HomeTheorem List (p. 498 of 503)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | catbas 49701 | The base of the category structure. (Contributed by Zhi Wang, 5-Nov-2025.) |
| ⊢ 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} & ⊢ 𝐵 ∈ V ⇒ ⊢ 𝐵 = (Base‘𝐶) | ||
| Theorem | cathomfval 49702 | The hom-sets of the category structure. (Contributed by Zhi Wang, 5-Nov-2025.) |
| ⊢ 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} & ⊢ 𝐻 ∈ V ⇒ ⊢ 𝐻 = (Hom ‘𝐶) | ||
| Theorem | catcofval 49703 | Composition of the category structure. (Contributed by Zhi Wang, 5-Nov-2025.) |
| ⊢ 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} & ⊢ · ∈ V ⇒ ⊢ · = (comp‘𝐶) | ||
| Theorem | natoppf 49704 | A natural transformation is natural between opposite functors. (Contributed by Zhi Wang, 18-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) & ⊢ 𝑁 = (𝐶 Nat 𝐷) & ⊢ 𝑀 = (𝑂 Nat 𝑃) & ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩)) ⇒ ⊢ (𝜑 → 𝐴 ∈ (⟨𝐾, tpos 𝐿⟩𝑀⟨𝐹, tpos 𝐺⟩)) | ||
| Theorem | natoppf2 49705 | A natural transformation is natural between opposite functors. (Contributed by Zhi Wang, 18-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) & ⊢ 𝑁 = (𝐶 Nat 𝐷) & ⊢ 𝑀 = (𝑂 Nat 𝑃) & ⊢ (𝜑 → 𝐾 = ( oppFunc ‘𝐹)) & ⊢ (𝜑 → 𝐿 = ( oppFunc ‘𝐺)) & ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺)) ⇒ ⊢ (𝜑 → 𝐴 ∈ (𝐿𝑀𝐾)) | ||
| Theorem | natoppfb 49706 | A natural transformation is natural between opposite functors, and vice versa. (Contributed by Zhi Wang, 18-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) & ⊢ 𝑁 = (𝐶 Nat 𝐷) & ⊢ 𝑀 = (𝑂 Nat 𝑃) & ⊢ (𝜑 → 𝐾 = ( oppFunc ‘𝐹)) & ⊢ (𝜑 → 𝐿 = ( oppFunc ‘𝐺)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐹𝑁𝐺) = (𝐿𝑀𝐾)) | ||
| Theorem | initoo2 49707 | An initial object is an object in the base set. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ 𝐵) | ||
| Theorem | termoo2 49708 | A terminal object is an object in the base set. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ 𝐵) | ||
| Theorem | zeroo2 49709 | A zero object is an object in the base set. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝑂 ∈ (ZeroO‘𝐶) → 𝑂 ∈ 𝐵) | ||
| Theorem | oppcinito 49710 | Initial objects are terminal in the opposite category. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ (TermO‘(oppCat‘𝐶))) | ||
| Theorem | oppctermo 49711 | Terminal objects are initial in the opposite category. Comments before Definition 7.4 in [Adamek] p. 102. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝐼 ∈ (TermO‘𝐶) ↔ 𝐼 ∈ (InitO‘(oppCat‘𝐶))) | ||
| Theorem | oppczeroo 49712 | Zero objects are zero in the opposite category. Remark 7.8 of [Adamek] p. 103. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝐼 ∈ (ZeroO‘𝐶) ↔ 𝐼 ∈ (ZeroO‘(oppCat‘𝐶))) | ||
| Theorem | termoeu2 49713 | Terminal objects are essentially unique; if 𝐴 is a terminal object, then so is every object that is isomorphic to 𝐴. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐴 ∈ (TermO‘𝐶)) & ⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵) ⇒ ⊢ (𝜑 → 𝐵 ∈ (TermO‘𝐶)) | ||
| Theorem | initopropdlemlem 49714 | Lemma for initopropdlem 49715, termopropdlem 49716, and zeroopropdlem 49717. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ 𝐹 Fn 𝑋 & ⊢ (𝜑 → ¬ 𝐴 ∈ 𝑌) & ⊢ 𝑋 ⊆ 𝑌 & ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐵) = ∅) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵)) | ||
| Theorem | initopropdlem 49715 | Lemma for initopropd 49718. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → ¬ 𝐶 ∈ V) ⇒ ⊢ (𝜑 → (InitO‘𝐶) = (InitO‘𝐷)) | ||
| Theorem | termopropdlem 49716 | Lemma for termopropd 49719. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → ¬ 𝐶 ∈ V) ⇒ ⊢ (𝜑 → (TermO‘𝐶) = (TermO‘𝐷)) | ||
| Theorem | zeroopropdlem 49717 | Lemma for zeroopropd 49720. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → ¬ 𝐶 ∈ V) ⇒ ⊢ (𝜑 → (ZeroO‘𝐶) = (ZeroO‘𝐷)) | ||
| Theorem | initopropd 49718 | Two structures with the same base, hom-sets and composition operation have the same initial objects. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → (InitO‘𝐶) = (InitO‘𝐷)) | ||
| Theorem | termopropd 49719 | Two structures with the same base, hom-sets and composition operation have the same terminal objects. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → (TermO‘𝐶) = (TermO‘𝐷)) | ||
| Theorem | zeroopropd 49720 | Two structures with the same base, hom-sets and composition operation have the same zero objects. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → (ZeroO‘𝐶) = (ZeroO‘𝐷)) | ||
| Theorem | reldmxpc 49721 | The binary product of categories is a proper operator, so it can be used with ovprc1 7406, elbasov 17186, strov2rcl 17187, and so on. See reldmxpcALT 49722 for an alternate proof with less "essential steps" but more "bytes". (Proposed by SN, 15-Oct-2025.) (Contributed by Zhi Wang, 15-Oct-2025.) |
| ⊢ Rel dom ×c | ||
| Theorem | reldmxpcALT 49722 | Alternate proof of reldmxpc 49721. (Contributed by Zhi Wang, 15-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ Rel dom ×c | ||
| Theorem | elxpcbasex1 49723 | A non-empty base set of the product category indicates the existence of the first factor of the product category. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof shortened by SN, 15-Oct-2025.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐶 ∈ V) | ||
| Theorem | elxpcbasex1ALT 49724 | Alternate proof of elxpcbasex1 49723. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐶 ∈ V) | ||
| Theorem | elxpcbasex2 49725 | A non-empty base set of the product category indicates the existence of the second factor of the product category. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof shortened by SN, 15-Oct-2025.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐷 ∈ V) | ||
| Theorem | elxpcbasex2ALT 49726 | Alternate proof of elxpcbasex2 49725. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐷 ∈ V) | ||
| Theorem | xpcfucbas 49727 | The base set of the product of two categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.) |
| ⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸)) ⇒ ⊢ ((𝐵 Func 𝐶) × (𝐷 Func 𝐸)) = (Base‘𝑇) | ||
| Theorem | xpcfuchomfval 49728* | Set of morphisms of the binary product of categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.) |
| ⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸)) & ⊢ 𝐴 = (Base‘𝑇) & ⊢ 𝐾 = (Hom ‘𝑇) ⇒ ⊢ 𝐾 = (𝑢 ∈ 𝐴, 𝑣 ∈ 𝐴 ↦ (((1st ‘𝑢)(𝐵 Nat 𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(𝐷 Nat 𝐸)(2nd ‘𝑣)))) | ||
| Theorem | xpcfuchom 49729 | Set of morphisms of the binary product of categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.) |
| ⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸)) & ⊢ 𝐴 = (Base‘𝑇) & ⊢ 𝐾 = (Hom ‘𝑇) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑋𝐾𝑌) = (((1st ‘𝑋)(𝐵 Nat 𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(𝐷 Nat 𝐸)(2nd ‘𝑌)))) | ||
| Theorem | xpcfuchom2 49730 | Value of the set of morphisms in the binary product of categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.) |
| ⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸)) & ⊢ (𝜑 → 𝑀 ∈ (𝐵 Func 𝐶)) & ⊢ (𝜑 → 𝑁 ∈ (𝐷 Func 𝐸)) & ⊢ (𝜑 → 𝑃 ∈ (𝐵 Func 𝐶)) & ⊢ (𝜑 → 𝑄 ∈ (𝐷 Func 𝐸)) & ⊢ 𝐾 = (Hom ‘𝑇) ⇒ ⊢ (𝜑 → (⟨𝑀, 𝑁⟩𝐾⟨𝑃, 𝑄⟩) = ((𝑀(𝐵 Nat 𝐶)𝑃) × (𝑁(𝐷 Nat 𝐸)𝑄))) | ||
| Theorem | xpcfucco2 49731 | Value of composition in the binary product of categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.) |
| ⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸)) & ⊢ 𝑂 = (comp‘𝑇) & ⊢ (𝜑 → 𝐹 ∈ (𝑀(𝐵 Nat 𝐶)𝑃)) & ⊢ (𝜑 → 𝐺 ∈ (𝑁(𝐷 Nat 𝐸)𝑄)) & ⊢ (𝜑 → 𝐾 ∈ (𝑃(𝐵 Nat 𝐶)𝑅)) & ⊢ (𝜑 → 𝐿 ∈ (𝑄(𝐷 Nat 𝐸)𝑆)) ⇒ ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝐾(⟨𝑀, 𝑃⟩(comp‘(𝐵 FuncCat 𝐶))𝑅)𝐹), (𝐿(⟨𝑁, 𝑄⟩(comp‘(𝐷 FuncCat 𝐸))𝑆)𝐺)⟩) | ||
| Theorem | xpcfuccocl 49732 | The composition of two natural transformations is a natural transformation. (Contributed by Zhi Wang, 1-Oct-2025.) |
| ⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸)) & ⊢ 𝑂 = (comp‘𝑇) & ⊢ (𝜑 → 𝐹 ∈ (𝑀(𝐵 Nat 𝐶)𝑃)) & ⊢ (𝜑 → 𝐺 ∈ (𝑁(𝐷 Nat 𝐸)𝑄)) & ⊢ (𝜑 → 𝐾 ∈ (𝑃(𝐵 Nat 𝐶)𝑅)) & ⊢ (𝜑 → 𝐿 ∈ (𝑄(𝐷 Nat 𝐸)𝑆)) ⇒ ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) ∈ ((𝑀(𝐵 Nat 𝐶)𝑅) × (𝑁(𝐷 Nat 𝐸)𝑆))) | ||
| Theorem | xpcfucco3 49733* | Value of composition in the binary product of categories of functors; expressed explicitly. (Contributed by Zhi Wang, 1-Oct-2025.) |
| ⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸)) & ⊢ 𝑂 = (comp‘𝑇) & ⊢ (𝜑 → 𝐹 ∈ (𝑀(𝐵 Nat 𝐶)𝑃)) & ⊢ (𝜑 → 𝐺 ∈ (𝑁(𝐷 Nat 𝐸)𝑄)) & ⊢ (𝜑 → 𝐾 ∈ (𝑃(𝐵 Nat 𝐶)𝑅)) & ⊢ (𝜑 → 𝐿 ∈ (𝑄(𝐷 Nat 𝐸)𝑆)) & ⊢ 𝑋 = (Base‘𝐵) & ⊢ 𝑌 = (Base‘𝐷) & ⊢ · = (comp‘𝐶) & ⊢ ∙ = (comp‘𝐸) ⇒ ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝑥 ∈ 𝑋 ↦ ((𝐾‘𝑥)(⟨((1st ‘𝑀)‘𝑥), ((1st ‘𝑃)‘𝑥)⟩ · ((1st ‘𝑅)‘𝑥))(𝐹‘𝑥))), (𝑦 ∈ 𝑌 ↦ ((𝐿‘𝑦)(⟨((1st ‘𝑁)‘𝑦), ((1st ‘𝑄)‘𝑦)⟩ ∙ ((1st ‘𝑆)‘𝑦))(𝐺‘𝑦)))⟩) | ||
| Syntax | cswapf 49734 | Extend class notation with the class of swap functors. |
| class swapF | ||
| Definition | df-swapf 49735* |
Define the swap functor from (𝐶 ×c 𝐷) to (𝐷
×c 𝐶) by
swapping all objects (swapf1 49747) and morphisms (swapf2 49749) .
Such functor is called a "swap functor" in https://arxiv.org/pdf/2302.07810 49749 or a "twist functor" in https://arxiv.org/pdf/2508.01886 49749, the latter of which finds its counterpart as "twisting map" in https://arxiv.org/pdf/2411.04102 49749 for tensor product of algebras. The "swap functor" or "twisting map" is often denoted as a small tau 𝜏 in literature. However, the term "twist functor" is defined differently in https://arxiv.org/pdf/1208.4046 49749 and thus not adopted here. tpos I depends on more mathbox theorems, and thus are not adopted here. See dfswapf2 49736 for an alternate definition. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ swapF = (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩) | ||
| Theorem | dfswapf2 49736* | Alternate definition of swapF (df-swapf 49735). (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ swapF = (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩) | ||
| Theorem | swapfval 49737* | Value of the swap functor. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆)) ⇒ ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩) | ||
| Theorem | swapfelvv 49738 | A swap functor is an ordered pair. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (V × V)) | ||
| Theorem | swapf2fvala 49739* | The morphism part of the swap functor. See also swapf2fval 49740. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆)) ⇒ ⊢ (𝜑 → (2nd ‘(𝐶 swapF 𝐷)) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))) | ||
| Theorem | swapf2fval 49740* | The morphism part of the swap functor. See also swapf2fvala 49739. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆)) & ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) ⇒ ⊢ (𝜑 → 𝑃 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))) | ||
| Theorem | swapf1vala 49741* | The object part of the swap functor. See also swapf1val 49742. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ (𝜑 → (1st ‘(𝐶 swapF 𝐷)) = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥})) | ||
| Theorem | swapf1val 49742* | The object part of the swap functor. See also swapf1vala 49741. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) ⇒ ⊢ (𝜑 → 𝑂 = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥})) | ||
| Theorem | swapf2fn 49743 | The morphism part of the swap functor is a function on the Cartesian square of the base set. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) ⇒ ⊢ (𝜑 → 𝑃 Fn (𝐵 × 𝐵)) | ||
| Theorem | swapf1a 49744 | The object part of the swap functor swaps the objects. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑂‘𝑋) = ⟨(2nd ‘𝑋), (1st ‘𝑋)⟩) | ||
| Theorem | swapf2vala 49745* | The morphism part of the swap functor swaps the morphisms. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆)) ⇒ ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓})) | ||
| Theorem | swapf2a 49746 | The morphism part of the swap functor swaps the morphisms. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆)) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐹) = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩) | ||
| Theorem | swapf1 49747 | The object part of the swap functor swaps the objects. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷)) ⇒ ⊢ (𝜑 → (𝑋𝑂𝑌) = ⟨𝑌, 𝑋⟩) | ||
| Theorem | swapf2val 49748* | The morphism part of the swap functor swaps the morphisms. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷)) & ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐷)) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆)) ⇒ ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑃⟨𝑍, 𝑊⟩) = (𝑓 ∈ (⟨𝑋, 𝑌⟩𝐻⟨𝑍, 𝑊⟩) ↦ ∪ ◡{𝑓})) | ||
| Theorem | swapf2 49749 | The morphism part of the swap functor swaps the morphisms. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷)) & ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐷)) & ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑍)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐷)𝑊)) ⇒ ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑌⟩𝑃⟨𝑍, 𝑊⟩)𝐺) = ⟨𝐺, 𝐹⟩) | ||
| Theorem | swapf1f1o 49750 | The object part of the swap functor is a bijection between base sets. (Contributed by Zhi Wang, 8-Oct-2025.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐴 = (Base‘𝑇) ⇒ ⊢ (𝜑 → 𝑂:𝐵–1-1-onto→𝐴) | ||
| Theorem | swapf2f1o 49751 | The morphism part of the swap functor is a bijection between hom-sets. (Contributed by Zhi Wang, 8-Oct-2025.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ 𝐻 = (Hom ‘𝑆) & ⊢ 𝐽 = (Hom ‘𝑇) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷)) & ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐷)) ⇒ ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑃⟨𝑍, 𝑊⟩):(⟨𝑋, 𝑌⟩𝐻⟨𝑍, 𝑊⟩)–1-1-onto→(⟨𝑌, 𝑋⟩𝐽⟨𝑊, 𝑍⟩)) | ||
| Theorem | swapf2f1oa 49752 | The morphism part of the swap functor is a bijection between hom-sets. (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ 𝐻 = (Hom ‘𝑆) & ⊢ 𝐽 = (Hom ‘𝑇) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝑃𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝑂‘𝑋)𝐽(𝑂‘𝑌))) | ||
| Theorem | swapf2f1oaALT 49753 | Alternate proof of swapf2f1oa 49752. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ 𝐻 = (Hom ‘𝑆) & ⊢ 𝐽 = (Hom ‘𝑇) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝑃𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝑂‘𝑋)𝐽(𝑂‘𝑌))) | ||
| Theorem | swapfid 49754 | Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also swapfida 49755. (Contributed by Zhi Wang, 8-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷)) & ⊢ 1 = (Id‘𝑆) & ⊢ 𝐼 = (Id‘𝑇) ⇒ ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩𝑃⟨𝑋, 𝑌⟩)‘( 1 ‘⟨𝑋, 𝑌⟩)) = (𝐼‘(𝑂‘⟨𝑋, 𝑌⟩))) | ||
| Theorem | swapfida 49755 | Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also swapfid 49754. (Contributed by Zhi Wang, 8-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 1 = (Id‘𝑆) & ⊢ 𝐼 = (Id‘𝑇) ⇒ ⊢ (𝜑 → ((𝑋𝑃𝑋)‘( 1 ‘𝑋)) = (𝐼‘(𝑂‘𝑋))) | ||
| Theorem | swapfcoa 49756 | Composition in the source category is mapped to composition in the target. (𝜑 → 𝐶 ∈ Cat) and (𝜑 → 𝐷 ∈ Cat) can be replaced by a weaker hypothesis (𝜑 → 𝑆 ∈ Cat). (Contributed by Zhi Wang, 8-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ 𝐻 = (Hom ‘𝑆) & ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐻𝑍)) & ⊢ · = (comp‘𝑆) & ⊢ ∙ = (comp‘𝑇) ⇒ ⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝑁(⟨𝑋, 𝑌⟩ · 𝑍)𝑀)) = (((𝑌𝑃𝑍)‘𝑁)(⟨(𝑂‘𝑋), (𝑂‘𝑌)⟩ ∙ (𝑂‘𝑍))((𝑋𝑃𝑌)‘𝑀))) | ||
| Theorem | swapffunc 49757 | The swap functor is a functor. (Contributed by Zhi Wang, 8-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) ⇒ ⊢ (𝜑 → 𝑂(𝑆 Func 𝑇)𝑃) | ||
| Theorem | swapfffth 49758 | The swap functor is a fully faithful functor. (Contributed by Zhi Wang, 8-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) ⇒ ⊢ (𝜑 → 𝑂((𝑆 Full 𝑇) ∩ (𝑆 Faith 𝑇))𝑃) | ||
| Theorem | swapffunca 49759 | The swap functor is a functor. (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) ⇒ ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (𝑆 Func 𝑇)) | ||
| Theorem | swapfiso 49760 | The swap functor is an isomorphism between product categories. (Contributed by Zhi Wang, 8-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ 𝐸 = (CatCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑈) & ⊢ (𝜑 → 𝑇 ∈ 𝑈) & ⊢ 𝐼 = (Iso‘𝐸) ⇒ ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (𝑆𝐼𝑇)) | ||
| Theorem | swapciso 49761 | The product category is categorically isomorphic to the swapped product category. (Contributed by Zhi Wang, 8-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ 𝐸 = (CatCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑈) & ⊢ (𝜑 → 𝑇 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝑆( ≃𝑐 ‘𝐸)𝑇) | ||
| Theorem | oppc1stflem 49762* | A utility theorem for proving theorems on projection functors of opposite categories. (Contributed by Zhi Wang, 19-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) & ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → ( oppFunc ‘(𝐶𝐹𝐷)) = (𝑂𝐹𝑃)) & ⊢ 𝐹 = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ 𝑌) ⇒ ⊢ (𝜑 → ( oppFunc ‘(𝐶𝐹𝐷)) = (𝑂𝐹𝑃)) | ||
| Theorem | oppc1stf 49763 | The opposite functor of the first projection functor is the first projection functor of opposite categories. (Contributed by Zhi Wang, 19-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) ⇒ ⊢ (𝜑 → ( oppFunc ‘(𝐶 1stF 𝐷)) = (𝑂 1stF 𝑃)) | ||
| Theorem | oppc2ndf 49764 | The opposite functor of the second projection functor is the second projection functor of opposite categories. (Contributed by Zhi Wang, 19-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) ⇒ ⊢ (𝜑 → ( oppFunc ‘(𝐶 2ndF 𝐷)) = (𝑂 2ndF 𝑃)) | ||
| Theorem | 1stfpropd 49765 | If two categories have the same set of objects, morphisms, and compositions, then they have same first projection functors. (Contributed by Zhi Wang, 20-Nov-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵)) & ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵)) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝐴 ∈ Cat) & ⊢ (𝜑 → 𝐵 ∈ Cat) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) ⇒ ⊢ (𝜑 → (𝐴 1stF 𝐶) = (𝐵 1stF 𝐷)) | ||
| Theorem | 2ndfpropd 49766 | If two categories have the same set of objects, morphisms, and compositions, then they have same second projection functors. (Contributed by Zhi Wang, 20-Nov-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵)) & ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵)) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝐴 ∈ Cat) & ⊢ (𝜑 → 𝐵 ∈ Cat) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) ⇒ ⊢ (𝜑 → (𝐴 2ndF 𝐶) = (𝐵 2ndF 𝐷)) | ||
| Theorem | diagpropd 49767 | If two categories have the same set of objects, morphisms, and compositions, then they have same diagonal functors. (Contributed by Zhi Wang, 20-Nov-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵)) & ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵)) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝐴 ∈ Cat) & ⊢ (𝜑 → 𝐵 ∈ Cat) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) ⇒ ⊢ (𝜑 → (𝐴Δfunc𝐶) = (𝐵Δfunc𝐷)) | ||
| Theorem | cofuswapfcl 49768 | The bifunctor pre-composed with a swap functor is a bifunctor. (Contributed by Zhi Wang, 10-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ (𝜑 → 𝐺 = (𝐹 ∘func (𝐶 swapF 𝐷))) ⇒ ⊢ (𝜑 → 𝐺 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) | ||
| Theorem | cofuswapf1 49769 | The object part of a bifunctor pre-composed with a swap functor. (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ (𝜑 → 𝐺 = (𝐹 ∘func (𝐶 swapF 𝐷))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋(1st ‘𝐺)𝑌) = (𝑌(1st ‘𝐹)𝑋)) | ||
| Theorem | cofuswapf2 49770 | The morphism part of a bifunctor pre-composed with a swap functor. (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ (𝜑 → 𝐺 = (𝐹 ∘func (𝐶 swapF 𝐷))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐴) & ⊢ (𝜑 → 𝑊 ∈ 𝐵) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑍)) & ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐽𝑊)) ⇒ ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘𝐺)⟨𝑍, 𝑊⟩)𝑁) = (𝑁(⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)𝑀)) | ||
| Theorem | tposcurf1cl 49771 | The partially evaluated transposed curry functor is a functor. (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 = ((1st ‘𝐺)‘𝑋)) ⇒ ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸)) | ||
| Theorem | tposcurf11 49772 | Value of the double evaluated transposed curry functor. (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 = ((1st ‘𝐺)‘𝑋)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = (𝑌(1st ‘𝐹)𝑋)) | ||
| Theorem | tposcurf12 49773 | The partially evaluated transposed curry functor at a morphism. (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 = ((1st ‘𝐺)‘𝑋)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐻 ∈ (𝑌𝐽𝑍)) ⇒ ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐻) = (𝐻(⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑍, 𝑋⟩)( 1 ‘𝑋))) | ||
| Theorem | tposcurf1 49774* | Value of the object part of the transposed curry functor. (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 = ((1st ‘𝐺)‘𝑋)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ 1 = (Id‘𝐶) ⇒ ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑦(1st ‘𝐹)𝑋)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋))))⟩) | ||
| Theorem | tposcurf2 49775* | Value of the transposed curry functor at a morphism. (Contributed by Zhi Wang, 10-Oct-2025.) |
| ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐼 = (Id‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)) ⇒ ⊢ (𝜑 → 𝐿 = (𝑧 ∈ 𝐵 ↦ ((𝐼‘𝑧)(⟨𝑧, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑌⟩)𝐾))) | ||
| Theorem | tposcurf2val 49776 | Value of a component of the transposed curry functor natural transformation. (Contributed by Zhi Wang, 10-Oct-2025.) |
| ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐼 = (Id‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐿‘𝑍) = ((𝐼‘𝑍)(⟨𝑍, 𝑋⟩(2nd ‘𝐹)⟨𝑍, 𝑌⟩)𝐾)) | ||
| Theorem | tposcurf2cl 49777 | The transposed curry functor at a morphism is a natural transformation. (Contributed by Zhi Wang, 10-Oct-2025.) |
| ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐼 = (Id‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)) & ⊢ 𝑁 = (𝐷 Nat 𝐸) ⇒ ⊢ (𝜑 → 𝐿 ∈ (((1st ‘𝐺)‘𝑋)𝑁((1st ‘𝐺)‘𝑌))) | ||
| Theorem | tposcurfcl 49778 | The transposed curry functor of a functor 𝐹:𝐷 × 𝐶⟶𝐸 is a functor tposcurry (𝐹):𝐶⟶(𝐷⟶𝐸). (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))) & ⊢ 𝑄 = (𝐷 FuncCat 𝐸) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) ⇒ ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝑄)) | ||
| Theorem | diag1 49779* | The constant functor of 𝑋. Example 3.20(2) of [Adamek] p. 30. (Contributed by Zhi Wang, 17-Oct-2025.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ 1 = (Id‘𝐶) ⇒ ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ 𝑋), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ (𝑦𝐽𝑧) ↦ ( 1 ‘𝑋)))⟩) | ||
| Theorem | diag1a 49780* | The constant functor of 𝑋. (Contributed by Zhi Wang, 19-Oct-2025.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ 1 = (Id‘𝐶) ⇒ ⊢ (𝜑 → 𝐾 = ⟨(𝐵 × {𝑋}), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ ((𝑦𝐽𝑧) × {( 1 ‘𝑋)}))⟩) | ||
| Theorem | diag1f1lem 49781 | The object part of the diagonal functor is 1-1 if 𝐵 is non-empty. Note that (𝜑 → (𝑀 = 𝑁 ↔ 𝑋 = 𝑌)) also holds because of diag1f1 49782 and f1fveq 7217. (Contributed by Zhi Wang, 19-Oct-2025.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ 𝑀 = ((1st ‘𝐿)‘𝑋) & ⊢ 𝑁 = ((1st ‘𝐿)‘𝑌) ⇒ ⊢ (𝜑 → (𝑀 = 𝑁 → 𝑋 = 𝑌)) | ||
| Theorem | diag1f1 49782 | The object part of the diagonal functor is 1-1 if 𝐵 is non-empty. (Contributed by Zhi Wang, 19-Oct-2025.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝐵 ≠ ∅) ⇒ ⊢ (𝜑 → (1st ‘𝐿):𝐴–1-1→(𝐷 Func 𝐶)) | ||
| Theorem | diag2f1lem 49783 | Lemma for diag2f1 49784. The converse is trivial (fveq2 6840). (Contributed by Zhi Wang, 21-Oct-2025.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → (((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = ((𝑋(2nd ‘𝐿)𝑌)‘𝐺) → 𝐹 = 𝐺)) | ||
| Theorem | diag2f1 49784 | If 𝐵 is non-empty, the morphism part of a diagonal functor is injective functions from hom-sets into sets of natural transformations. (Contributed by Zhi Wang, 21-Oct-2025.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ 𝑁 = (𝐷 Nat 𝐶) ⇒ ⊢ (𝜑 → (𝑋(2nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–1-1→(((1st ‘𝐿)‘𝑋)𝑁((1st ‘𝐿)‘𝑌))) | ||
| Theorem | fucofulem1 49785 | Lemma for proving functor theorems. (Contributed by Zhi Wang, 25-Sep-2025.) |
| ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏))) & ⊢ ((𝜑 ∧ (𝜃 ∧ 𝜏)) → 𝜂) & ⊢ 𝜒 & ⊢ ((𝜑 ∧ 𝜂) → 𝜃) & ⊢ ((𝜑 ∧ 𝜂) → 𝜏) ⇒ ⊢ (𝜑 → (𝜓 ↔ 𝜂)) | ||
| Theorem | fucofulem2 49786* | Lemma for proving functor theorems. Maybe consider eufnfv 7184 to prove the uniqueness of a functor. (Contributed by Zhi Wang, 25-Sep-2025.) |
| ⊢ 𝐵 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) & ⊢ 𝐻 = (Hom ‘((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))) ⇒ ⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))(𝐶 Nat 𝐸)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑢𝐺𝑣)) ∧ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝑚𝐺𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))))) | ||
| Theorem | fuco2el 49787 | Equivalence of product functor. (Contributed by Zhi Wang, 29-Sep-2025.) |
| ⊢ (⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩ ∈ (𝑆 × 𝑅) ↔ (𝐾𝑆𝐿 ∧ 𝐹𝑅𝐺)) | ||
| Theorem | fuco2eld 49788 | Equivalence of product functor. (Contributed by Zhi Wang, 29-Sep-2025.) |
| ⊢ (𝜑 → 𝑊 = (𝑆 × 𝑅)) & ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩) & ⊢ (𝜑 → 𝐾𝑆𝐿) & ⊢ (𝜑 → 𝐹𝑅𝐺) ⇒ ⊢ (𝜑 → 𝑈 ∈ 𝑊) | ||
| Theorem | fuco2eld2 49789 | Equivalence of product functor. (Contributed by Zhi Wang, 29-Sep-2025.) |
| ⊢ (𝜑 → 𝑊 = (𝑆 × 𝑅)) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ Rel 𝑆 & ⊢ Rel 𝑅 ⇒ ⊢ (𝜑 → 𝑈 = ⟨⟨(1st ‘(1st ‘𝑈)), (2nd ‘(1st ‘𝑈))⟩, ⟨(1st ‘(2nd ‘𝑈)), (2nd ‘(2nd ‘𝑈))⟩⟩) | ||
| Theorem | fuco2eld3 49790 | Equivalence of product functor. (Contributed by Zhi Wang, 29-Sep-2025.) |
| ⊢ (𝜑 → 𝑊 = (𝑆 × 𝑅)) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ Rel 𝑆 & ⊢ Rel 𝑅 ⇒ ⊢ (𝜑 → ((1st ‘(1st ‘𝑈))𝑆(2nd ‘(1st ‘𝑈)) ∧ (1st ‘(2nd ‘𝑈))𝑅(2nd ‘(2nd ‘𝑈)))) | ||
| Syntax | cfuco 49791 | Extend class notation with functor composition bifunctors. |
| class ∘F | ||
| Definition | df-fuco 49792* | Definition of functor composition bifunctors. Given three categories 𝐶, 𝐷, and 𝐸, (⟨𝐶, 𝐷⟩ ∘F 𝐸) is a functor from the product category of two categories of functors to a category of functors (fucofunc 49834). The object part maps two functors to their composition (fuco11 49801 and fuco11b 49812). The morphism part defines the "composition" of two natural transformations (fuco22 49814) into another natural transformation (fuco22nat 49821) such that a "cube-like" diagram commutes. The naturality property also gives an alternate definition (fuco23a 49827). Note that such "composition" is different from fucco 17932 because they "compose" along different "axes". (Contributed by Zhi Wang, 29-Sep-2025.) |
| ⊢ ∘F = (𝑝 ∈ V, 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌⦋((𝑑 Func 𝑒) × (𝑐 Func 𝑑)) / 𝑤⦌⟨( ∘func ↾ 𝑤), (𝑢 ∈ 𝑤, 𝑣 ∈ 𝑤 ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝑑 Nat 𝑒)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝑐 Nat 𝑑)(2nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩) | ||
| Theorem | fucofvalg 49793* | Value of the function giving the functor composition bifunctor. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → 𝑃 ∈ 𝑈) & ⊢ (𝜑 → (1st ‘𝑃) = 𝐶) & ⊢ (𝜑 → (2nd ‘𝑃) = 𝐷) & ⊢ (𝜑 → 𝐸 ∈ 𝑉) & ⊢ (𝜑 → (𝑃 ∘F 𝐸) = ⚬ ) & ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ⇒ ⊢ (𝜑 → ⚬ = ⟨( ∘func ↾ 𝑊), (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝐷 Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝐶 Nat 𝐷)(2nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩) | ||
| Theorem | fucofval 49794* | Value of the function giving the functor composition bifunctor. Hypotheses fucofval.c and fucofval.d are not redundant (fucofvalne 49800). (Contributed by Zhi Wang, 29-Sep-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑇) & ⊢ (𝜑 → 𝐷 ∈ 𝑈) & ⊢ (𝜑 → 𝐸 ∈ 𝑉) & ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⚬ ) & ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ⇒ ⊢ (𝜑 → ⚬ = ⟨( ∘func ↾ 𝑊), (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝐷 Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝐶 Nat 𝐷)(2nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩) | ||
| Theorem | fucoelvv 49795 | A functor composition bifunctor is an ordered pair. Enables 1st2ndb 7982. (Contributed by Zhi Wang, 29-Sep-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑇) & ⊢ (𝜑 → 𝐷 ∈ 𝑈) & ⊢ (𝜑 → 𝐸 ∈ 𝑉) & ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⚬ ) ⇒ ⊢ (𝜑 → ⚬ ∈ (V × V)) | ||
| Theorem | fuco1 49796 | The object part of the functor composition bifunctor. (Contributed by Zhi Wang, 29-Sep-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑇) & ⊢ (𝜑 → 𝐷 ∈ 𝑈) & ⊢ (𝜑 → 𝐸 ∈ 𝑉) & ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩) & ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ⇒ ⊢ (𝜑 → 𝑂 = ( ∘func ↾ 𝑊)) | ||
| Theorem | fucof1 49797 | The object part of the functor composition bifunctor maps ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) into (𝐶 Func 𝐸). (Contributed by Zhi Wang, 29-Sep-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑇) & ⊢ (𝜑 → 𝐷 ∈ 𝑈) & ⊢ (𝜑 → 𝐸 ∈ 𝑉) & ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩) & ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ⇒ ⊢ (𝜑 → 𝑂:𝑊⟶(𝐶 Func 𝐸)) | ||
| Theorem | fuco2 49798* | The morphism part of the functor composition bifunctor. (Contributed by Zhi Wang, 29-Sep-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑇) & ⊢ (𝜑 → 𝐷 ∈ 𝑈) & ⊢ (𝜑 → 𝐸 ∈ 𝑉) & ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩) & ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ⇒ ⊢ (𝜑 → 𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝐷 Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝐶 Nat 𝐷)(2nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))) | ||
| Theorem | fucofn2 49799 | The morphism part of the functor composition bifunctor is a function on the Cartesian square of the base set. (Contributed by Zhi Wang, 29-Sep-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑇) & ⊢ (𝜑 → 𝐷 ∈ 𝑈) & ⊢ (𝜑 → 𝐸 ∈ 𝑉) & ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩) & ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ⇒ ⊢ (𝜑 → 𝑃 Fn (𝑊 × 𝑊)) | ||
| Theorem | fucofvalne 49800* | Value of the function giving the functor composition bifunctor, if 𝐶 or 𝐷 are not sets. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → ¬ (𝐶 ∈ V ∧ 𝐷 ∈ V)) & ⊢ (𝜑 → 𝐸 ∈ Cat) & ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⚬ ) & ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ⇒ ⊢ (𝜑 → ⚬ ≠ ⟨( ∘func ↾ 𝑊), (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝐷 Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝐶 Nat 𝐷)(2nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩) | ||
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