P. 178 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 17701-17800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mrieqv2d 17701*	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.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑠(𝑠 ⊊ 𝑆 → (𝑁‘𝑠) ⊊ (𝑁‘𝑆))))
Theorem	mrissmrcd 17702	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 17689, and so are equal by mrieqv2d 17701.) (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 17703	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 17704*	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 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 17705*	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 17706*	This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 17710. (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 17707*	Used in mreexexlem4d 17709 to prove the induction step in mreexexd 17710. 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 17708*	Base case of the induction in mreexexd 17710. (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 17709*	Induction step of the induction in mreexexd 17710. (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 17710*	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 17708 for the base case and mreexexlem4d 17709 for the induction step. (Contributed by David Moews, 1-May-2017.) Remove dependencies on ax-rep 5237 and ax-ac2 10453. (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 17711*	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 17710. (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 17712*	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 17711 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)
7.2.3  Algebraic closure systems
Theorem	isacs 17713*	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 17714	Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
Theorem	isacs2 17715*	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 17716*	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 17717*	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 17718	An algebraic closure system is also a Moore system. Deduction form of acsmre 17714. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    ⇒   ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
Theorem	isacs1i 17719*	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 17720	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 17721*	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 17722*	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 17723*	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 17724*	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 17725*	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))
PART 8  BASIC CATEGORY THEORY
8.1  Categories
8.1.1  Categories
Syntax	ccat 17726	Extend class notation with the class of categories.
class Cat
Syntax	ccid 17727	Extend class notation with the identity arrow of a category.
class Id
Syntax	chomf 17728	Extend class notation to include functionalized Hom-set extractor.
class Homf
Syntax	ccomf 17729	Extend class notation to include functionalized composition operation.
class compf
Definition	df-cat 17730*	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, without the axiom CAT 1, i.e., pairwise disjointness of hom-sets (cat1 18160). See setc2obas 18157 and setc2ohom 18158 for a counterexample. 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 17731. (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 17731*	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 17732*	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 17733*	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 17734*	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 17735*	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 17736*	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 17737*	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 17738*	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 17739*	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 17740	The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.)
⊢ Id Fn Cat
Theorem	cidfn 17741	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 17742*	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 17743*	Version of iscatd 17735 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 17744	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 17745	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 17746	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 17747	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 17748	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	catcone0 17749	Composition of non-empty hom-sets is non-empty. (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑋𝐻𝑌) ≠ ∅)    &   ⊢ (𝜑 → (𝑌𝐻𝑍) ≠ ∅)    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑍) ≠ ∅)
Theorem	0catg 17750	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 17751	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 17752*	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 17753	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 17754	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 17755	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 17756*	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 17757	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 17758	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 17759	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 17760*	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 17761*	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 17762	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 17763*	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 17764*	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 17765	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 17766	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 17767	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 17768*	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 17769	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 17770	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 17771	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 17772	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))
8.1.2  Opposite category
Syntax	coppc 17773	The opposite category operation.
class oppCat
Definition	df-oppc 17774*	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 17775*	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 17776	Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)
⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ tpos 𝐻 = (Hom ‘𝑂)
Theorem	oppchom 17777	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 17778	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 17779	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 17780	Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 18-Oct-2024.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ 𝐵 = (Base‘𝑂)
Theorem	oppccatid 17781	Lemma for oppccat 17784. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ (𝐶 ∈ Cat → (𝑂 ∈ Cat ∧ (Id‘𝑂) = (Id‘𝐶)))
Theorem	oppchomf 17782	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 17783	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 17784	An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
Theorem	2oppcbas 17785	The double opposite category has the same objects as the original category. Intended for use with property lemmas such as monpropd 17800. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ 𝐵 = (Base‘(oppCat‘𝑂))
Theorem	2oppchomf 17786	The double opposite category has the same morphisms as the original category. Intended for use with property lemmas such as monpropd 17800. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))
Theorem	2oppccomf 17787	The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 17800. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂))
Theorem	oppchomfpropd 17788	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 17789	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)))
Theorem	oppccatf 17790	oppCat restricted to Cat is a function from Cat to Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
⊢ (oppCat ↾ Cat):Cat⟶Cat
8.1.3  Monomorphisms and epimorphisms
Syntax	cmon 17791	Extend class notation with the class of all monomorphisms.
class Mono
Syntax	cepi 17792	Extend class notation with the class of all epimorphisms.
class Epi
Definition	df-mon 17793*	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 17794	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 17795*	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 17796*	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 17797*	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 17798	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 17799	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 17800	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))