P. 227 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 22601-22700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	leordtval 22601	The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞))    &   ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥))    &   ⊢ 𝐶 = ran (,)    ⇒   ⊢ (ordTop‘ ≤ ) = (topGen‘((𝐴 ∪ 𝐵) ∪ 𝐶))
Theorem	iccordt 22602	A closed interval is closed in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝐴[,]𝐵) ∈ (Clsd‘(ordTop‘ ≤ ))
Theorem	iocpnfordt 22603	An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝐴(,]+∞) ∈ (ordTop‘ ≤ )
Theorem	icomnfordt 22604	An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (-∞[,)𝐴) ∈ (ordTop‘ ≤ )
Theorem	iooordt 22605	An open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝐴(,)𝐵) ∈ (ordTop‘ ≤ )
Theorem	reordt 22606	The real numbers are an open set in the topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ ℝ ∈ (ordTop‘ ≤ )
Theorem	lecldbas 22607	The set of closed intervals forms a closed subbasis for the topology on the extended reals. Since our definition of a basis is in terms of open sets, we express this by showing that the complements of closed intervals form an open subbasis for the topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ ran [,] ↦ (ℝ* ∖ 𝑥))    ⇒   ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘ran 𝐹))
Theorem	pnfnei 22608*	A neighborhood of +∞ contains an unbounded interval based at a real number. Together with xrtgioo 24206 (which describes neighborhoods of ℝ) and mnfnei 22609, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 22605 and similar ensure that it has all the sets we want). (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ ((𝐴 ∈ (ordTop‘ ≤ ) ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
Theorem	mnfnei 22609*	A neighborhood of -∞ contains an unbounded interval based at a real number. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ ((𝐴 ∈ (ordTop‘ ≤ ) ∧ -∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴)
Theorem	ordtrestixx 22610*	The restriction of the less than order to an interval gives the same topology as the subspace topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝐴 ⊆ ℝ*    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴)    ⇒   ⊢ ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))
Theorem	ordtresticc 22611	The restriction of the less than order to a closed interval gives the same topology as the subspace topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ ((ordTop‘ ≤ ) ↾t (𝐴[,]𝐵)) = (ordTop‘( ≤ ∩ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))
12.1.9  Limits and continuity in topological spaces
Syntax	ccn 22612	Extend class notation with the class of continuous functions between topologies.
class Cn
Syntax	ccnp 22613	Extend class notation with the class of functions between topologies continuous at a given point.
class CnP
Syntax	clm 22614	Extend class notation with a function on topological spaces whose value is the convergence relation for limit sequences in the space.
class ⇝𝑡
Definition	df-cn 22615*	Define a function on two topologies whose value is the set of continuous mappings from the first topology to the second. Based on definition of continuous function in [Munkres] p. 102. See iscn 22623 for the predicate form. (Contributed by NM, 17-Oct-2006.)
⊢ Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 (◡𝑓 “ 𝑦) ∈ 𝑗})
Definition	df-cnp 22616*	Define a function on two topologies whose value is the set of continuous mappings at a specified point in the first topology. Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 17-Oct-2006.)
⊢ CnP = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑥 ∈ ∪ 𝑗 ↦ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}))
Definition	df-lm 22617*	Define a function on topologies whose value is the convergence relation for sequences into the given topological space. Although 𝑓 is typically a sequence (a function from an upperset of integers) with values in the topological space, it need not be. Note, however, that the limit property concerns only values at integers, so that the real-valued function (𝑥 ∈ ℝ ↦ (sin‘(π · 𝑥))) converges to zero (in the standard topology on the reals) with this definition. (Contributed by NM, 7-Sep-2006.)
⊢ ⇝𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})
Theorem	lmrel 22618	The topological space convergence relation is a relation. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
⊢ Rel (⇝𝑡‘𝐽)
Theorem	lmrcl 22619	Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)
⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top)
Theorem	lmfval 22620*	The relation "sequence 𝑓 converges to point 𝑦 " in a metric space. (Contributed by NM, 7-Sep-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝑋) → (⇝𝑡‘𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})
Theorem	cnfval 22621*	The set of all continuous functions from topology 𝐽 to topology 𝐾. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽})
Theorem	cnpfval 22622*	The function mapping the points in a topology 𝐽 to the set of all functions from 𝐽 to topology 𝐾 continuous at that point. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 CnP 𝐾) = (𝑥 ∈ 𝑋 ↦ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑤 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑤 → ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ (𝑓 “ 𝑣) ⊆ 𝑤))}))
Theorem	iscn 22623*	The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾". Definition of continuous function in [Munkres] p. 102. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))
Theorem	cnpval 22624*	The set of all functions from topology 𝐽 to topology 𝐾 that are continuous at a point 𝑃. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → ((𝐽 CnP 𝐾)‘𝑃) = {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))})
Theorem	iscnp 22625*	The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃". Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))
Theorem	iscn2 22626*	The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾". Definition of continuous function in [Munkres] p. 102. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))
Theorem	iscnp2 22627*	The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃". Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))
Theorem	cntop1 22628	Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
Theorem	cntop2 22629	Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
Theorem	cnptop1 22630	Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
Theorem	cnptop2 22631	Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top)
Theorem	iscnp3 22632*	The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃". (Contributed by NM, 15-May-2007.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))))
Theorem	cnprcl 22633	Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 ∈ 𝑋)
Theorem	cnf 22634	A continuous function is a mapping. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌)
Theorem	cnpf 22635	A continuous function at point 𝑃 is a mapping. (Contributed by FL, 17-Nov-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋⟶𝑌)
Theorem	cnpcl 22636	The value of a continuous function from 𝐽 to 𝐾 at point 𝑃 belongs to the underlying set of topology 𝐾. (Contributed by FL, 27-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ 𝑌)
Theorem	cnf2 22637	A continuous function is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌)
Theorem	cnpf2 22638	A continuous function at point 𝑃 is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌)
Theorem	cnprcl2 22639	Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ 𝑋)
Theorem	tgcn 22640*	The continuity predicate when the range is given by a basis for a topology. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 = (topGen‘𝐵))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽)))
Theorem	tgcnp 22641*	The "continuous at a point" predicate when the range is given by a basis for a topology. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 = (topGen‘𝐵))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝑃 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))
Theorem	subbascn 22642*	The continuity predicate when the range is given by a subbasis for a topology. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐾 = (topGen‘(fi‘𝐵)))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽)))
Theorem	ssidcn 22643	The identity function is a continuous function from one topology to another topology on the same set iff the domain is finer than the codomain. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ 𝐾 ⊆ 𝐽))
Theorem	cnpimaex 22644*	Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.)
⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝐴) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝐴))
Theorem	idcn 22645	A restricted identity function is a continuous function. (Contributed by FL, 27-Dec-2006.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
Theorem	lmbr 22646*	Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a topological space. Definition 1.4-1 of [Kreyszig] p. 25. The condition 𝐹 ⊆ (ℂ × 𝑋) allows to use objects more general than sequences when convenient; see the comment in df-lm 22617. (Contributed by Mario Carneiro, 14-Nov-2013.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    ⇒   ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝐹 ↾ 𝑦):𝑦⟶𝑢))))
Theorem	lmbr2 22647*	Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space using an arbitrary upper set of integers. (Contributed by Mario Carneiro, 14-Nov-2013.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))))
Theorem	lmbrf 22648*	Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space using an arbitrary upper set of integers. This version of lmbr2 22647 presupposes that 𝐹 is a function. (Contributed by Mario Carneiro, 14-Nov-2013.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹:𝑍⟶𝑋)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    ⇒   ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝐴 ∈ 𝑢))))
Theorem	lmconst 22649	A constant sequence converges to its value. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝑃})(⇝𝑡‘𝐽)𝑃)
Theorem	lmcvg 22650*	Convergence property of a converging sequence. (Contributed by Mario Carneiro, 14-Nov-2013.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑃 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐽)    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑈)
Theorem	iscnp4 22651*	The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃 " in terms of neighborhoods. (Contributed by FL, 18-Jul-2011.) (Revised by Mario Carneiro, 10-Sep-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹 “ 𝑥) ⊆ 𝑦)))
Theorem	cnpnei 22652*	A condition for continuity at a point in terms of neighborhoods. (Contributed by Jeff Hankins, 7-Sep-2009.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴})))
Theorem	cnima 22653	An open subset of the codomain of a continuous function has an open preimage. (Contributed by FL, 15-Dec-2006.)
⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝐾) → (◡𝐹 “ 𝐴) ∈ 𝐽)
Theorem	cnco 22654	The composition of two continuous functions is a continuous function. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐿))
Theorem	cnpco 22655	The composition of a function 𝐹 continuous at 𝑃 with a function continuous at (𝐹‘𝑃) is continuous at 𝑃. Proposition 2 of [BourbakiTop1] p. I.9. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → (𝐺 ∘ 𝐹) ∈ ((𝐽 CnP 𝐿)‘𝑃))
Theorem	cnclima 22656	A closed subset of the codomain of a continuous function has a closed preimage. (Contributed by NM, 15-Mar-2007.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽))
Theorem	iscncl 22657*	A characterization of a continuity function using closed sets. Theorem 1(d) of [BourbakiTop1] p. I.9. (Contributed by FL, 19-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))))
Theorem	cncls2i 22658	Property of the preimage of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐽)‘(◡𝐹 “ 𝑆)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑆)))
Theorem	cnntri 22659	Property of the preimage of an interior. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → (◡𝐹 “ ((int‘𝐾)‘𝑆)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑆)))
Theorem	cnclsi 22660	Property of the image of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → (𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑆)))
Theorem	cncls2 22661*	Continuity in terms of closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥)))))
Theorem	cncls 22662*	Continuity in terms of closure. (Contributed by Jeff Hankins, 1-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥)))))
Theorem	cnntr 22663*	Continuity in terms of interior. (Contributed by Jeff Hankins, 2-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌(◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥)))))
Theorem	cnss1 22664	If the topology 𝐾 is finer than 𝐽, then there are more continuous functions from 𝐾 than from 𝐽. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐽 Cn 𝐿) ⊆ (𝐾 Cn 𝐿))
Theorem	cnss2 22665	If the topology 𝐾 is finer than 𝐽, then there are fewer continuous functions into 𝐾 than into 𝐽 from some other space. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn 𝐿))
Theorem	cncnpi 22666	A continuous function is continuous at all points. One direction of Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
Theorem	cnsscnp 22667	The set of continuous functions is a subset of the set of continuous functions at a point. (Contributed by Raph Levien, 21-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝑃 ∈ 𝑋 → (𝐽 Cn 𝐾) ⊆ ((𝐽 CnP 𝐾)‘𝑃))
Theorem	cncnp 22668*	A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 15-May-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
Theorem	cncnp2 22669*	A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ (𝑋 ≠ ∅ → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
Theorem	cnnei 22670*	Continuity in terms of neighborhoods. (Contributed by Thierry Arnoux, 3-Jan-2018.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑝 ∈ 𝑋 ∀𝑤 ∈ ((nei‘𝐾)‘{(𝐹‘𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹 “ 𝑣) ⊆ 𝑤))
Theorem	cnconst2 22671	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 22672	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 22673	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 22674	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 22675	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	cnpresti 22676	One direction of cnprest 22677 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 Mario Carneiro, 1-May-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃))
Theorem	cnprest 22677	Equivalence of continuity at a point and continuity of the restricted function at a point. (Contributed by Mario Carneiro, 8-Aug-2014.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃)))
Theorem	cnprest2 22678	Equivalence of point-continuity in the parent topology and point-continuity in a subspace. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ ((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)))
Theorem	cndis 22679	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	cnindis 22680	Every function is continuous when the codomain is indiscrete (trivial). (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐽 Cn {∅, 𝐴}) = (𝐴 ↑m 𝑋))
Theorem	cnpdis 22681	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	paste 22682	Pasting lemma. If 𝐴 and 𝐵 are closed sets in 𝑋 with 𝐴 ∪ 𝐵 = 𝑋, then any function whose restrictions to 𝐴 and 𝐵 are continuous is continuous on all of 𝑋. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽))    &   ⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽))    &   ⊢ (𝜑 → (𝐴 ∪ 𝐵) = 𝑋)    &   ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)    &   ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾))    &   ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ ((𝐽 ↾t 𝐵) Cn 𝐾))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
Theorem	lmfpm 22683	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 22684	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 22685	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 22686	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 22687	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 22688	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 22689*	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	lmcls 22690*	Any convergent sequence of points in a subset of a topological space converges to a point in the closure of the subset. (Contributed by Mario Carneiro, 30-Dec-2013.) (Revised by Mario Carneiro, 1-May-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → 𝑃 ∈ ((cls‘𝐽)‘𝑆))
Theorem	lmcld 22691*	Any convergent sequence of points in a closed subset of a topological space converges to a point in the set. (Contributed by Mario Carneiro, 30-Dec-2013.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ 𝑆)    &   ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽))    ⇒   ⊢ (𝜑 → 𝑃 ∈ 𝑆)
Theorem	lmcnp 22692	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 CnP 𝐾)‘𝑃))    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹)(⇝𝑡‘𝐾)(𝐺‘𝑃))
Theorem	lmcn 22693	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))
12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...
Syntax	ct0 22694	Extend class notation with the class of all T0 spaces.
class Kol2
Syntax	ct1 22695	Extend class notation to include T1 spaces (also called Fréchet spaces).
class Fre
Syntax	cha 22696	Extend class notation with the class of all Hausdorff spaces.
class Haus
Syntax	creg 22697	Extend class notation with the class of all regular topologies.
class Reg
Syntax	cnrm 22698	Extend class notation with the class of all normal topologies.
class Nrm
Syntax	ccnrm 22699	Extend class notation with the class of all completely normal topologies.
class CNrm
Syntax	cpnrm 22700	Extend class notation with the class of all perfectly normal topologies.
class PNrm