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Theorem List for Metamath Proof Explorer - 5901-6000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnffn 5901 Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
𝑥𝐹    &   𝑥𝐴       𝑥 𝐹 Fn 𝐴

Theoremfnfun 5902 A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
(𝐹 Fn 𝐴 → Fun 𝐹)

Theoremfnrel 5903 A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
(𝐹 Fn 𝐴 → Rel 𝐹)

Theoremfndm 5904 The domain of a function. (Contributed by NM, 2-Aug-1994.)
(𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)

Theoremfunfni 5905 Inference to convert a function and domain antecedent. (Contributed by NM, 22-Apr-2004.)
((Fun 𝐹𝐵 ∈ dom 𝐹) → 𝜑)       ((𝐹 Fn 𝐴𝐵𝐴) → 𝜑)

Theoremfndmu 5906 A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
((𝐹 Fn 𝐴𝐹 Fn 𝐵) → 𝐴 = 𝐵)

Theoremfnbr 5907 The first argument of binary relation on a function belongs to the function's domain. (Contributed by NM, 7-May-2004.)
((𝐹 Fn 𝐴𝐵𝐹𝐶) → 𝐵𝐴)

Theoremfnop 5908 The first argument of an ordered pair in a function belongs to the function's domain. (Contributed by NM, 8-Aug-1994.)
((𝐹 Fn 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝐹) → 𝐵𝐴)

Theoremfneu 5909* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦 𝐵𝐹𝑦)

Theoremfneu2 5910* There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)
((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦𝐵, 𝑦⟩ ∈ 𝐹)

Theoremfnun 5911 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
(((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵))

Theoremfnunsn 5912 Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
(𝜑𝑋 ∈ V)    &   (𝜑𝑌 ∈ V)    &   (𝜑𝐹 Fn 𝐷)    &   𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})    &   𝐸 = (𝐷 ∪ {𝑋})    &   (𝜑 → ¬ 𝑋𝐷)       (𝜑𝐺 Fn 𝐸)

Theoremfnco 5913 Composition of two functions. (Contributed by NM, 22-May-2006.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn 𝐵)

Theoremfnresdm 5914 A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
(𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)

Theoremfnresdisj 5915 A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)
(𝐹 Fn 𝐴 → ((𝐴𝐵) = ∅ ↔ (𝐹𝐵) = ∅))

Theorem2elresin 5916 Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)))))

Theoremfnssresb 5917 Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)
(𝐹 Fn 𝐴 → ((𝐹𝐵) Fn 𝐵𝐵𝐴))

Theoremfnssres 5918 Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)

Theoremfnresin1 5919 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
(𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))

Theoremfnresin2 5920 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
(𝐹 Fn 𝐴 → (𝐹 ↾ (𝐵𝐴)) Fn (𝐵𝐴))

Theoremfnres 5921* An equivalence for functionality of a restriction. Compare dffun8 5831. (Contributed by Mario Carneiro, 20-May-2015.)
((𝐹𝐴) Fn 𝐴 ↔ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦)

Theoremfnresi 5922 Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
( I ↾ 𝐴) Fn 𝐴

Theoremfnima 5923 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 → (𝐹𝐴) = ran 𝐹)

Theoremfn0 5924 A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹 Fn ∅ ↔ 𝐹 = ∅)

Theoremfnimadisj 5925 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 ∧ (𝐴𝐶) = ∅) → (𝐹𝐶) = ∅)

Theoremfnimaeq0 5926 Images under a function never map nonempty sets to empty sets. EDITORIAL: usable in fnwe2lem2 36639. (Contributed by Stefan O'Rear, 21-Jan-2015.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = ∅ ↔ 𝐵 = ∅))

Theoremdfmpt3 5927 Alternate definition for the "maps to" notation df-mpt 4645. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝑥𝐴𝐵) = 𝑥𝐴 ({𝑥} × {𝐵})

Theoremmptfnf 5928 The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Thierry Arnoux, 10-May-2017.)
𝑥𝐴       (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)

Theoremfnmptf 5929 The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)
𝑥𝐴       (∀𝑥𝐴 𝐵𝑉 → (𝑥𝐴𝐵) Fn 𝐴)

Theoremfnopabg 5930* 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⟩ ∣ (𝑥𝐴𝜑)}       (∀𝑥𝐴 ∃!𝑦𝜑𝐹 Fn 𝐴)

Theoremfnopab 5931* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.)
(𝑥𝐴 → ∃!𝑦𝜑)    &   𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}       𝐹 Fn 𝐴

Theoremmptfng 5932* The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.)
𝐹 = (𝑥𝐴𝐵)       (∀𝑥𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴)

Theoremfnmpt 5933* The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.)
𝐹 = (𝑥𝐴𝐵)       (∀𝑥𝐴 𝐵𝑉𝐹 Fn 𝐴)

Theoremmpt0 5934 A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝑥 ∈ ∅ ↦ 𝐴) = ∅

Theoremfnmpti 5935* 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 = (𝑥𝐴𝐵)       𝐹 Fn 𝐴

Theoremdmmpti 5936* Domain of the mapping operation. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐵 ∈ V    &   𝐹 = (𝑥𝐴𝐵)       dom 𝐹 = 𝐴

Theoremdmmptd 5937* The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴 = (𝑥𝐵𝐶)    &   ((𝜑𝑥𝐵) → 𝐶𝑉)       (𝜑 → dom 𝐴 = 𝐵)

Theoremmptun 5938 Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝑥 ∈ (𝐴𝐵) ↦ 𝐶) = ((𝑥𝐴𝐶) ∪ (𝑥𝐵𝐶))

Theoremfeq1 5939 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹:𝐴𝐵𝐺:𝐴𝐵))

Theoremfeq2 5940 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹:𝐴𝐶𝐹:𝐵𝐶))

Theoremfeq3 5941 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹:𝐶𝐴𝐹:𝐶𝐵))

Theoremfeq23 5942 Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐴 = 𝐶𝐵 = 𝐷) → (𝐹:𝐴𝐵𝐹:𝐶𝐷))

Theoremfeq1d 5943 Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
(𝜑𝐹 = 𝐺)       (𝜑 → (𝐹:𝐴𝐵𝐺:𝐴𝐵))

Theoremfeq2d 5944 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹:𝐴𝐶𝐹:𝐵𝐶))

Theoremfeq3d 5945 Equality deduction for functions. (Contributed by AV, 1-Jan-2020.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹:𝑋𝐴𝐹:𝑋𝐵))

Theoremfeq12d 5946 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐶))

Theoremfeq123d 5947 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐷))

Theoremfeq123 5948 Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → (𝐹:𝐴𝐵𝐺:𝐶𝐷))

Theoremfeq1i 5949 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐹 = 𝐺       (𝐹:𝐴𝐵𝐺:𝐴𝐵)

Theoremfeq2i 5950 Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
𝐴 = 𝐵       (𝐹:𝐴𝐶𝐹:𝐵𝐶)

Theoremfeq12i 5951 Equality inference for functions. (Contributed by AV, 7-Feb-2021.)
𝐹 = 𝐺    &   𝐴 = 𝐵       (𝐹:𝐴𝐶𝐺:𝐵𝐶)

Theoremfeq23i 5952 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐴 = 𝐶    &   𝐵 = 𝐷       (𝐹:𝐴𝐵𝐹:𝐶𝐷)

Theoremfeq23d 5953 Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
(𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → (𝐹:𝐴𝐵𝐹:𝐶𝐷))

Theoremnff 5954 Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐹    &   𝑥𝐴    &   𝑥𝐵       𝑥 𝐹:𝐴𝐵

Theoremsbcfng 5955* Distribute proper substitution through the function predicate with a domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
(𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))

Theoremsbcfg 5956* Distribute proper substitution through the function predicate with domain and codomain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
(𝑋𝑉 → ([𝑋 / 𝑥]𝐹:𝐴𝐵𝑋 / 𝑥𝐹:𝑋 / 𝑥𝐴𝑋 / 𝑥𝐵))

Theoremelimf 5957 Eliminate a mapping hypothesis for the weak deduction theorem dedth 4089, when a special case 𝐺:𝐴𝐵 is provable, in order to convert 𝐹:𝐴𝐵 from a hypothesis to an antecedent. (Contributed by NM, 24-Aug-2006.)
𝐺:𝐴𝐵       if(𝐹:𝐴𝐵, 𝐹, 𝐺):𝐴𝐵

Theoremffn 5958 A mapping is a function with domain. (Contributed by NM, 2-Aug-1994.)
(𝐹:𝐴𝐵𝐹 Fn 𝐴)

Theoremffnd 5959 A mapping is a function with domain, deduction form. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐹:𝐴𝐵)       (𝜑𝐹 Fn 𝐴)

Theoremdffn2 5960 Any function is a mapping into V. (Contributed by NM, 31-Oct-1995.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹 Fn 𝐴𝐹:𝐴⟶V)

Theoremffun 5961 A mapping is a function. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴𝐵 → Fun 𝐹)

Theoremffund 5962 A mapping is a function, deduction version. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → Fun 𝐹)

Theoremfrel 5963 A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴𝐵 → Rel 𝐹)

Theoremfdm 5964 The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
(𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)

Theoremfdmi 5965 The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
𝐹:𝐴𝐵       dom 𝐹 = 𝐴

Theoremfrn 5966 The range of a mapping. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴𝐵 → ran 𝐹𝐵)

Theoremdffn3 5967 A function maps to its range. (Contributed by NM, 1-Sep-1999.)
(𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)

Theoremffrn 5968 A function maps to its range. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝐹:𝐴𝐵𝐹:𝐴⟶ran 𝐹)

Theoremfss 5969 Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹:𝐴𝐵𝐵𝐶) → 𝐹:𝐴𝐶)

Theoremfssd 5970 Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐹:𝐴𝐶)

Theoremfco 5971 Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹:𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)

Theoremfco2 5972 Functionality of a composition with weakened out of domain condition on the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.)
(((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)

Theoremfssxp 5973 A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))

Theoremfunssxp 5974 Two ways of specifying a partial function from 𝐴 to 𝐵. (Contributed by NM, 13-Nov-2007.)
((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))

Theoremffdm 5975 A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
(𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))

Theoremffdmd 5976 The domain of a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑𝐹:dom 𝐹𝐵)

Theoremfdmrn 5977 A different way to write 𝐹 is a function. (Contributed by Thierry Arnoux, 7-Dec-2016.)
(Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)

Theoremopelf 5978 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:𝐴𝐵 ∧ ⟨𝐶, 𝐷⟩ ∈ 𝐹) → (𝐶𝐴𝐷𝐵))

Theoremfun 5979 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
(((𝐹:𝐴𝐶𝐺:𝐵𝐷) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷))

Theoremfun2 5980 The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)
(((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)

Theoremfun2d 5981 The union of functions with disjoint domains is a function, deduction version of fun2 5980. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.)
(𝜑𝐹:𝐴𝐶)    &   (𝜑𝐺:𝐵𝐶)    &   (𝜑 → (𝐴𝐵) = ∅)       (𝜑 → (𝐹𝐺):(𝐴𝐵)⟶𝐶)

Theoremfnfco 5982 Composition of two functions. (Contributed by NM, 22-May-2006.)
((𝐹 Fn 𝐴𝐺:𝐵𝐴) → (𝐹𝐺) Fn 𝐵)

Theoremfssres 5983 Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)
((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)

Theoremfssresd 5984 Restriction of a function with a subclass of its domain, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → (𝐹𝐶):𝐶𝐵)

Theoremfssres2 5985 Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)
(((𝐹𝐴):𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)

Theoremfresin 5986 An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)
(𝐹:𝐴𝐵 → (𝐹𝑋):(𝐴𝑋)⟶𝐵)

Theoremresasplit 5987 If two functions agree on their common domain, express their union as a union of three functions with pairwise disjoint domains. (Contributed by Stefan O'Rear, 9-Oct-2014.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺) = ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))))

Theoremfresaun 5988 The union of two functions which agree on their common domain is a function. (Contributed by Stefan O'Rear, 9-Oct-2014.)
((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)

Theoremfresaunres2 5989 From the union of two functions that agree on the domain overlap, either component can be recovered by restriction. (Contributed by Stefan O'Rear, 9-Oct-2014.)
((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)

Theoremfresaunres1 5990 From the union of two functions that agree on the domain overlap, either component can be recovered by restriction. (Contributed by Mario Carneiro, 16-Feb-2015.)
((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐴) = 𝐹)

Theoremfcoi1 5991 Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)

Theoremfcoi2 5992 Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)

Theoremfeu 5993* There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)
((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦𝐵𝐶, 𝑦⟩ ∈ 𝐹)

Theoremfimass 5994 The image of a class is a subset of its codomain. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐹:𝐴𝐵 → (𝐹𝑋) ⊆ 𝐵)

Theoremfcnvres 5995 The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
(𝐹:𝐴𝐵(𝐹𝐴) = (𝐹𝐵))

Theoremfimacnvdisj 5996 The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)
((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐹𝐶) = ∅)

Theoremfint 5997* Function into an intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
𝐵 ≠ ∅       (𝐹:𝐴 𝐵 ↔ ∀𝑥𝐵 𝐹:𝐴𝑥)

Theoremfin 5998 Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴⟶(𝐵𝐶) ↔ (𝐹:𝐴𝐵𝐹:𝐴𝐶))

Theoremf0 5999 The empty function. (Contributed by NM, 14-Aug-1999.)
∅:∅⟶𝐴

Theoremf00 6000 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 = ∅))

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