | ||
Syntax | ccid 16926 | Extend class notation with the identity arrow of a category. |
class Id | ||
Syntax | chomf 16927 | Extend class notation to include functionalized Hom-set extractor. |
class Homf | ||
Syntax | ccomf 16928 | Extend class notation to include functionalized composition operation. |
class compf | ||
Definition | df-cat 16929* | 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 16930. (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) / ℎ][(comp‘𝑐) / 𝑜]∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓))))} | ||
Definition | df-cid 16930* | 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) / ℎ⦌⦋(comp‘𝑐) / 𝑜⦌(𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓)))) | ||
Definition | df-homf 16931* | 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))) | ||
Definition | df-comf 16932* | 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 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓)))) | ||
Theorem | iscat 16933* | The predicate "is a category". (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ Cat ↔ ∀𝑥 ∈ 𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓)))))) | ||
Theorem | iscatd 16934* | Properties that determine a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → · = (comp‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ (𝑥𝐻𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(〈𝑥, 𝑥〉 · 𝑦) 1 ) = 𝑓) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) & ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
Theorem | catidex 16935* | 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) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓)) | ||
Theorem | catideu 16936* | 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) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓)) | ||
Theorem | cidfval 16937* | 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) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 1 = (Id‘𝐶) ⇒ ⊢ (𝜑 → 1 = (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓)))) | ||
Theorem | cidval 16938* | 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) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) = (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓))) | ||
Theorem | cidffn 16939 | The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.) |
⊢ Id Fn Cat | ||
Theorem | cidfn 16940 | 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) | ||
Theorem | catidd 16941* | Deduce the identity arrow in a category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → · = (comp‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ (𝑥𝐻𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(〈𝑥, 𝑥〉 · 𝑦) 1 ) = 𝑓) ⇒ ⊢ (𝜑 → (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ 1 )) | ||
Theorem | iscatd2 16942* | Version of iscatd 16934 with a uniform assumption list, for increased proof sharing capabilities. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → · = (comp‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜓 ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 1 ∈ (𝑦𝐻𝑦)) & ⊢ ((𝜑 ∧ 𝜓) → ( 1 (〈𝑥, 𝑦〉 · 𝑦)𝑓) = 𝑓) & ⊢ ((𝜑 ∧ 𝜓) → (𝑔(〈𝑦, 𝑦〉 · 𝑧) 1 ) = 𝑔) & ⊢ ((𝜑 ∧ 𝜓) → (𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) & ⊢ ((𝜑 ∧ 𝜓) → ((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))) ⇒ ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ 1 ))) | ||
Theorem | catidcl 16943 | 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) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋)) | ||
Theorem | catlid 16944 | Left identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝐹) = 𝐹) | ||
Theorem | catrid 16945 | Right identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → (𝐹(〈𝑋, 𝑋〉 · 𝑌)( 1 ‘𝑋)) = 𝐹) | ||
Theorem | catcocl 16946 | Closure of a composition arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ (𝑋𝐻𝑍)) | ||
Theorem | catass 16947 | Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) & ⊢ (𝜑 → 𝑊 ∈ 𝐵) & ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑊)) ⇒ ⊢ (𝜑 → ((𝐾(〈𝑌, 𝑍〉 · 𝑊)𝐺)(〈𝑋, 𝑌〉 · 𝑊)𝐹) = (𝐾(〈𝑋, 𝑍〉 · 𝑊)(𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹))) | ||
Theorem | 0catg 16948 | Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat) | ||
Theorem | 0cat 16949 | 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 | ||
Theorem | homffval 16950* | Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.) |
⊢ 𝐹 = (Homf ‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) | ||
Theorem | fnhomeqhomf 16951 | 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 = (Homf ‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻) | ||
Theorem | homfval 16952 | Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝐹 = (Homf ‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌)) | ||
Theorem | homffn 16953 | The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝐹 = (Homf ‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ 𝐹 Fn (𝐵 × 𝐵) | ||
Theorem | homfeq 16954* | 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) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) ⇒ ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦))) | ||
Theorem | homfeqd 16955 | 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 → (Homf ‘𝐶) = (Homf ‘𝐷)) | ||
Theorem | homfeqbas 16956 | Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) ⇒ ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) | ||
Theorem | homfeqval 16957 | Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐽𝑌)) | ||
Theorem | comfffval 16958* | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.) |
⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) ⇒ ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) | ||
Theorem | comffval 16959* | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓))) | ||
Theorem | comfval 16960 | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) | ||
Theorem | comfffval2 16961* | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Homf ‘𝐶) & ⊢ · = (comp‘𝐶) ⇒ ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) | ||
Theorem | comffval2 16962* | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Homf ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓))) | ||
Theorem | comfval2 16963 | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Homf ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) | ||
Theorem | comfffn 16964 | The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ 𝑂 Fn ((𝐵 × 𝐵) × 𝐵) | ||
Theorem | comffn 16965 | The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌))) | ||
Theorem | comfeq 16966* | 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) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) ⇒ ⊢ (𝜑 → ((compf‘𝐶) = (compf‘𝐷) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉 ∙ 𝑧)𝑓))) | ||
Theorem | comfeqd 16967 | 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 → (Homf ‘𝐶) = (Homf ‘𝐷)) ⇒ ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) | ||
Theorem | comfeqval 16968 | Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ ∙ = (comp‘𝐷) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 ∙ 𝑍)𝐹)) | ||
Theorem | catpropd 16969 | 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 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat)) | ||
Theorem | cidpropd 16970 | Two structures with the same base, hom-sets and composition operation have the same identity function. (Contributed by Mario Carneiro, 17-Jan-2017.) |
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) ⇒ ⊢ (𝜑 → (Id‘𝐶) = (Id‘𝐷)) | ||
Syntax | coppc 16971 | The opposite category operation. |
class oppCat | ||
Definition | df-oppc 16972* | 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, (2nd ‘𝑢)〉(comp‘𝑓)(1st ‘𝑢)))〉)) | ||
Theorem | oppcval 16973* | Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (𝐶 ∈ 𝑉 → 𝑂 = ((𝐶 sSet 〈(Hom ‘ndx), tpos 𝐻〉) sSet 〈(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (〈𝑧, (2nd ‘𝑢)〉 · (1st ‘𝑢)))〉)) | ||
Theorem | oppchomfval 16974 | Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ tpos 𝐻 = (Hom ‘𝑂) | ||
Theorem | oppchom 16975 | Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (𝑋(Hom ‘𝑂)𝑌) = (𝑌𝐻𝑋) | ||
Theorem | oppccofval 16976 | Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (〈𝑋, 𝑌〉(comp‘𝑂)𝑍) = tpos (〈𝑍, 𝑌〉 · 𝑋)) | ||
Theorem | oppcco 16977 | Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉(comp‘𝑂)𝑍)𝐹) = (𝐹(〈𝑍, 𝑌〉 · 𝑋)𝐺)) | ||
Theorem | oppcbas 16978 | Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ 𝐵 = (Base‘𝑂) | ||
Theorem | oppccatid 16979 | Lemma for oppccat 16982. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (𝐶 ∈ Cat → (𝑂 ∈ Cat ∧ (Id‘𝑂) = (Id‘𝐶))) | ||
Theorem | oppchomf 16980 | Hom-sets of the opposite category. (Contributed by Mario Carneiro, 17-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐻 = (Homf ‘𝐶) ⇒ ⊢ tpos 𝐻 = (Homf ‘𝑂) | ||
Theorem | oppcid 16981 | Identity function of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐵 = (Id‘𝐶) ⇒ ⊢ (𝐶 ∈ Cat → (Id‘𝑂) = 𝐵) | ||
Theorem | oppccat 16982 | An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) | ||
Theorem | 2oppcbas 16983 | The double opposite category has the same objects as the original category. Intended for use with property lemmas such as monpropd 16997. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ 𝐵 = (Base‘(oppCat‘𝑂)) | ||
Theorem | 2oppchomf 16984 | The double opposite category has the same morphisms as the original category. Intended for use with property lemmas such as monpropd 16997. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)) | ||
Theorem | 2oppccomf 16985 | The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 16997. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂)) | ||
Theorem | oppchomfpropd 16986 | If two categories have the same hom-sets, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.) |
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) ⇒ ⊢ (𝜑 → (Homf ‘(oppCat‘𝐶)) = (Homf ‘(oppCat‘𝐷))) | ||
Theorem | oppccomfpropd 16987 | If two categories have the same hom-sets and composition, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.) |
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → (compf‘(oppCat‘𝐶)) = (compf‘(oppCat‘𝐷))) | ||
Syntax | cmon 16988 | Extend class notation with the class of all monomorphisms. |
class Mono | ||
Syntax | cepi 16989 | Extend class notation with the class of all epimorphisms. |
class Epi | ||
Definition | df-mon 16990* | Function returning the monomorphisms of the category 𝑐. JFM CAT1 def. 10. (Contributed by FL, 5-Dec-2007.) (Revised by Mario Carneiro, 2-Jan-2017.) |
⊢ Mono = (𝑐 ∈ Cat ↦ ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom ‘𝑐) / ℎ⦌(𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (𝑥ℎ𝑦) ∣ ∀𝑧 ∈ 𝑏 Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(〈𝑧, 𝑥〉(comp‘𝑐)𝑦)𝑔))})) | ||
Definition | df-epi 16991 | Function returning the epimorphisms of the category 𝑐. JFM CAT1 def. 11. (Contributed by FL, 8-Aug-2008.) (Revised by Mario Carneiro, 2-Jan-2017.) |
⊢ Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐))) | ||
Theorem | monfval 16992* | Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑀 = (Mono‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) ⇒ ⊢ (𝜑 → 𝑀 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(〈𝑧, 𝑥〉 · 𝑦)𝑔))})) | ||
Theorem | ismon 16993* | Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑀 = (Mono‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(〈𝑧, 𝑋〉 · 𝑌)𝑔))))) | ||
Theorem | ismon2 16994* | Write out the monomorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑀 = (Mono‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑧𝐻𝑋)∀ℎ ∈ (𝑧𝐻𝑋)((𝐹(〈𝑧, 𝑋〉 · 𝑌)𝑔) = (𝐹(〈𝑧, 𝑋〉 · 𝑌)ℎ) → 𝑔 = ℎ)))) | ||
Theorem | monhom 16995 | A monomorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑀 = (Mono‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝑀𝑌) ⊆ (𝑋𝐻𝑌)) | ||
Theorem | moni 16996 | Property of a monomorphism. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑀 = (Mono‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝑀𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑍𝐻𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑋)) ⇒ ⊢ (𝜑 → ((𝐹(〈𝑍, 𝑋〉 · 𝑌)𝐺) = (𝐹(〈𝑍, 𝑋〉 · 𝑌)𝐾) ↔ 𝐺 = 𝐾)) | ||
Theorem | monpropd 16997 | If two categories have the same set of objects, morphisms, and compositions, then they have the same monomorphisms. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) ⇒ ⊢ (𝜑 → (Mono‘𝐶) = (Mono‘𝐷)) | ||
Theorem | oppcmon 16998 | A monomorphism in the opposite category is an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 𝑀 = (Mono‘𝑂) & ⊢ 𝐸 = (Epi‘𝐶) ⇒ ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋)) | ||
Theorem | oppcepi 16999 | An epimorphism in the opposite category is a monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 𝐸 = (Epi‘𝑂) & ⊢ 𝑀 = (Mono‘𝐶) ⇒ ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑌𝑀𝑋)) | ||
Theorem | isepi 17000* | Definition of an epimorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝐸 = (Epi‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑌𝐻𝑧) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑧)𝐹))))) |
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