P. 142 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14101-14200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iscn 14101*	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 14102*	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 14103*	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 14104*	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	cntop1 14105	Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
Theorem	cntop2 14106	Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
Theorem	iscnp3 14107*	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	cnf 14108	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	cnf2 14109	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	cnprcl2k 14110	Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ 𝑋)
Theorem	cnpf2 14111	A continuous function at point 𝑃 is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌)
Theorem	tgcn 14112*	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 14113*	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	ssidcn 14114	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	icnpimaex 14115*	Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.) (Revised by Jim Kingdon, 28-Mar-2023.)
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝐴)) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝐴))
Theorem	idcn 14116	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 14117*	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 us to use objects more general than sequences when convenient; see the comment in df-lm 14094. (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 14118*	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 14119*	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 14118 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 14120	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 14121*	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 14122*	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 14123*	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 14124	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 14125	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	cnptopco 14126	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.)
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐿 ∈ Top) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))) → (𝐺 ∘ 𝐹) ∈ ((𝐽 CnP 𝐿)‘𝑃))
Theorem	cnclima 14127	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	cnntri 14128	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	cnntr 14129*	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 14130	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 14131	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 14132	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 14133	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 14134*	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	cncnp2m 14135*	A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Jim Kingdon, 30-Mar-2023.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦 ∈ 𝑋) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
Theorem	cnnei 14136*	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 14137	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 14138	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 14139	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 14140	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 14141	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 14142	One direction of cnptoprest 14143 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 14143	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 14144	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 14145	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 14146	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 14147	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 14148	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 14149	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 14150	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 14151	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 14152	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 14153*	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 14154	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 14155	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))
8.1.8  Product topologies
Syntax	ctx 14156	Extend class notation with the binary topological product operation.
class ×t
Definition	df-tx 14157*	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 14158	Existence of the binary topological product. If 𝑅 and 𝑆 are known to be topologies, see txtop 14164. (Contributed by Jim Kingdon, 3-Aug-2023.)
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) ∈ V)
Theorem	txval 14159*	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 14160*	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 14161*	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 14162*	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 14163*	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 14164	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 14165	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 14166	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 14167	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 14168	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 14169	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 14170	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 14171	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 14172	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 14173	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 14174	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 14175*	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 14176*	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 14177*	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 14178*	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 14179	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 14180	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 14181	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 14182*	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 14183*	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 14184*	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))
8.1.9  Continuous function-builders
Theorem	cnmptid 14185*	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 14186*	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 14187*	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 14188*	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 14189*	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 14190*	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 14191*	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 14192*	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 14193*	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 14194*	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 14195*	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 14196*	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 14197*	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 14198*	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 14199*	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 14200*	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))