HomeTheorem List (p. 500 of 509)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | posjidm 49901 | Poset join is idempotent. latjidm 18553 could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) ⇒ ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 𝑋) = 𝑋) | ||
| Theorem | posmidm 49902 | Poset meet is idempotent. latmidm 18565 could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 𝑋) = 𝑋) | ||
| Theorem | resiposbas 49903 | Construct a poset (resipos 49904) for any base set. (Contributed by Zhi Wang, 20-Oct-2025.) |
| ⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩} ⇒ ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐾)) | ||
| Theorem | resipos 49904 | A set equipped with an order where no distinct elements are comparable is a poset. (Contributed by Zhi Wang, 20-Oct-2025.) |
| ⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩} ⇒ ⊢ (𝐵 ∈ 𝑉 → 𝐾 ∈ Poset) | ||
| Theorem | exbaspos 49905* | There exists a poset for any base set. (Contributed by Zhi Wang, 20-Oct-2025.) |
| ⊢ (𝐵 ∈ 𝑉 → ∃𝑘 ∈ Poset 𝐵 = (Base‘𝑘)) | ||
| Theorem | exbasprs 49906* | There exists a preordered set for any base set. (Contributed by Zhi Wang, 20-Oct-2025.) |
| ⊢ (𝐵 ∈ 𝑉 → ∃𝑘 ∈ Proset 𝐵 = (Base‘𝑘)) | ||
| Theorem | basresposfo 49907 | The base function restricted to the class of posets maps the class of posets onto the universal class. (Contributed by Zhi Wang, 20-Oct-2025.) |
| ⊢ (Base ↾ Poset):Poset–onto→V | ||
| Theorem | basresprsfo 49908 | The base function restricted to the class of preordered sets maps the class of preordered sets onto the universal class. (Contributed by Zhi Wang, 20-Oct-2025.) |
| ⊢ (Base ↾ Proset ): Proset –onto→V | ||
| Theorem | posnex 49909 | The class of posets is a proper class. (Contributed by Zhi Wang, 20-Oct-2025.) |
| ⊢ Poset ∉ V | ||
| Theorem | prsnex 49910 | The class of preordered sets is a proper class. (Contributed by Zhi Wang, 20-Oct-2025.) |
| ⊢ Proset ∉ V | ||
| Theorem | toslat 49911 | A toset is a lattice. (Contributed by Zhi Wang, 26-Sep-2024.) |
| ⊢ (𝐾 ∈ Toset → 𝐾 ∈ Lat) | ||
| Theorem | isclatd 49912* | The predicate "is a complete lattice" (deduction form). (Contributed by Zhi Wang, 29-Sep-2024.) |
| ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝑈 = (lub‘𝐾)) & ⊢ (𝜑 → 𝐺 = (glb‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Poset) & ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝑈) & ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝐺) ⇒ ⊢ (𝜑 → 𝐾 ∈ CLat) | ||
| Theorem | intubeu 49913* | Existential uniqueness of the least upper bound. (Contributed by Zhi Wang, 28-Sep-2024.) |
| ⊢ (𝐶 ∈ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) ↔ 𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥})) | ||
| Theorem | unilbeu 49914* | Existential uniqueness of the greatest lower bound. (Contributed by Zhi Wang, 29-Sep-2024.) |
| ⊢ (𝐶 ∈ 𝐵 → ((𝐶 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) ↔ 𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})) | ||
| Theorem | ipolublem 49915* | Lemma for ipolubdm 49916 and ipolub 49917. (Contributed by Zhi Wang, 28-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐹) & ⊢ ≤ = (le‘𝐼) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐹) → ((∪ 𝑆 ⊆ 𝑋 ∧ ∀𝑧 ∈ 𝐹 (∪ 𝑆 ⊆ 𝑧 → 𝑋 ⊆ 𝑧)) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑋 ∧ ∀𝑧 ∈ 𝐹 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑋 ≤ 𝑧)))) | ||
| Theorem | ipolubdm 49916* | The domain of the LUB of the inclusion poset. (Contributed by Zhi Wang, 28-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐹) & ⊢ (𝜑 → 𝑈 = (lub‘𝐼)) & ⊢ (𝜑 → 𝑇 = ∩ {𝑥 ∈ 𝐹 ∣ ∪ 𝑆 ⊆ 𝑥}) ⇒ ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ 𝑇 ∈ 𝐹)) | ||
| Theorem | ipolub 49917* | The LUB of the inclusion poset. (hypotheses "ipolub.s" and "ipolub.t" could be eliminated with 𝑆 ∈ dom 𝑈.) Could be significantly shortened if poslubdg 18503 is in quantified form. mrelatlub 18653 could potentially be shortened using this. See mrelatlubALT 49924. (Contributed by Zhi Wang, 28-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐹) & ⊢ (𝜑 → 𝑈 = (lub‘𝐼)) & ⊢ (𝜑 → 𝑇 = ∩ {𝑥 ∈ 𝐹 ∣ ∪ 𝑆 ⊆ 𝑥}) & ⊢ (𝜑 → 𝑇 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝑈‘𝑆) = 𝑇) | ||
| Theorem | ipoglblem 49918* | Lemma for ipoglbdm 49919 and ipoglb 49920. (Contributed by Zhi Wang, 29-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐹) & ⊢ ≤ = (le‘𝐼) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐹) → ((𝑋 ⊆ ∩ 𝑆 ∧ ∀𝑧 ∈ 𝐹 (𝑧 ⊆ ∩ 𝑆 → 𝑧 ⊆ 𝑋)) ↔ (∀𝑦 ∈ 𝑆 𝑋 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐹 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑋)))) | ||
| Theorem | ipoglbdm 49919* | The domain of the GLB of the inclusion poset. (Contributed by Zhi Wang, 29-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐹) & ⊢ (𝜑 → 𝐺 = (glb‘𝐼)) & ⊢ (𝜑 → 𝑇 = ∪ {𝑥 ∈ 𝐹 ∣ 𝑥 ⊆ ∩ 𝑆}) ⇒ ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ 𝑇 ∈ 𝐹)) | ||
| Theorem | ipoglb 49920* | The GLB of the inclusion poset. (hypotheses "ipolub.s" and "ipoglb.t" could be eliminated with 𝑆 ∈ dom 𝐺.) Could be significantly shortened if posglbdg 18504 is in quantified form. mrelatglb 18651 could potentially be shortened using this. See mrelatglbALT 49925. (Contributed by Zhi Wang, 29-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐹) & ⊢ (𝜑 → 𝐺 = (glb‘𝐼)) & ⊢ (𝜑 → 𝑇 = ∪ {𝑥 ∈ 𝐹 ∣ 𝑥 ⊆ ∩ 𝑆}) & ⊢ (𝜑 → 𝑇 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐺‘𝑆) = 𝑇) | ||
| Theorem | ipolub0 49921 | The LUB of the empty set is the intersection of the base. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝑈 = (lub‘𝐼)) & ⊢ (𝜑 → ∩ 𝐹 ∈ 𝐹) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑈‘∅) = ∩ 𝐹) | ||
| Theorem | ipolub00 49922 | The LUB of the empty set is the empty set if it is contained. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝑈 = (lub‘𝐼)) & ⊢ (𝜑 → ∅ ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝑈‘∅) = ∅) | ||
| Theorem | ipoglb0 49923 | The GLB of the empty set is the union of the base. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝐺 = (glb‘𝐼)) & ⊢ (𝜑 → ∪ 𝐹 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐺‘∅) = ∪ 𝐹) | ||
| Theorem | mrelatlubALT 49924 | Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Zhi Wang, 29-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐼 = (toInc‘𝐶) & ⊢ 𝐹 = (mrCls‘𝐶) & ⊢ 𝐿 = (lub‘𝐼) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈)) | ||
| Theorem | mrelatglbALT 49925 | Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Zhi Wang, 29-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐼 = (toInc‘𝐶) & ⊢ 𝐺 = (glb‘𝐼) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → (𝐺‘𝑈) = ∩ 𝑈) | ||
| Theorem | mreclat 49926 | A Moore space is a complete lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐶) ⇒ ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ CLat) | ||
| Theorem | topclat 49927 | A topology is a complete lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) ⇒ ⊢ (𝐽 ∈ Top → 𝐼 ∈ CLat) | ||
| Theorem | toplatglb0 49928 | The empty intersection in a topology is realized by the base set. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ 𝐺 = (glb‘𝐼) ⇒ ⊢ (𝜑 → (𝐺‘∅) = ∪ 𝐽) | ||
| Theorem | toplatlub 49929 | Least upper bounds in a topology are realized by unions. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝑆 ⊆ 𝐽) & ⊢ 𝑈 = (lub‘𝐼) ⇒ ⊢ (𝜑 → (𝑈‘𝑆) = ∪ 𝑆) | ||
| Theorem | toplatglb 49930 | Greatest lower bounds in a topology are realized by the interior of the intersection. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝑆 ⊆ 𝐽) & ⊢ 𝐺 = (glb‘𝐼) & ⊢ (𝜑 → 𝑆 ≠ ∅) ⇒ ⊢ (𝜑 → (𝐺‘𝑆) = ((int‘𝐽)‘∩ 𝑆)) | ||
| Theorem | toplatjoin 49931 | Joins in a topology are realized by unions. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝐴 ∈ 𝐽) & ⊢ (𝜑 → 𝐵 ∈ 𝐽) & ⊢ ∨ = (join‘𝐼) ⇒ ⊢ (𝜑 → (𝐴 ∨ 𝐵) = (𝐴 ∪ 𝐵)) | ||
| Theorem | toplatmeet 49932 | Meets in a topology are realized by intersections. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝐴 ∈ 𝐽) & ⊢ (𝜑 → 𝐵 ∈ 𝐽) & ⊢ ∧ = (meet‘𝐼) ⇒ ⊢ (𝜑 → (𝐴 ∧ 𝐵) = (𝐴 ∩ 𝐵)) | ||
| Theorem | topdlat 49933 | A topology is a distributive lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) ⇒ ⊢ (𝐽 ∈ Top → 𝐼 ∈ DLat) | ||
| Theorem | elmgpcntrd 49934* | The center of a ring. (Contributed by Zhi Wang, 11-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝑍 = (Cntr‘𝑀) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑋(.r‘𝑅)𝑦) = (𝑦(.r‘𝑅)𝑋)) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝑍) | ||
| Theorem | asclelbasALT 49935 | Alternate proof of asclelbas 22101. (Contributed by Zhi Wang, 11-Sep-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ AssAlg) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐴‘𝐶) ∈ (Base‘𝑊)) | ||
| Theorem | asclcntr 49936 | The algebra scalar lifting function maps into the center of the algebra. Equivalently, a lifted scalar is a center of the algebra. (Contributed by Zhi Wang, 11-Sep-2025.) |
| ⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ AssAlg) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ 𝑀 = (mulGrp‘𝑊) ⇒ ⊢ (𝜑 → (𝐴‘𝐶) ∈ (Cntr‘𝑀)) | ||
| Theorem | asclcom 49937 |
Scalars are commutative after being lifted.
However, the scalars themselves are not necessarily commutative if the algebra is not a faithful module. For example, Let 𝐹 be the 2 by 2 upper triangular matrix algebra over a commutative ring 𝑊. It is provable that 𝐹 is in general non-commutative. Define scalar multiplication 𝐶 · 𝑋 as multipying the top-left entry, which is a "vector" element of 𝑊, of the "scalar" 𝐶, which is now an upper triangular matrix, with the "vector" 𝑋 ∈ (Base‘𝑊). Equivalently, the algebra scalar lifting function is not necessarily injective unless the algebra is faithful. Therefore, all "scalar injection" was renamed. Alternate proof involves assa2ass 22081, assa2ass2 22082, and asclval 22097, by setting 𝑋 and 𝑌 the multiplicative identity of the algebra. (Contributed by Zhi Wang, 11-Sep-2025.) |
| ⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ AssAlg) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ ∗ = (.r‘𝐹) & ⊢ (𝜑 → 𝐷 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐴‘(𝐶 ∗ 𝐷)) = (𝐴‘(𝐷 ∗ 𝐶))) | ||
| Theorem | homf0 49938 | The base is empty iff the functionalized Hom-set operation is empty. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ ((Base‘𝐶) = ∅ ↔ (Homf ‘𝐶) = ∅) | ||
| Theorem | catprslem 49939* | Lemma for catprs 49940. (Contributed by Zhi Wang, 18-Sep-2024.) |
| ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)) | ||
| Theorem | catprs 49940* | A preorder can be extracted from a category. See catprs2 49941 for more details. (Contributed by Zhi Wang, 18-Sep-2024.) |
| ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ Cat) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍))) | ||
| Theorem | catprs2 49941* | A category equipped with the induced preorder, where an object 𝑥 is defined to be "less than or equal to" 𝑦 iff there is a morphism from 𝑥 to 𝑦, is a preordered set, or a proset. The category might not be thin. See catprsc 49942 and catprsc2 49943 for constructions satisfying the hypothesis "catprs.1". See catprs 49940 for a more primitive version. See prsthinc 50393 for constructing a thin category from a proset. (Contributed by Zhi Wang, 18-Sep-2024.) |
| ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → ≤ = (le‘𝐶)) ⇒ ⊢ (𝜑 → 𝐶 ∈ Proset ) | ||
| Theorem | catprsc 49942* | A construction of the preorder induced by a category. See catprs2 49941 for details. See also catprsc2 49943 for an alternate construction. (Contributed by Zhi Wang, 18-Sep-2024.) |
| ⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}) ⇒ ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅)) | ||
| Theorem | catprsc2 49943* | An alternate construction of the preorder induced by a category. See catprs2 49941 for details. See also catprsc 49942 for a different construction. The two constructions are different because df-cat 17759 does not require the domain of 𝐻 to be 𝐵 × 𝐵. (Contributed by Zhi Wang, 23-Sep-2024.) |
| ⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅}) ⇒ ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅)) | ||
| Theorem | endmndlem 49944 | A diagonal hom-set in a category equipped with the restriction of the composition has a structure of monoid. See also df-mndtc 50507 for converting a monoid to a category. Lemma for bj-endmnd 38073. (Contributed by Zhi Wang, 25-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑋𝐻𝑋) = (Base‘𝑀)) & ⊢ (𝜑 → (⟨𝑋, 𝑋⟩ · 𝑋) = (+g‘𝑀)) ⇒ ⊢ (𝜑 → 𝑀 ∈ Mnd) | ||
| Theorem | oppccatb 49945 | An opposite category is a category. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐶 ∈ Cat ↔ 𝑂 ∈ Cat)) | ||
| Theorem | oppcmndclem 49946 | Lemma for oppcmndc 49948. Everything is true for two distinct elements in a singleton or an empty set (since it is impossible). Note that if this theorem and oppcendc 49947 are in ¬ 𝑥 = 𝑦 form, then both proofs should be one step shorter. (Contributed by Zhi Wang, 16-Oct-2025.) |
| ⊢ (𝜑 → 𝐵 = {𝐴}) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ≠ 𝑌 → 𝜓)) | ||
| Theorem | oppcendc 49947* | The opposite category of a category whose morphisms are all endomorphisms has the same base and hom-sets as the original category. (Contributed by Zhi Wang, 16-Oct-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ≠ 𝑦 → (𝑥𝐻𝑦) = ∅)) ⇒ ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝑂)) | ||
| Theorem | oppcmndc 49948 | The opposite category of a category whose base set is a singleton or an empty set has the same base and hom-sets as the original category. (Contributed by Zhi Wang, 16-Oct-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐵 = {𝑋}) ⇒ ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝑂)) | ||
| Theorem | idmon 49949 | An identity arrow, or an identity morphism, is a monomorphism. (Contributed by Zhi Wang, 21-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝑀 = (Mono‘𝐶) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝑀𝑋)) | ||
| Theorem | idepi 49950 | An identity arrow, or an identity morphism, is an epimorphism. (Contributed by Zhi Wang, 21-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝐸 = (Epi‘𝐶) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐸𝑋)) | ||
| Theorem | sectrcl 49951 | Reverse closure for section relations. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
| Theorem | sectrcl2 49952 | Reverse closure for section relations. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) | ||
| Theorem | invrcl 49953 | Reverse closure for inverse relations. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑁 = (Inv‘𝐶) & ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
| Theorem | invrcl2 49954 | Reverse closure for inverse relations. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑁 = (Inv‘𝐶) & ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) | ||
| Theorem | isinv2 49955 | The property "𝐹 is an inverse of 𝐺". (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑁 = (Inv‘𝐶) & ⊢ 𝑆 = (Sect‘𝐶) ⇒ ⊢ (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐺(𝑌𝑆𝑋)𝐹)) | ||
| Theorem | isisod 49956 | The predicate "is an isomorphism" (deduction form). (Contributed by Zhi Wang, 16-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝐼 = (Iso‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑋)) & ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋)) & ⊢ (𝜑 → (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝐺) = ( 1 ‘𝑌)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) | ||
| Theorem | upeu2lem 49957* | Lemma for upeu2 50101. There exists a unique morphism from 𝑌 to 𝑍 that commutes if 𝐹:𝑋⟶𝑌 is an isomorphism. (Contributed by Zhi Wang, 20-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝐼 = (Iso‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑍)) ⇒ ⊢ (𝜑 → ∃!𝑘 ∈ (𝑌𝐻𝑍)𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)) | ||
| Theorem | sectfn 49958 | The function value of the function returning the sections of a category is a function over the Cartesian square of the base set of the category. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝐶 ∈ Cat → (Sect‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) | ||
| Theorem | invfn 49959 | The function value of the function returning the inverses of a category is a function over the Cartesian square of the base set of the category. Simplifies isofn 17867 (see isofnALT 49960). (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) | ||
| Theorem | isofnALT 49960 | The function value of the function returning the isomorphisms of a category is a function over the Cartesian square of the base set of the category. (Contributed by AV, 5-Apr-2020.) (Proof shortened by Zhi Wang, 3-Nov-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) | ||
| Theorem | isofval2 49961* | Function value of the function returning the isomorphisms of a category. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝑁 = (Inv‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 𝐼 = (Iso‘𝐶) ⇒ ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥𝑁𝑦))) | ||
| Theorem | isorcl 49962 | Reverse closure for isomorphism relations. (Contributed by Zhi Wang, 17-Nov-2025.) |
| ⊢ 𝐼 = (Iso‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
| Theorem | isorcl2 49963 | Reverse closure for isomorphism relations. (Contributed by Zhi Wang, 17-Nov-2025.) |
| ⊢ 𝐼 = (Iso‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) | ||
| Theorem | isoval2 49964 | The isomorphisms are the domain of the inverse relation. (Contributed by Zhi Wang, 17-Nov-2025.) |
| ⊢ 𝑁 = (Inv‘𝐶) & ⊢ 𝐼 = (Iso‘𝐶) ⇒ ⊢ (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌) | ||
| Theorem | sectpropdlem 49965 | Lemma for sectpropd 49966. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ (Sect‘𝐶)) → 𝑃 ∈ (Sect‘𝐷)) | ||
| Theorem | sectpropd 49966 | Two structures with the same base, hom-sets and composition operation have the same sections. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → (Sect‘𝐶) = (Sect‘𝐷)) | ||
| Theorem | invpropdlem 49967 | Lemma for invpropd 49968. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → 𝑃 ∈ (Inv‘𝐷)) | ||
| Theorem | invpropd 49968 | Two structures with the same base, hom-sets and composition operation have the same inverses. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → (Inv‘𝐶) = (Inv‘𝐷)) | ||
| Theorem | isopropdlem 49969 | Lemma for isopropd 49970. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → 𝑃 ∈ (Iso‘𝐷)) | ||
| Theorem | isopropd 49970 | Two structures with the same base, hom-sets and composition operation have the same isomorphisms. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → (Iso‘𝐶) = (Iso‘𝐷)) | ||
| Theorem | cicfn 49971 | ≃𝑐 is a function on Cat. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ ≃𝑐 Fn Cat | ||
| Theorem | cicrcl2 49972 | Isomorphism implies the structure being a category. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝑅( ≃𝑐 ‘𝐶)𝑆 → 𝐶 ∈ Cat) | ||
| Theorem | oppccic 49973 | Isomorphic objects are isomorphic in the opposite category. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ (𝜑 → 𝑅( ≃𝑐 ‘𝐶)𝑆) ⇒ ⊢ (𝜑 → 𝑅( ≃𝑐 ‘𝑂)𝑆) | ||
| Theorem | relcic 49974 | The set of isomorphic objects is a relation. Simplifies cicer 17898 (see cicerALT 49975). (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝐶 ∈ Cat → Rel ( ≃𝑐 ‘𝐶)) | ||
| Theorem | cicerALT 49975 | Isomorphism is an equivalence relation on objects of a category. Remark 3.16 in [Adamek] p. 29. (Contributed by AV, 5-Apr-2020.) (Proof shortened by Zhi Wang, 3-Nov-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) Er (Base‘𝐶)) | ||
| Theorem | cic1st2nd 49976 | Reconstruction of a pair of isomorphic objects in terms of its ordered pair components. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩) | ||
| Theorem | cic1st2ndbr 49977 | Rewrite the predicate of isomorphic objects with separated parts. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → (1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃)) | ||
| Theorem | cicpropdlem 49978 | Lemma for cicpropd 49979. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝑃 ∈ ( ≃𝑐 ‘𝐷)) | ||
| Theorem | cicpropd 49979 | Two structures with the same base, hom-sets and composition operation have the same isomorphic objects. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → ( ≃𝑐 ‘𝐶) = ( ≃𝑐 ‘𝐷)) | ||
| Theorem | oppccicb 49980 | Isomorphic objects are isomorphic in the opposite category. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (𝑅( ≃𝑐 ‘𝐶)𝑆 ↔ 𝑅( ≃𝑐 ‘𝑂)𝑆) | ||
| Theorem | oppcciceq 49981 | The opposite category has the same isomorphic objects as the original category. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ ( ≃𝑐 ‘𝐶) = ( ≃𝑐 ‘𝑂) | ||
| Theorem | dmdm 49982 | The double domain of a function on a Cartesian square. (Contributed by Zhi Wang, 1-Nov-2025.) |
| ⊢ (𝐴 Fn (𝐵 × 𝐵) → 𝐵 = dom dom 𝐴) | ||
| Theorem | iinfssclem1 49983* | Lemma for iinfssc 49986. (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 ⊆cat 𝐽) & ⊢ (𝜑 → 𝐾 = (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 (𝐻‘𝑦))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 = dom dom 𝐻) & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝜑 → 𝐾 = (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝑆, 𝑤 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 ↦ ∩ 𝑥 ∈ 𝐴 (𝑧𝐻𝑤))) | ||
| Theorem | iinfssclem2 49984* | Lemma for iinfssc 49986. (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 ⊆cat 𝐽) & ⊢ (𝜑 → 𝐾 = (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 (𝐻‘𝑦))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 = dom dom 𝐻) & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝜑 → 𝐾 Fn (∩ 𝑥 ∈ 𝐴 𝑆 × ∩ 𝑥 ∈ 𝐴 𝑆)) | ||
| Theorem | iinfssclem3 49985* | Lemma for iinfssc 49986. (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 ⊆cat 𝐽) & ⊢ (𝜑 → 𝐾 = (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 (𝐻‘𝑦))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 = dom dom 𝐻) & ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑋 ∈ ∩ 𝑥 ∈ 𝐴 𝑆) & ⊢ (𝜑 → 𝑌 ∈ ∩ 𝑥 ∈ 𝐴 𝑆) ⇒ ⊢ (𝜑 → (𝑋𝐾𝑌) = ∩ 𝑥 ∈ 𝐴 (𝑋𝐻𝑌)) | ||
| Theorem | iinfssc 49986* | Indexed intersection of subcategories is a subcategory (the category-agnostic version). (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 ⊆cat 𝐽) & ⊢ (𝜑 → 𝐾 = (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 (𝐻‘𝑦))) ⇒ ⊢ (𝜑 → 𝐾 ⊆cat 𝐽) | ||
| Theorem | iinfsubc 49987* | Indexed intersection of subcategories is a subcategory. (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 ∈ (Subcat‘𝐶)) & ⊢ (𝜑 → 𝐾 = (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 (𝐻‘𝑦))) ⇒ ⊢ (𝜑 → 𝐾 ∈ (Subcat‘𝐶)) | ||
| Theorem | iinfprg 49988* | Indexed intersection of functions with an unordered pair index. (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑥) ∩ (𝐵‘𝑥))) = (𝑥 ∈ ∩ 𝑦 ∈ {𝐴, 𝐵}dom 𝑦 ↦ ∩ 𝑦 ∈ {𝐴, 𝐵} (𝑦‘𝑥))) | ||
| Theorem | infsubc 49989* | The intersection of two subcategories is a subcategory. (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑥) ∩ (𝐵‘𝑥))) ∈ (Subcat‘𝐶)) | ||
| Theorem | infsubc2 49990* | The intersection of two subcategories is a subcategory. (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑥 ∈ (dom dom 𝐴 ∩ dom dom 𝐵), 𝑦 ∈ (dom dom 𝐴 ∩ dom dom 𝐵) ↦ ((𝑥𝐴𝑦) ∩ (𝑥𝐵𝑦))) ∈ (Subcat‘𝐶)) | ||
| Theorem | infsubc2d 49991* | The intersection of two subcategories is a subcategory. (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) & ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇)) & ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶)) & ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) ⇒ ⊢ (𝜑 → (𝑥 ∈ (𝑆 ∩ 𝑇), 𝑦 ∈ (𝑆 ∩ 𝑇) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) ∈ (Subcat‘𝐶)) | ||
| Theorem | discsubclem 49992* | Lemma for discsubc 49993. (Contributed by Zhi Wang, 1-Nov-2025.) |
| ⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅)) ⇒ ⊢ 𝐽 Fn (𝑆 × 𝑆) | ||
| Theorem | discsubc 49993* | A discrete category, whose only morphisms are the identity morphisms, is a subcategory. (Contributed by Zhi Wang, 1-Nov-2025.) |
| ⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅)) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐼 = (Id‘𝐶) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ Cat) ⇒ ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) | ||
| Theorem | iinfconstbaslem 49994* | Lemma for iinfconstbas 49995. (Contributed by Zhi Wang, 1-Nov-2025.) |
| ⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅)) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐼 = (Id‘𝐶) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐴 = ((Subcat‘𝐶) ∩ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)})) ⇒ ⊢ (𝜑 → 𝐽 ∈ 𝐴) | ||
| Theorem | iinfconstbas 49995* | The discrete category is the indexed intersection of all subcategories with the same base. (Contributed by Zhi Wang, 1-Nov-2025.) |
| ⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅)) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐼 = (Id‘𝐶) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐴 = ((Subcat‘𝐶) ∩ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)})) ⇒ ⊢ (𝜑 → 𝐽 = (𝑧 ∈ ∩ ℎ ∈ 𝐴 dom ℎ ↦ ∩ ℎ ∈ 𝐴 (ℎ‘𝑧))) | ||
| Theorem | nelsubclem 49996* | Lemma for nelsubc 49997. (Contributed by Zhi Wang, 5-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝑆 ≠ ∅) & ⊢ (𝜑 → 𝐽 = ((𝑆 × 𝑆) × {∅})) & ⊢ 𝐻 = (Homf ‘𝐶) ⇒ ⊢ (𝜑 → (𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ (¬ ∀𝑥 ∈ 𝑆 𝐼 ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓)))) | ||
| Theorem | nelsubc 49997* | An empty "hom-set" for non-empty base satisfies all conditions for a subcategory but the existence of identity morphisms. (Contributed by Zhi Wang, 5-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝑆 ≠ ∅) & ⊢ (𝜑 → 𝐽 = ((𝑆 × 𝑆) × {∅})) & ⊢ 𝐻 = (Homf ‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝜑 → (𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ (¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))))) | ||
| Theorem | nelsubc2 49998 | An empty "hom-set" for non-empty base is not a subcategory. (Contributed by Zhi Wang, 5-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝑆 ≠ ∅) & ⊢ (𝜑 → 𝐽 = ((𝑆 × 𝑆) × {∅})) & ⊢ (𝜑 → 𝐶 ∈ Cat) ⇒ ⊢ (𝜑 → ¬ 𝐽 ∈ (Subcat‘𝐶)) | ||
| Theorem | nelsubc3lem 49999* | Lemma for nelsubc3 50000. (Contributed by Zhi Wang, 5-Nov-2025.) |
| ⊢ 𝐶 ∈ Cat & ⊢ 𝐽 ∈ V & ⊢ 𝑆 ∈ V & ⊢ (𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat (Homf ‘𝐶) ∧ (¬ ∀𝑥 ∈ 𝑆 ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))) ⇒ ⊢ ∃𝑐 ∈ Cat ∃𝑗∃𝑠(𝑗 Fn (𝑠 × 𝑠) ∧ (𝑗 ⊆cat (Homf ‘𝑐) ∧ (¬ ∀𝑥 ∈ 𝑠 ((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))) | ||
| Theorem | nelsubc3 50000* |
Remark 4.2(2) of [Adamek] p. 48. There exists
a set satisfying all
conditions for a subcategory but the existence of identity morphisms.
Therefore such condition in df-subc 17904 is necessary.
Note that this theorem cheated a little bit because (𝐶 ↾cat 𝐽) is not a category. In fact (𝐶 ↾cat 𝐽) ∈ Cat is a stronger statement than the condition (d) of Definition 4.1(1) of [Adamek] p. 48, as stated here (see the proof of issubc3 17941). To construct such a category, see setc1onsubc 50531 and cnelsubc 50533. (Contributed by Zhi Wang, 5-Nov-2025.) |
| ⊢ ∃𝑐 ∈ Cat ∃𝑗∃𝑠(𝑗 Fn (𝑠 × 𝑠) ∧ (𝑗 ⊆cat (Homf ‘𝑐) ∧ (¬ ∀𝑥 ∈ 𝑠 ((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))) | ||
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