P. 145 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14401-14500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cnconst2 14401	A constant function is continuous. (Contributed by Mario Carneiro, 19-Mar-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))
Theorem	cnconst 14402	A constant function is continuous. (Contributed by FL, 15-Jan-2007.) (Proof shortened by Mario Carneiro, 19-Mar-2015.)
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐵 ∈ 𝑌 ∧ 𝐹:𝑋⟶{𝐵})) → 𝐹 ∈ (𝐽 Cn 𝐾))
Theorem	cnrest 14403	Continuity of a restriction from a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾))
Theorem	cnrest2 14404	Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Hankins, 10-Jul-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵))))
Theorem	cnrest2r 14405	Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Jun-2014.)
⊢ (𝐾 ∈ Top → (𝐽 Cn (𝐾 ↾t 𝐵)) ⊆ (𝐽 Cn 𝐾))
Theorem	cnptopresti 14406	One direction of cnptoprest 14407 under the weaker condition that the point is in the subset rather than the interior of the subset. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 31-Mar-2023.)
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃))
Theorem	cnptoprest 14407	Equivalence of continuity at a point and continuity of the restricted function at a point. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 5-Apr-2023.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃)))
Theorem	cnptoprest2 14408	Equivalence of point-continuity in the parent topology and point-continuity in a subspace. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 6-Apr-2023.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)))
Theorem	cndis 14409	Every function is continuous when the domain is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝒫 𝐴 Cn 𝐽) = (𝑋 ↑𝑚 𝐴))
Theorem	cnpdis 14410	If 𝐴 is an isolated point in 𝑋 (or equivalently, the singleton {𝐴} is open in 𝑋), then every function is continuous at 𝐴. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → ((𝐽 CnP 𝐾)‘𝐴) = (𝑌 ↑𝑚 𝑋))
Theorem	lmfpm 14411	If 𝐹 converges, then 𝐹 is a partial function. (Contributed by Mario Carneiro, 23-Dec-2013.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝐹 ∈ (𝑋 ↑pm ℂ))
Theorem	lmfss 14412	Inclusion of a function having a limit (used to ensure the limit relation is a set, under our definition). (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝐹 ⊆ (ℂ × 𝑋))
Theorem	lmcl 14413	Closure of a limit. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝑃 ∈ 𝑋)
Theorem	lmss 14414	Limit on a subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
⊢ 𝐾 = (𝐽 ↾t 𝑌)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑃 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹:𝑍⟶𝑌)    ⇒   ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹(⇝𝑡‘𝐾)𝑃))
Theorem	sslm 14415	A finer topology has fewer convergent sequences (but the sequences that do converge, converge to the same value). (Contributed by Mario Carneiro, 15-Sep-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (⇝𝑡‘𝐾) ⊆ (⇝𝑡‘𝐽))
Theorem	lmres 14416	A function converges iff its restriction to an upper integers set converges. (Contributed by Mario Carneiro, 31-Dec-2013.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋 ↑pm ℂ))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ↾ (ℤ≥‘𝑀))(⇝𝑡‘𝐽)𝑃))
Theorem	lmff 14417*	If 𝐹 converges, there is some upper integer set on which 𝐹 is a total function. (Contributed by Mario Carneiro, 31-Dec-2013.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘𝐽))    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)
Theorem	lmtopcnp 14418	The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.) (Revised by Jim Kingdon, 6-Apr-2023.)
⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃)    &   ⊢ (𝜑 → 𝐾 ∈ Top)    &   ⊢ (𝜑 → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝑃))    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹)(⇝𝑡‘𝐾)(𝐺‘𝑃))
Theorem	lmcn 14419	The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.)
⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹)(⇝𝑡‘𝐾)(𝐺‘𝑃))
9.1.8  Product topologies
Syntax	ctx 14420	Extend class notation with the binary topological product operation.
class ×t
Definition	df-tx 14421*	Define the binary topological product, which is homeomorphic to the general topological product over a two element set, but is more convenient to use. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ×t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))))
Theorem	txvalex 14422	Existence of the binary topological product. If 𝑅 and 𝑆 are known to be topologies, see txtop 14428. (Contributed by Jim Kingdon, 3-Aug-2023.)
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) ∈ V)
Theorem	txval 14423*	Value of the binary topological product operation. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 30-Aug-2015.)
⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘𝐵))
Theorem	txuni2 14424*	The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))    &   ⊢ 𝑋 = ∪ 𝑅    &   ⊢ 𝑌 = ∪ 𝑆    ⇒   ⊢ (𝑋 × 𝑌) = ∪ 𝐵
Theorem	txbasex 14425*	The basis for the product topology is a set. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝐵 ∈ V)
Theorem	txbas 14426*	The set of Cartesian products of elements from two topological bases is a basis. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))    ⇒   ⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → 𝐵 ∈ TopBases)
Theorem	eltx 14427*	A set in a product is open iff each point is surrounded by an open rectangle. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑝 ∈ 𝑆 ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
Theorem	txtop 14428	The product of two topologies is a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
Theorem	txtopi 14429	The product of two topologies is a topology. (Contributed by Jeff Madsen, 15-Jun-2010.)
⊢ 𝑅 ∈ Top    &   ⊢ 𝑆 ∈ Top    ⇒   ⊢ (𝑅 ×t 𝑆) ∈ Top
Theorem	txtopon 14430	The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 22-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.)
⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
Theorem	txuni 14431	The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ 𝑋 = ∪ 𝑅    &   ⊢ 𝑌 = ∪ 𝑆    ⇒   ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆))
Theorem	txunii 14432	The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 15-Jun-2010.)
⊢ 𝑅 ∈ Top    &   ⊢ 𝑆 ∈ Top    &   ⊢ 𝑋 = ∪ 𝑅    &   ⊢ 𝑌 = ∪ 𝑆    ⇒   ⊢ (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆)
Theorem	txopn 14433	The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴 × 𝐵) ∈ (𝑅 ×t 𝑆))
Theorem	txss12 14434	Subset property of the topological product. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ (((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)) → (𝐴 ×t 𝐶) ⊆ (𝐵 ×t 𝐷))
Theorem	txbasval 14435	It is sufficient to consider products of the bases for the topologies in the topological product. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((topGen‘𝑅) ×t (topGen‘𝑆)) = (𝑅 ×t 𝑆))
Theorem	neitx 14436	The Cartesian product of two neighborhoods is a neighborhood in the product topology. (Contributed by Thierry Arnoux, 13-Jan-2018.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ∈ ((nei‘(𝐽 ×t 𝐾))‘(𝐶 × 𝐷)))
Theorem	tx1cn 14437	Continuity of the first projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
Theorem	tx2cn 14438	Continuity of the second projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
Theorem	txcnp 14439*	If two functions are continuous at 𝐷, then the ordered pair of them is continuous at 𝐷 into the product topology. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 CnP (𝐾 ×t 𝐿))‘𝐷))
Theorem	upxp 14440*	Universal property of the Cartesian product considered as a categorical product in the category of sets. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
⊢ 𝑃 = (1st ↾ (𝐵 × 𝐶))    &   ⊢ 𝑄 = (2nd ↾ (𝐵 × 𝐶))    ⇒   ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → ∃!ℎ(ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))
Theorem	txcnmpt 14441*	A map into the product of two topological spaces is continuous if both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ 𝑊 = ∪ 𝑈    &   ⊢ 𝐻 = (𝑥 ∈ 𝑊 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)    ⇒   ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐻 ∈ (𝑈 Cn (𝑅 ×t 𝑆)))
Theorem	uptx 14442*	Universal property of the binary topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
⊢ 𝑇 = (𝑅 ×t 𝑆)    &   ⊢ 𝑋 = ∪ 𝑅    &   ⊢ 𝑌 = ∪ 𝑆    &   ⊢ 𝑍 = (𝑋 × 𝑌)    &   ⊢ 𝑃 = (1st ↾ 𝑍)    &   ⊢ 𝑄 = (2nd ↾ 𝑍)    ⇒   ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃!ℎ ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))
Theorem	txcn 14443	A map into the product of two topological spaces is continuous iff both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
⊢ 𝑋 = ∪ 𝑅    &   ⊢ 𝑌 = ∪ 𝑆    &   ⊢ 𝑍 = (𝑋 × 𝑌)    &   ⊢ 𝑊 = ∪ 𝑈    &   ⊢ 𝑃 = (1st ↾ 𝑍)    &   ⊢ 𝑄 = (2nd ↾ 𝑍)    ⇒   ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ↔ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))))
Theorem	txrest 14444	The subspace of a topological product space induced by a subset with a Cartesian product representation is a topological product of the subspaces induced by the subspaces of the terms of the products. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((𝑅 ↾t 𝐴) ×t (𝑆 ↾t 𝐵)))
Theorem	txdis 14445	The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) = 𝒫 (𝐴 × 𝐵))
Theorem	txdis1cn 14446*	A function is jointly continuous on a discrete left topology iff it is continuous as a function of its right argument, for each fixed left value. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐾 ∈ Top)    &   ⊢ (𝜑 → 𝐹 Fn (𝑋 × 𝑌))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → 𝐹 ∈ ((𝒫 𝑋 ×t 𝐽) Cn 𝐾))
Theorem	txlm 14447*	Two sequences converge iff the sequence of their ordered pairs converges. Proposition 14-2.6 of [Gleason] p. 230. (Contributed by NM, 16-Jul-2007.) (Revised by Mario Carneiro, 5-May-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐹:𝑍⟶𝑋)    &   ⊢ (𝜑 → 𝐺:𝑍⟶𝑌)    &   ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩)    ⇒   ⊢ (𝜑 → ((𝐹(⇝𝑡‘𝐽)𝑅 ∧ 𝐺(⇝𝑡‘𝐾)𝑆) ↔ 𝐻(⇝𝑡‘(𝐽 ×t 𝐾))⟨𝑅, 𝑆⟩))
Theorem	lmcn2 14448*	The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 15-May-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐹:𝑍⟶𝑋)    &   ⊢ (𝜑 → 𝐺:𝑍⟶𝑌)    &   ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑅)    &   ⊢ (𝜑 → 𝐺(⇝𝑡‘𝐾)𝑆)    &   ⊢ (𝜑 → 𝑂 ∈ ((𝐽 ×t 𝐾) Cn 𝑁))    &   ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)𝑂(𝐺‘𝑛)))    ⇒   ⊢ (𝜑 → 𝐻(⇝𝑡‘𝑁)(𝑅𝑂𝑆))
9.1.9  Continuous function-builders
Theorem	cnmptid 14449*	The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
Theorem	cnmptc 14450*	A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝑃 ∈ 𝑌)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑃) ∈ (𝐽 Cn 𝐾))
Theorem	cnmpt11 14451*	The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 𝐵) ∈ (𝐾 Cn 𝐿))    &   ⊢ (𝑦 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿))
Theorem	cnmpt11f 14452*	The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐾 Cn 𝐿))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝐽 Cn 𝐿))
Theorem	cnmpt1t 14453*	The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
Theorem	cnmpt12f 14454*	The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿))    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐾 ×t 𝐿) Cn 𝑀))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀))
Theorem	cnmpt12 14455*	The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))    &   ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀))    &   ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐷) ∈ (𝐽 Cn 𝑀))
Theorem	cnmpt1st 14456*	The projection onto the first coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
Theorem	cnmpt2nd 14457*	The projection onto the second coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
Theorem	cnmpt2c 14458*	A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))    &   ⊢ (𝜑 → 𝑃 ∈ 𝑍)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
Theorem	cnmpt21 14459*	The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))    &   ⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀))    &   ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
Theorem	cnmpt21f 14460*	The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐿 Cn 𝑀))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐹‘𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
Theorem	cnmpt2t 14461*	The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
Theorem	cnmpt22 14462*	The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))    &   ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑊))    &   ⊢ (𝜑 → (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))    &   ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐷) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
Theorem	cnmpt22f 14463*	The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
Theorem	cnmpt1res 14464*	The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 5-Jun-2014.)
⊢ 𝐾 = (𝐽 ↾t 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
Theorem	cnmpt2res 14465*	The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
⊢ 𝐾 = (𝐽 ↾t 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ 𝑁 = (𝑀 ↾t 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑍))    &   ⊢ (𝜑 → 𝑊 ⊆ 𝑍)    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ∈ ((𝐽 ×t 𝑀) Cn 𝐿))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴) ∈ ((𝐾 ×t 𝑁) Cn 𝐿))
Theorem	cnmptcom 14466*	The argument converse of a continuous function is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))    ⇒   ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
Theorem	imasnopn 14467	If a relation graph is open, then an image set of a singleton is also open. Corollary of Proposition 4 of [BourbakiTop1] p. I.26. (Contributed by Thierry Arnoux, 14-Jan-2018.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ∈ 𝐾)
9.1.10  Homeomorphisms
Syntax	chmeo 14468	Extend class notation with the class of all homeomorphisms.
class Homeo
Definition	df-hmeo 14469*	Function returning all the homeomorphisms from topology 𝑗 to topology 𝑘. (Contributed by FL, 14-Feb-2007.)
⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)})
Theorem	hmeofn 14470	The set of homeomorphisms is a function on topologies. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ Homeo Fn (Top × Top)
Theorem	hmeofvalg 14471*	The set of all the homeomorphisms between two topologies. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)})
Theorem	ishmeo 14472	The predicate F is a homeomorphism between topology 𝐽 and topology 𝐾. Proposition of [BourbakiTop1] p. I.2. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽)))
Theorem	hmeocn 14473	A homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)
⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
Theorem	hmeocnvcn 14474	The converse of a homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)
⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
Theorem	hmeocnv 14475	The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾Homeo𝐽))
Theorem	hmeof1o2 14476	A homeomorphism is a 1-1-onto mapping. (Contributed by Mario Carneiro, 22-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽Homeo𝐾)) → 𝐹:𝑋–1-1-onto→𝑌)
Theorem	hmeof1o 14477	A homeomorphism is a 1-1-onto mapping. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→𝑌)
Theorem	hmeoima 14478	The image of an open set by a homeomorphism is an open set. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ∈ 𝐽) → (𝐹 “ 𝐴) ∈ 𝐾)
Theorem	hmeoopn 14479	Homeomorphisms preserve openness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐹 “ 𝐴) ∈ 𝐾))
Theorem	hmeocld 14480	Homeomorphisms preserve closedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝐹 “ 𝐴) ∈ (Clsd‘𝐾)))
Theorem	hmeontr 14481	Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴)))
Theorem	hmeoimaf1o 14482*	The function mapping open sets to their images under a homeomorphism is a bijection of topologies. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ 𝐺 = (𝑥 ∈ 𝐽 ↦ (𝐹 “ 𝑥))    ⇒   ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐺:𝐽–1-1-onto→𝐾)
Theorem	hmeores 14483	The restriction of a homeomorphism is a homeomorphism. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌)Homeo(𝐾 ↾t (𝐹 “ 𝑌))))
Theorem	hmeoco 14484	The composite of two homeomorphisms is a homeomorphism. (Contributed by FL, 9-Mar-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽Homeo𝐿))
Theorem	idhmeo 14485	The identity function is a homeomorphism. (Contributed by FL, 14-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽Homeo𝐽))
Theorem	hmeocnvb 14486	The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
⊢ (Rel 𝐹 → (◡𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹 ∈ (𝐾Homeo𝐽)))
Theorem	txhmeo 14487*	Lift a pair of homeomorphisms on the factors to a homeomorphism of product topologies. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽Homeo𝐿))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾Homeo𝑀))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐿 ×t 𝑀)))
Theorem	txswaphmeolem 14488*	Show inverse for the "swap components" operation on a Cartesian product. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌))
Theorem	txswaphmeo 14489*	There is a homeomorphism from 𝑋 × 𝑌 to 𝑌 × 𝑋. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐾 ×t 𝐽)))
9.2  Metric spaces
9.2.1  Pseudometric spaces
Theorem	psmetrel 14490	The class of pseudometrics is a relation. (Contributed by Jim Kingdon, 24-Apr-2023.)
⊢ Rel PsMet
Theorem	ispsmet 14491*	Express the predicate "𝐷 is a pseudometric". (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
Theorem	psmetdmdm 14492	Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
Theorem	psmetf 14493	The distance function of a pseudometric as a function. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
Theorem	psmetcl 14494	Closure of the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*)
Theorem	psmet0 14495	The distance function of a pseudometric space is zero if its arguments are equal. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0)
Theorem	psmettri2 14496	Triangle inequality for the distance function of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))
Theorem	psmetsym 14497	The distance function of a pseudometric is symmetrical. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
Theorem	psmettri 14498	Triangle inequality for the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 11-Feb-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐶𝐷𝐵)))
Theorem	psmetge0 14499	The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.) (Revised by Jim Kingdon, 19-Apr-2023.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵))
Theorem	psmetxrge0 14500	The distance function of a pseudometric space is a function into the nonnegative extended real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))