P. 54 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5301-5400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fnssres 5301	Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵)
Theorem	fnresin1 5302	Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵))
Theorem	fnresin2 5303	Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐵 ∩ 𝐴)) Fn (𝐵 ∩ 𝐴))
Theorem	fnres 5304*	An equivalence for functionality of a restriction. Compare dffun8 5216. (Contributed by Mario Carneiro, 20-May-2015.)
⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦)
Theorem	fnresi 5305	Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
⊢ ( I ↾ 𝐴) Fn 𝐴
Theorem	fnima 5306	The image of a function's domain is its range. (Contributed by NM, 4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)
Theorem	fn0 5307	A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
Theorem	fnimadisj 5308	A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.)
⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 “ 𝐶) = ∅)
Theorem	fnimaeq0 5309	Images under a function never map nonempty sets to empty sets. (Contributed by Stefan O'Rear, 21-Jan-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝐹 “ 𝐵) = ∅ ↔ 𝐵 = ∅))
Theorem	dfmpt3 5310	Alternate definition for the maps-to notation df-mpt 4045. (Contributed by Mario Carneiro, 30-Dec-2016.)
⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵})
Theorem	fnopabg 5311*	Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ↔ 𝐹 Fn 𝐴)
Theorem	fnopab 5312*	Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.)
⊢ (𝑥 ∈ 𝐴 → ∃!𝑦𝜑)    &   ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}    ⇒   ⊢ 𝐹 Fn 𝐴
Theorem	mptfng 5313*	The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴)
Theorem	fnmpt 5314*	The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴)
Theorem	mpt0 5315	A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ (𝑥 ∈ ∅ ↦ 𝐴) = ∅
Theorem	fnmpti 5316*	Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ 𝐹 Fn 𝐴
Theorem	dmmpti 5317*	Domain of an ordered-pair class abstraction that specifies a function. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ dom 𝐹 = 𝐴
Theorem	dmmptd 5318*	The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → dom 𝐴 = 𝐵)
Theorem	mptun 5319	Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ∪ (𝑥 ∈ 𝐵 ↦ 𝐶))
Theorem	feq1 5320	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹 = 𝐺 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵))
Theorem	feq2 5321	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐴 = 𝐵 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶))
Theorem	feq3 5322	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐴 = 𝐵 → (𝐹:𝐶⟶𝐴 ↔ 𝐹:𝐶⟶𝐵))
Theorem	feq23 5323	Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷))
Theorem	feq1d 5324	Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
⊢ (𝜑 → 𝐹 = 𝐺)    ⇒   ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵))
Theorem	feq2d 5325	Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶))
Theorem	feq3d 5326	Equality deduction for functions. (Contributed by AV, 1-Jan-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹:𝑋⟶𝐴 ↔ 𝐹:𝑋⟶𝐵))
Theorem	feq12d 5327	Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶))
Theorem	feq123d 5328	Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐷))
Theorem	feq123 5329	Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐶⟶𝐷))
Theorem	feq1i 5330	Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 𝐹 = 𝐺    ⇒   ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵)
Theorem	feq2i 5331	Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶)
Theorem	feq23i 5332	Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)
Theorem	feq23d 5333	Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷))
Theorem	nff 5334	Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐹:𝐴⟶𝐵
Theorem	sbcfng 5335*	Distribute proper substitution through the function predicate with a domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴 ↔ ⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴))
Theorem	sbcfg 5336*	Distribute proper substitution through the function predicate with domain and codomain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹:𝐴⟶𝐵 ↔ ⦋𝑋 / 𝑥⦌𝐹:⦋𝑋 / 𝑥⦌𝐴⟶⦋𝑋 / 𝑥⦌𝐵))
Theorem	ffn 5337	A mapping is a function. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
Theorem	ffnd 5338	A mapping is a function with domain, deduction form. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → 𝐹 Fn 𝐴)
Theorem	dffn2 5339	Any function is a mapping into V. (Contributed by NM, 31-Oct-1995.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V)
Theorem	ffun 5340	A mapping is a function. (Contributed by NM, 3-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
Theorem	ffund 5341	A mapping is a function, deduction version. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → Fun 𝐹)
Theorem	frel 5342	A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹)
Theorem	fdm 5343	The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
Theorem	fdmd 5344	Deduction form of fdm 5343. The domain of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → dom 𝐹 = 𝐴)
Theorem	fdmi 5345	The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
⊢ 𝐹:𝐴⟶𝐵    ⇒   ⊢ dom 𝐹 = 𝐴
Theorem	frn 5346	The range of a mapping. (Contributed by NM, 3-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
Theorem	frnd 5347	Deduction form of frn 5346. The range of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
Theorem	dffn3 5348	A function maps to its range. (Contributed by NM, 1-Sep-1999.)
⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
Theorem	fss 5349	Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶)
Theorem	fssd 5350	Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
Theorem	fssdmd 5351	Expressing that a class is a subclass of the domain of a function expressed in maps-to notation, deduction form. (Contributed by AV, 21-Aug-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐷 ⊆ dom 𝐹)    ⇒   ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
Theorem	fssdm 5352	Expressing that a class is a subclass of the domain of a function expressed in maps-to notation, semi-deduction form. (Contributed by AV, 21-Aug-2022.)
⊢ 𝐷 ⊆ dom 𝐹    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
Theorem	fco 5353	Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
Theorem	fco2 5354	Functionality of a composition with weakened out of domain condition on the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.)
⊢ (((𝐹 ↾ 𝐵):𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
Theorem	fssxp 5355	A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
Theorem	fex2 5356	A function with bounded domain and range is a set. This version is proven without the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
Theorem	funssxp 5357	Two ways of specifying a partial function from 𝐴 to 𝐵. (Contributed by NM, 13-Nov-2007.)
⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))
Theorem	ffdm 5358	A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))
Theorem	opelf 5359	The members of an ordered pair element of a mapping belong to the mapping's domain and codomain. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝐶, 𝐷⟩ ∈ 𝐹) → (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))
Theorem	fun 5360	The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷))
Theorem	fun2 5361	The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)
⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)
Theorem	fnfco 5362	Composition of two functions. (Contributed by NM, 22-May-2006.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)
Theorem	fssres 5363	Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
Theorem	fssresd 5364	Restriction of a function with a subclass of its domain, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
Theorem	fssres2 5365	Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)
⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
Theorem	fresin 5366	An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)
⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ 𝑋):(𝐴 ∩ 𝑋)⟶𝐵)
Theorem	resasplitss 5367	If two functions agree on their common domain, their union contains a union of three functions with pairwise disjoint domains. If we assumed the law of the excluded middle, this would be equality rather than subset. (Contributed by Jim Kingdon, 28-Dec-2018.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) ⊆ (𝐹 ∪ 𝐺))
Theorem	fcoi1 5368	Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
Theorem	fcoi2 5369	Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ (𝐹:𝐴⟶𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)
Theorem	feu 5370*	There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 ⟨𝐶, 𝑦⟩ ∈ 𝐹)
Theorem	fcnvres 5371	The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
⊢ (𝐹:𝐴⟶𝐵 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ 𝐵))
Theorem	fimacnvdisj 5372	The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (◡𝐹 “ 𝐶) = ∅)
Theorem	fintm 5373*	Function into an intersection. (Contributed by Jim Kingdon, 28-Dec-2018.)
⊢ ∃𝑥 𝑥 ∈ 𝐵    ⇒   ⊢ (𝐹:𝐴⟶∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐹:𝐴⟶𝑥)
Theorem	fin 5374	Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ (𝐹:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝐹:𝐴⟶𝐵 ∧ 𝐹:𝐴⟶𝐶))
Theorem	fabexg 5375*	Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)
⊢ 𝐹 = {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)}    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V)
Theorem	fabex 5376*	Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐹 = {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)}    ⇒   ⊢ 𝐹 ∈ V
Theorem	dmfex 5377	If a mapping is a set, its domain is a set. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ ((𝐹 ∈ 𝐶 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V)
Theorem	f0 5378	The empty function. (Contributed by NM, 14-Aug-1999.)
⊢ ∅:∅⟶𝐴
Theorem	f00 5379	A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.)
⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Theorem	f0bi 5380	A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅)
Theorem	f0dom0 5381	A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.)
⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅))
Theorem	f0rn0 5382*	If there is no element in the range of a function, its domain must be empty. (Contributed by Alexander van der Vekens, 12-Jul-2018.)
⊢ ((𝐸:𝑋⟶𝑌 ∧ ¬ ∃𝑦 ∈ 𝑌 𝑦 ∈ ran 𝐸) → 𝑋 = ∅)
Theorem	fconst 5383	A cross product with a singleton is a constant function. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 × {𝐵}):𝐴⟶{𝐵}
Theorem	fconstg 5384	A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
Theorem	fnconstg 5385	A cross product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}) Fn 𝐴)
Theorem	fconst6g 5386	Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}):𝐴⟶𝐶)
Theorem	fconst6 5387	A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)
⊢ 𝐵 ∈ 𝐶    ⇒   ⊢ (𝐴 × {𝐵}):𝐴⟶𝐶
Theorem	f1eq1 5388	Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1→𝐵 ↔ 𝐺:𝐴–1-1→𝐵))
Theorem	f1eq2 5389	Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐹:𝐵–1-1→𝐶))
Theorem	f1eq3 5390	Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵))
Theorem	nff1 5391	Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐹:𝐴–1-1→𝐵
Theorem	dff12 5392*	Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)
⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦))
Theorem	f1f 5393	A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
Theorem	f1rn 5394	The range of a one-to-one mapping. (Contributed by BJ, 6-Jul-2022.)
⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵)
Theorem	f1fn 5395	A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.)
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
Theorem	f1fun 5396	A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹)
Theorem	f1rel 5397	A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
⊢ (𝐹:𝐴–1-1→𝐵 → Rel 𝐹)
Theorem	f1dm 5398	The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴)
Theorem	f1ss 5399	A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Mario Carneiro, 12-Jan-2013.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴–1-1→𝐶)
Theorem	f1ssr 5400	Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹:𝐴–1-1→𝐶)