P. 462 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 46101-46200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	i0oii 46101	(0[,)𝐴) is open in II. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (𝐴 ≤ 1 → (0[,)𝐴) ∈ II)
Theorem	io1ii 46102	(𝐴(,]1) is open in II. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (0 ≤ 𝐴 → (𝐴(,]1) ∈ II)
20.42.9.6  Separated sets
Theorem	sepnsepolem1 46103*	Lemma for sepnsepo 46105. (Contributed by Zhi Wang, 1-Sep-2024.)
⊢ (∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥 ∈ 𝐽 (𝜑 ∧ ∃𝑦 ∈ 𝐽 (𝜓 ∧ 𝜒)))
Theorem	sepnsepolem2 46104*	Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 46105. Proof could be shortened by 1 step using ssdisjdr 46042. (Contributed by Zhi Wang, 1-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    ⇒   ⊢ (𝜑 → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
Theorem	sepnsepo 46105*	Open neighborhood and neighborhood is equivalent regarding disjointness for both sides. Namely, separatedness by open neighborhoods is equivalent to separatedness by neighborhoods. (Contributed by Zhi Wang, 1-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝐶)∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
Theorem	sepdisj 46106	Separated sets are disjoint. Note that in general separatedness also requires 𝑇 ⊆ ∪ 𝐽 and (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ as well but they are unnecessary here. (Contributed by Zhi Wang, 7-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)    ⇒   ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
Theorem	seposep 46107*	If two sets are separated by (open) neighborhoods, then they are separated subsets of the underlying set. Note that separatedness by open neighborhoods is equivalent to separatedness by neighborhoods. See sepnsepo 46105. The relationship between separatedness and closure is also seen in isnrm 22394, isnrm2 22417, isnrm3 22418. (Contributed by Zhi Wang, 7-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))    ⇒   ⊢ (𝜑 → ((𝑆 ⊆ ∪ 𝐽 ∧ 𝑇 ⊆ ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))
Theorem	sepcsepo 46108*	If two sets are separated by closed neighborhoods, then they are separated by (open) neighborhoods. See sepnsepo 46105 for the equivalence between separatedness by open neighborhoods and separatedness by neighborhoods. Although 𝐽 ∈ Top might be redundant here, it is listed for explicitness. 𝐽 ∈ Top can be obtained from neircl 46086, adantr 480, and rexlimiva 3209. (Contributed by Zhi Wang, 8-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅))    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))
Theorem	sepfsepc 46109*	If two sets are separated by a continuous function, then they are separated by closed neighborhoods. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (◡𝑓 “ {0}) ∧ 𝑇 ⊆ (◡𝑓 “ {1})))    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅))
Theorem	seppsepf 46110	If two sets are precisely separated by a continuous function, then they are separated by the continuous function. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1})))    ⇒   ⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (◡𝑓 “ {0}) ∧ 𝑇 ⊆ (◡𝑓 “ {1})))
Theorem	seppcld 46111*	If two sets are precisely separated by a continuous function, then they are closed. An alternate proof involves II ∈ Fre. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1})))    ⇒   ⊢ (𝜑 → (𝑆 ∈ (Clsd‘𝐽) ∧ 𝑇 ∈ (Clsd‘𝐽)))
20.42.9.7  Separated spaces: T0, T1, T2 (Hausdorff) ...
Theorem	isnrm4 46112*	A topological space is normal iff any two disjoint closed sets are separated by neighborhoods. (Contributed by Zhi Wang, 1-Sep-2024.)
⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝐽)‘𝑐)∃𝑦 ∈ ((nei‘𝐽)‘𝑑)(𝑥 ∩ 𝑦) = ∅)))
Theorem	dfnrm2 46113*	A topological space is normal if any disjoint closed sets can be separated by open neighborhoods. An alternate definition of df-nrm 22376. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ Nrm = {𝑗 ∈ Top ∣ ∀𝑐 ∈ (Clsd‘𝑗)∀𝑑 ∈ (Clsd‘𝑗)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝑗 ∃𝑦 ∈ 𝑗 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))}
Theorem	dfnrm3 46114*	A topological space is normal if any disjoint closed sets can be separated by neighborhoods. An alternate definition of df-nrm 22376. (Contributed by Zhi Wang, 2-Sep-2024.)
⊢ Nrm = {𝑗 ∈ Top ∣ ∀𝑐 ∈ (Clsd‘𝑗)∀𝑑 ∈ (Clsd‘𝑗)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝑗)‘𝑐)∃𝑦 ∈ ((nei‘𝑗)‘𝑑)(𝑥 ∩ 𝑦) = ∅)}
Theorem	iscnrm3lem1 46115*	Lemma for iscnrm3 46134. Subspace topology is a topology. (Contributed by Zhi Wang, 3-Sep-2024.)
⊢ (𝐽 ∈ Top → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ((𝐽 ↾t 𝑥) ∈ Top ∧ 𝜑)))
Theorem	iscnrm3lem2 46116*	Lemma for iscnrm3 46134 proving a biconditional on restricted universal quantifications. (Contributed by Zhi Wang, 3-Sep-2024.)
⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓 → ((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒)))    &   ⊢ (𝜑 → (∀𝑤 ∈ 𝐷 ∀𝑣 ∈ 𝐸 𝜒 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜓)))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓 ↔ ∀𝑤 ∈ 𝐷 ∀𝑣 ∈ 𝐸 𝜒))
Theorem	iscnrm3lem3 46117	Lemma for iscnrm3lem4 46118. (Contributed by Zhi Wang, 4-Sep-2024.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜓))
Theorem	iscnrm3lem4 46118	Lemma for iscnrm3lem5 46119 and iscnrm3r 46130. (Contributed by Zhi Wang, 4-Sep-2024.)
⊢ (𝜂 → (𝜓 → 𝜁))    &   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜂)    &   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → (𝜁 → 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏))))
Theorem	iscnrm3lem5 46119*	Lemma for iscnrm3l 46133. (Contributed by Zhi Wang, 3-Sep-2024.)
⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑇) → (𝜑 ↔ 𝜓))    &   ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑇) → (𝜒 ↔ 𝜃))    &   ⊢ ((𝜏 ∧ 𝜂 ∧ 𝜁) → (𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊))    &   ⊢ ((𝜏 ∧ 𝜂 ∧ 𝜁) → ((𝜓 → 𝜃) → 𝜎))    ⇒   ⊢ (𝜏 → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 (𝜑 → 𝜒) → (𝜂 → (𝜁 → 𝜎))))
Theorem	iscnrm3lem6 46120*	Lemma for iscnrm3lem7 46121. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑊 𝜓 → 𝜒))
Theorem	iscnrm3lem7 46121*	Lemma for iscnrm3rlem8 46129 and iscnrm3llem2 46132 involving restricted existential quantifications. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝑧 = 𝑍 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑤 = 𝑊 → (𝜃 ↔ 𝜏))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → (𝑍 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝜏))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜒))
Theorem	iscnrm3rlem1 46122	Lemma for iscnrm3rlem2 46123. The hypothesis could be generalized to (𝜑 → (𝑆 ∖ 𝑇) ⊆ 𝑋). (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝜑 → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑆 ∖ 𝑇) = (𝑆 ∩ (𝑋 ∖ (𝑆 ∩ 𝑇))))
Theorem	iscnrm3rlem2 46123	Lemma for iscnrm3rlem3 46124. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)    ⇒   ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∖ 𝑇) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ 𝑇)))))
Theorem	iscnrm3rlem3 46124	Lemma for iscnrm3r 46130. The designed subspace is a subset of the original set; the two sets are closed sets in the subspace. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽)) → ((∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 ∪ 𝐽 ∧ (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))))
Theorem	iscnrm3rlem4 46125	Lemma for iscnrm3rlem8 46129. Given two disjoint subsets 𝑆 and 𝑇 of the underlying set of a topology 𝐽, if 𝑁 is a superset of (((cls‘𝐽)‘𝑆) ∖ 𝑇), then it is a superset of 𝑆. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)    &   ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∖ 𝑇) ⊆ 𝑁)    ⇒   ⊢ (𝜑 → 𝑆 ⊆ 𝑁)
Theorem	iscnrm3rlem5 46126	Lemma for iscnrm3rlem6 46127. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽)    ⇒   ⊢ (𝜑 → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝐽)
Theorem	iscnrm3rlem6 46127	Lemma for iscnrm3rlem7 46128. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → 𝑂 ⊆ (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))    ⇒   ⊢ (𝜑 → (𝑂 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ↔ 𝑂 ∈ 𝐽))
Theorem	iscnrm3rlem7 46128	Lemma for iscnrm3rlem8 46129. Open neighborhoods in the subspace topology are open neighborhoods in the original topology given that the subspace is an open set in the original topology. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽)    &   ⊢ (𝜑 → 𝑂 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))    ⇒   ⊢ (𝜑 → 𝑂 ∈ 𝐽)
Theorem	iscnrm3rlem8 46129*	Lemma for iscnrm3r 46130. Disjoint open neighborhoods in the subspace topology are disjoint open neighborhoods in the original topology given that the subspace is an open set in the original topology. Therefore, given any two sets separated in the original topology and separated by open neighborhoods in the subspace topology, they must be separated by open neighborhoods in the original topology. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) → (∃𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))
Theorem	iscnrm3r 46130*	Lemma for iscnrm3 46134. If all subspaces of a topology are normal, i.e., two disjoint closed sets can be separated by open neighborhoods, then in the original topology two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝐽 ∈ Top → (∀𝑧 ∈ 𝒫 ∪ 𝐽∀𝑐 ∈ (Clsd‘(𝐽 ↾t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽 ↾t 𝑧))((𝑐 ∩ 𝑑) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑧)∃𝑘 ∈ (𝐽 ↾t 𝑧)(𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → ((𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) → (((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))))
Theorem	iscnrm3llem1 46131	Lemma for iscnrm3l 46133. Closed sets in the subspace are subsets of the underlying set of the original topology. (Contributed by Zhi Wang, 4-Sep-2024.)
⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (𝐶 ∈ 𝒫 ∪ 𝐽 ∧ 𝐷 ∈ 𝒫 ∪ 𝐽))
Theorem	iscnrm3llem2 46132*	Lemma for iscnrm3l 46133. If there exist disjoint open neighborhoods in the original topology for two disjoint closed sets in a subspace, then they can be separated by open neighborhoods in the subspace topology. (Could shorten proof with ssin0 42492.) (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) → ∃𝑙 ∈ (𝐽 ↾t 𝑍)∃𝑘 ∈ (𝐽 ↾t 𝑍)(𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))
Theorem	iscnrm3l 46133*	Lemma for iscnrm3 46134. Given a topology 𝐽, if two separated sets can be separated by open neighborhoods, then all subspaces of the topology 𝐽 are normal, i.e., two disjoint closed sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝐽 ∈ Top → (∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) → ((𝐶 ∩ 𝐷) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑍)∃𝑘 ∈ (𝐽 ↾t 𝑍)(𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))))
Theorem	iscnrm3 46134*	A completely normal topology is a topology in which two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
Theorem	iscnrm3v 46135*	A topology is completely normal iff two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 10-Sep-2024.)
⊢ (𝐽 ∈ Top → (𝐽 ∈ CNrm ↔ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
Theorem	iscnrm4 46136*	A completely normal topology is a topology in which two separated sets can be separated by neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑠)∃𝑚 ∈ ((nei‘𝐽)‘𝑡)(𝑛 ∩ 𝑚) = ∅)))
20.42.10  Preordered sets and directed sets using extensible structures
Theorem	isprsd 46137*	Property of being a preordered set (deduction form). (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → ≤ = (le‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐾 ∈ Proset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
20.42.11  Posets and lattices using extensible structures
20.42.11.1  Posets
Theorem	lubeldm2 46138*	Member of the domain of the least upper bound function of a poset. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))    &   ⊢ (𝜑 → 𝐾 ∈ Poset)    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
Theorem	glbeldm2 46139*	Member of the domain of the greatest lower bound function of a poset. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))    &   ⊢ (𝜑 → 𝐾 ∈ Poset)    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
Theorem	lubeldm2d 46140*	Member of the domain of the least upper bound function of a poset. (Contributed by Zhi Wang, 28-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → ≤ = (le‘𝐾))    &   ⊢ (𝜑 → 𝑈 = (lub‘𝐾))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))    &   ⊢ (𝜑 → 𝐾 ∈ Poset)    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
Theorem	glbeldm2d 46141*	Member of the domain of the greatest lower bound function of a poset. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → ≤ = (le‘𝐾))    &   ⊢ (𝜑 → 𝐺 = (glb‘𝐾))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))    &   ⊢ (𝜑 → 𝐾 ∈ Poset)    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
Theorem	lubsscl 46142	If a subset of 𝑆 contains the LUB of 𝑆, then the two sets have the same LUB. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑆)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ dom 𝑈)    &   ⊢ (𝜑 → (𝑈‘𝑆) ∈ 𝑇)    ⇒   ⊢ (𝜑 → (𝑇 ∈ dom 𝑈 ∧ (𝑈‘𝑇) = (𝑈‘𝑆)))
Theorem	glbsscl 46143	If a subset of 𝑆 contains the GLB of 𝑆, then the two sets have the same GLB. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑆)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ dom 𝐺)    &   ⊢ (𝜑 → (𝐺‘𝑆) ∈ 𝑇)    ⇒   ⊢ (𝜑 → (𝑇 ∈ dom 𝐺 ∧ (𝐺‘𝑇) = (𝐺‘𝑆)))
Theorem	lubprlem 46144	Lemma for lubprdm 46145 and lubpr 46146. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌})    &   ⊢ 𝑈 = (lub‘𝐾)    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ∧ (𝑈‘𝑆) = 𝑌))
Theorem	lubprdm 46145	The set of two comparable elements in a poset has LUB. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌})    &   ⊢ 𝑈 = (lub‘𝐾)    ⇒   ⊢ (𝜑 → 𝑆 ∈ dom 𝑈)
Theorem	lubpr 46146	The LUB of the set of two comparable elements in a poset is the greater one of the two. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌})    &   ⊢ 𝑈 = (lub‘𝐾)    ⇒   ⊢ (𝜑 → (𝑈‘𝑆) = 𝑌)
Theorem	glbprlem 46147	Lemma for glbprdm 46148 and glbpr 46149. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌})    &   ⊢ 𝐺 = (glb‘𝐾)    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ∧ (𝐺‘𝑆) = 𝑋))
Theorem	glbprdm 46148	The set of two comparable elements in a poset has GLB. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌})    &   ⊢ 𝐺 = (glb‘𝐾)    ⇒   ⊢ (𝜑 → 𝑆 ∈ dom 𝐺)
Theorem	glbpr 46149	The GLB of the set of two comparable elements in a poset is the less one of the two. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌})    &   ⊢ 𝐺 = (glb‘𝐾)    ⇒   ⊢ (𝜑 → (𝐺‘𝑆) = 𝑋)
Theorem	joindm2 46150*	The join of any two elements always exists iff all unordered pairs have LUB. (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝑈))
Theorem	joindm3 46151*	The join of any two elements always exists iff all unordered pairs have LUB (expanded version). (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ∧ ∀𝑤 ∈ 𝐵 ((𝑥 ≤ 𝑤 ∧ 𝑦 ≤ 𝑤) → 𝑧 ≤ 𝑤))))
Theorem	meetdm2 46152*	The meet of any two elements always exists iff all unordered pairs have GLB. (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺))
Theorem	meetdm3 46153*	The meet of any two elements always exists iff all unordered pairs have GLB (expanded version). (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑥 ∧ 𝑧 ≤ 𝑦) ∧ ∀𝑤 ∈ 𝐵 ((𝑤 ≤ 𝑥 ∧ 𝑤 ≤ 𝑦) → 𝑤 ≤ 𝑧))))
Theorem	posjidm 46154	Poset join is idempotent. latjidm 18095 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 46155	Poset meet is idempotent. latmidm 18107 could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 𝑋) = 𝑋)
20.42.11.2  Lattices
Theorem	toslat 46156	A toset is a lattice. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝐾 ∈ Toset → 𝐾 ∈ Lat)
Theorem	isclatd 46157*	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)
20.42.11.3  Subset order structures
Theorem	intubeu 46158*	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 46159*	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 46160*	Lemma for ipolubdm 46161 and ipolub 46162. (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 46161*	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 46162*	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 18047 is in quantified form. mrelatlub 18195 could potentially be shortened using this. See mrelatlubALT 46169. (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 46163*	Lemma for ipoglbdm 46164 and ipoglb 46165. (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 46164*	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 46165*	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 18048 is in quantified form. mrelatglb 18193 could potentially be shortened using this. See mrelatglbALT 46170. (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 46166	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 46167	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 46168	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 46169	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 46170	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 46171	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 46172	A topology is a complete lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    ⇒   ⊢ (𝐽 ∈ Top → 𝐼 ∈ CLat)
Theorem	toplatglb0 46173	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 46174	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 46175	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 46176	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 46177	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 46178	A topology is a distributive lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    ⇒   ⊢ (𝐽 ∈ Top → 𝐼 ∈ DLat)
20.42.12  Categories
20.42.12.1  Categories
Theorem	catprslem 46179*	Lemma for catprs 46180. (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 46180*	A preorder can be extracted from a category. See catprs2 46181 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 46181*	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 46182 and catprsc2 46183 for constructions satisfying the hypothesis "catprs.1". See catprs 46180 for a more primitive version. See prsthinc 46223 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 46182*	A construction of the preorder induced by a category. See catprs2 46181 for details. See also catprsc2 46183 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 46183*	An alternate construction of the preorder induced by a category. See catprs2 46181 for details. See also catprsc 46182 for a different construction. The two constructions are different because df-cat 17294 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 46184	A diagonal hom-set in a category equipped with the restriction of the composition has a structure of monoid. See also df-mndtc 46251 for converting a monoid to a category. Lemma for bj-endmnd 35416. (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)
20.42.12.2  Monomorphisms and epimorphisms
Theorem	idmon 46185	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 46186	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))
20.42.12.3  Functors
Theorem	funcf2lem 46187*	A utility theorem for proving equivalence of "is a functor". (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
20.42.13  Examples of categories
20.42.13.1  Thin categories
Syntax	cthinc 46188	Extend class notation with the class of thin categories.
class ThinCat
Definition	df-thinc 46189*	Definition of the class of thin categories, or posetal categories, whose hom-sets each contain at most one morphism. Example 3.26(2) of [Adamek] p. 33. "ThinCat" was taken instead of "PosCat" because the latter might mean the category of posets. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)}
Theorem	isthinc 46190*	The predicate "is a thin category". (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
Theorem	isthinc2 46191*	A thin category is a category in which all hom-sets have cardinality less than or equal to the cardinality of 1o. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ≼ 1o))
Theorem	isthinc3 46192*	A thin category is a category in which, given a pair of objects 𝑥 and 𝑦 and any two morphisms 𝑓, 𝑔 from 𝑥 to 𝑦, the morphisms are equal. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑥𝐻𝑦)𝑓 = 𝑔))
Theorem	thincc 46193	A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝐶 ∈ ThinCat → 𝐶 ∈ Cat)
Theorem	thinccd 46194	A thin category is a category (deduction form). (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    ⇒   ⊢ (𝜑 → 𝐶 ∈ Cat)
Theorem	thincssc 46195	A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ ThinCat ⊆ Cat
Theorem	isthincd2lem1 46196*	Lemma for isthincd2 46207 and thincmo2 46197. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))    ⇒   ⊢ (𝜑 → 𝐹 = 𝐺)
Theorem	thincmo2 46197	Morphisms in the same hom-set are identical. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ ThinCat)    ⇒   ⊢ (𝜑 → 𝐹 = 𝐺)
Theorem	thincmo 46198*	There is at most one morphism in each hom-set. (Contributed by Zhi Wang, 21-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
Theorem	thincmoALT 46199*	Alternate proof for thincmo 46198. (Contributed by Zhi Wang, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
Theorem	thincmod 46200*	At most one morphism in each hom-set (deduction form). (Contributed by Zhi Wang, 21-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))