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Theorem List for Metamath Proof Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremidref 6901* Two ways to state that a relation is reflexive on a class. (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Revised by NM, 30-Mar-2016.)
(( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥𝐴 𝑥𝑅𝑥)

Theoremfuniun 6902* A function is a union of singletons of ordered pairs indexed by its domain. (Contributed by AV, 18-Sep-2020.)
(Fun 𝐹𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩})

Theoremfunopsn 6903* If a function is an ordered pair then it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) (Proof shortened by AV, 15-Jul-2021.) A function is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a function is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is funsng 6395, as relsnopg 5664 is to relop 5709. (New usage is discouraged.)
𝑋 ∈ V    &   𝑌 ∈ V       ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))

Theoremfunop 6904* An ordered pair is a function iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) A function is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a function is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is funsng 6395, as relsnopg 5664 is to relop 5709. (New usage is discouraged.)
𝑋 ∈ V    &   𝑌 ∈ V       (Fun ⟨𝑋, 𝑌⟩ ↔ ∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}))

Theoremfunopdmsn 6905 The domain of a function which is an ordered pair is a singleton. (Contributed by AV, 15-Nov-2021.) (Avoid depending on this detail.)
𝐺 = ⟨𝑋, 𝑌    &   𝑋𝑉    &   𝑌𝑊       ((Fun 𝐺𝐴 ∈ dom 𝐺𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵)

Theoremfunsndifnop 6906 A singleton of an ordered pair is not an ordered pair if the components are different. (Contributed by AV, 23-Sep-2020.) (Avoid depending on this detail.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐺 = {⟨𝐴, 𝐵⟩}       (𝐴𝐵 → ¬ 𝐺 ∈ (V × V))

Theoremfunsneqopb 6907 A singleton of an ordered pair is an ordered pair iff the components are equal. (Contributed by AV, 24-Sep-2020.) (Avoid depending on this detail.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐺 = {⟨𝐴, 𝐵⟩}       (𝐴 = 𝐵𝐺 ∈ (V × V))

Theoremressnop0 6908 If 𝐴 is not in 𝐶, then the restriction of a singleton of 𝐴, 𝐵 to 𝐶 is null. (Contributed by Scott Fenton, 15-Apr-2011.)
𝐴𝐶 → ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅)

Theoremfpr 6909 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       (𝐴𝐵 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}⟶{𝐶, 𝐷})

Theoremfprg 6910 A function with a domain of two elements. (Contributed by FL, 2-Feb-2014.)
(((𝐴𝐸𝐵𝐹) ∧ (𝐶𝐺𝐷𝐻) ∧ 𝐴𝐵) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}⟶{𝐶, 𝐷})

Theoremftpg 6911 A function with a domain of three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
(((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴𝐹𝐵𝐺𝐶𝐻) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩}:{𝑋, 𝑌, 𝑍}⟶{𝐴, 𝐵, 𝐶})

Theoremftp 6912 A function with a domain of three elements. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by Alexander van der Vekens, 23-Jan-2018.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝑋 ∈ V    &   𝑌 ∈ V    &   𝑍 ∈ V    &   𝐴𝐵    &   𝐴𝐶    &   𝐵𝐶       {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩, ⟨𝐶, 𝑍⟩}:{𝐴, 𝐵, 𝐶}⟶{𝑋, 𝑌, 𝑍}

Theoremfnressn 6913 A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩})

Theoremfunressn 6914 A function restricted to a singleton. (Contributed by Mario Carneiro, 16-Nov-2014.)
(Fun 𝐹 → (𝐹 ↾ {𝐵}) ⊆ {⟨𝐵, (𝐹𝐵)⟩})

Theoremfressnfv 6915 The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹𝐵) ∈ 𝐶))

Theoremfvrnressn 6916 If the value of a function is in the range of the function restricted to the singleton containing the argument, then the value of the function is in the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
(𝑋𝑉 → ((𝐹𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹𝑋) ∈ ran 𝐹))

Theoremfvressn 6917 The value of a function restricted to the singleton containing the argument equals the value of the function for this argument. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
(𝑋𝑉 → ((𝐹 ↾ {𝑋})‘𝑋) = (𝐹𝑋))

Theoremfvn0fvelrn 6918 If the value of a function is not null, the value is an element of the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
((𝐹𝑋) ≠ ∅ → (𝐹𝑋) ∈ ran 𝐹)

Theoremfvconst 6919 The value of a constant function. (Contributed by NM, 30-May-1999.)
((𝐹:𝐴⟶{𝐵} ∧ 𝐶𝐴) → (𝐹𝐶) = 𝐵)

Theoremfnsnr 6920 If a class belongs to a function on a singleton, then that class is the obvious ordered pair. Note that this theorem also holds when 𝐴 is a proper class, but its meaning is then different. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.)
(𝐹 Fn {𝐴} → (𝐵𝐹𝐵 = ⟨𝐴, (𝐹𝐴)⟩))

Theoremfnsnb 6921 A function whose domain is a singleton can be represented as a singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Revised to add reverse implication. (Revised by NM, 29-Dec-2018.)
𝐴 ∈ V       (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})

Theoremfmptsn 6922* Express a singleton function in maps-to notation. (Contributed by NM, 6-Jun-2006.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 28-Feb-2015.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))

Theoremfmptsng 6923* Express a singleton function in maps-to notation. Version of fmptsn 6922 allowing the value 𝐵 to depend on the variable 𝑥. (Contributed by AV, 27-Feb-2019.)
(𝑥 = 𝐴𝐵 = 𝐶)       ((𝐴𝑉𝐶𝑊) → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))

Theoremfmptsnd 6924* Express a singleton function in maps-to notation. Deduction form of fmptsng 6923. (Contributed by AV, 4-Aug-2019.)
((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))

Theoremfmptap 6925* Append an additional value to a function. (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑅 ∪ {𝐴}) = 𝑆    &   (𝑥 = 𝐴𝐶 = 𝐵)       ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶)

Theoremfmptapd 6926* Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)    &   ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐵)       (𝜑 → ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶))

Theoremfmptpr 6927* Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)    &   ((𝜑𝑥 = 𝐴) → 𝐸 = 𝐶)    &   ((𝜑𝑥 = 𝐵) → 𝐸 = 𝐷)       (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸))

Theoremfvresi 6928 The value of a restricted identity function. (Contributed by NM, 19-May-2004.)
(𝐵𝐴 → (( I ↾ 𝐴)‘𝐵) = 𝐵)

Theoremfninfp 6929* Express the class of fixed points of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥𝐴 ∣ (𝐹𝑥) = 𝑥})

Theoremfnelfp 6930 Property of a fixed point of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
((𝐹 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹 ∩ I ) ↔ (𝐹𝑋) = 𝑋))

Theoremfndifnfp 6931* Express the class of non-fixed points of a function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥})

Theoremfnelnfp 6932 Property of a non-fixed point of a function. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐹 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑋) ≠ 𝑋))

Theoremfnnfpeq0 6933 A function is the identity iff it moves no points. (Contributed by Stefan O'Rear, 25-Aug-2015.)
(𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ 𝐹 = ( I ↾ 𝐴)))

Theoremfvunsn 6934 Remove an ordered pair not participating in a function value. (Contributed by NM, 1-Oct-2013.) (Revised by Mario Carneiro, 28-May-2014.)
(𝐵𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = (𝐴𝐷))

Theoremfvsng 6935 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.) (Proof shortened by BJ, 25-Feb-2023.)
((𝐴𝑉𝐵𝑊) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)

Theoremfvsn 6936 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.) (Proof shortened by BJ, 25-Feb-2023.)
𝐴 ∈ V    &   𝐵 ∈ V       ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵

Theoremfvsnun1 6937 The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 6938. (Contributed by NM, 23-Sep-2007.) Put in deduction form. (Revised by BJ, 25-Feb-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))       (𝜑 → (𝐺𝐴) = 𝐵)

Theoremfvsnun2 6938 The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 6937. (Contributed by NM, 23-Sep-2007.) Put in deduction form. (Revised by BJ, 25-Feb-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))    &   (𝜑𝐷 ∈ (𝐶 ∖ {𝐴}))       (𝜑 → (𝐺𝐷) = (𝐹𝐷))

Theoremfnsnsplit 6939 Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.)
((𝐹 Fn 𝐴𝑋𝐴) → 𝐹 = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))

Theoremfsnunf 6940 Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇)

Theoremfsnunf2 6941 Adjoining a point to a punctured function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):𝑆𝑇)

Theoremfsnunfv 6942 Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.)
((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋) = 𝑌)

Theoremfsnunres 6943 Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = 𝐹)

Theoremfunresdfunsn 6944 Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself. (Contributed by AV, 2-Dec-2018.)
((Fun 𝐹𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}) = 𝐹)

Theoremfvpr1 6945 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
𝐴 ∈ V    &   𝐶 ∈ V       (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)

Theoremfvpr2 6946 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
𝐵 ∈ V    &   𝐷 ∈ V       (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷)

Theoremfvpr1g 6947 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
((𝐴𝑉𝐶𝑊𝐴𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)

Theoremfvpr2g 6948 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
((𝐵𝑉𝐷𝑊𝐴𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷)

Theoremfprb 6949* A condition for functionhood over a pair. (Contributed by Scott Fenton, 16-Sep-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵 → (𝐹:{𝐴, 𝐵}⟶𝑅 ↔ ∃𝑥𝑅𝑦𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}))

Theoremfvtp1 6950 The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
𝐴 ∈ V    &   𝐷 ∈ V       ((𝐴𝐵𝐴𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐴) = 𝐷)

Theoremfvtp2 6951 The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
𝐵 ∈ V    &   𝐸 ∈ V       ((𝐴𝐵𝐵𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐵) = 𝐸)

Theoremfvtp3 6952 The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
𝐶 ∈ V    &   𝐹 ∈ V       ((𝐴𝐶𝐵𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐶) = 𝐹)

Theoremfvtp1g 6953 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
(((𝐴𝑉𝐷𝑊) ∧ (𝐴𝐵𝐴𝐶)) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐴) = 𝐷)

Theoremfvtp2g 6954 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
(((𝐵𝑉𝐸𝑊) ∧ (𝐴𝐵𝐵𝐶)) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐵) = 𝐸)

Theoremfvtp3g 6955 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
(((𝐶𝑉𝐹𝑊) ∧ (𝐴𝐶𝐵𝐶)) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐶) = 𝐹)

Theoremtpres 6956 An unordered triple of ordered pairs restricted to all but one first components of the pairs is an unordered pair of ordered pairs. (Contributed by AV, 14-Mar-2020.)
(𝜑𝑇 = {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐸𝑉)    &   (𝜑𝐹𝑉)    &   (𝜑𝐵𝐴)    &   (𝜑𝐶𝐴)       (𝜑 → (𝑇 ↾ (V ∖ {𝐴})) = {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})

Theoremfvconst2g 6957 The value of a constant function. (Contributed by NM, 20-Aug-2005.)
((𝐵𝐷𝐶𝐴) → ((𝐴 × {𝐵})‘𝐶) = 𝐵)

Theoremfconst2g 6958 A constant function expressed as a Cartesian product. (Contributed by NM, 27-Nov-2007.)
(𝐵𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))

Theoremfvconst2 6959 The value of a constant function. (Contributed by NM, 16-Apr-2005.)
𝐵 ∈ V       (𝐶𝐴 → ((𝐴 × {𝐵})‘𝐶) = 𝐵)

Theoremfconst2 6960 A constant function expressed as a Cartesian product. (Contributed by NM, 20-Aug-1999.)
𝐵 ∈ V       (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵}))

Theoremfconst5 6961 Two ways to express that a function is constant. (Contributed by NM, 27-Nov-2007.)
((𝐹 Fn 𝐴𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))

Theoremrnmptc 6962* Range of a constant function in maps-to notation. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Remove extra hypothesis. (Revised by SN, 17-Apr-2024.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐴 ≠ ∅)       (𝜑 → ran 𝐹 = {𝐵})

TheoremrnmptcOLD 6963* Obsolete version of rnmptc 6962 as of 17-Apr-2024. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝐶)    &   (𝜑𝐴 ≠ ∅)       (𝜑 → ran 𝐹 = {𝐵})

Theoremfnprb 6964 A function whose domain has at most two elements can be represented as a set of at most two ordered pairs. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Scott Fenton, 12-Oct-2017.) Eliminate unnecessary antecedent 𝐴𝐵. (Revised by NM, 29-Dec-2018.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})

Theoremfntpb 6965 A function whose domain has at most three elements can be represented as a set of at most three ordered pairs. (Contributed by AV, 26-Jan-2021.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩})

Theoremfnpr2g 6966 A function whose domain has at most two elements can be represented as a set of at most two ordered pairs. (Contributed by Thierry Arnoux, 12-Jul-2020.)
((𝐴𝑉𝐵𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))

Theoremfpr2g 6967 A function that maps a pair to a class is a pair of ordered pairs. (Contributed by Thierry Arnoux, 12-Jul-2020.)
((𝐴𝑉𝐵𝑊) → (𝐹:{𝐴, 𝐵}⟶𝐶 ↔ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})))

Theoremfconstfv 6968* A constant function expressed in terms of its functionality, domain, and value. See also fconst2 6960. (Contributed by NM, 27-Aug-2004.) (Proof shortened by OpenAI, 25-Mar-2020.)
(𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵))

Theoremfconst3 6969 Two ways to express a constant function. (Contributed by NM, 15-Mar-2007.)
(𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))

Theoremfconst4 6970 Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
(𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (𝐹 “ {𝐵}) = 𝐴))

Theoremresfunexg 6971 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)

Theoremresiexd 6972 The restriction of the identity relation to a set is a set. (Contributed by AV, 15-Feb-2020.)
(𝜑𝐵𝑉)       (𝜑 → ( I ↾ 𝐵) ∈ V)

Theoremfnex 6973 If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 6971. See fnexALT 7649 for alternate proof. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)

Theoremfnexd 6974 If the domain of a function is a set, the function is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐴𝑉)       (𝜑𝐹 ∈ V)

Theoremfunex 6975 If the domain of a function exists, so does the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 6973. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995.)
((Fun 𝐹 ∧ dom 𝐹𝐵) → 𝐹 ∈ V)

Theoremopabex 6976* Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)
𝐴 ∈ V    &   (𝑥𝐴 → ∃*𝑦𝜑)       {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V

Theoremmptexg 6977* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)

Theoremmptexgf 6978 If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) (Revised by Thierry Arnoux, 17-May-2020.)
𝑥𝐴       (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)

Theoremmptex 6979* If the domain of a function given by maps-to notation is a set, the function is a set. Inference version of mptexg 6977. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.)
𝐴 ∈ V       (𝑥𝐴𝐵) ∈ V

Theoremmptexd 6980* If the domain of a function given by maps-to notation is a set, the function is a set. Deduction version of mptexg 6977. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴𝑉)       (𝜑 → (𝑥𝐴𝐵) ∈ V)

Theoremmptrabex 6981* If the domain of a function given by maps-to notation is a class abstraction based on a set, the function is a set. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
𝐴 ∈ V       (𝑥 ∈ {𝑦𝐴𝜑} ↦ 𝐵) ∈ V

Theoremfex 6982 If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)
((𝐹:𝐴𝐵𝐴𝐶) → 𝐹 ∈ V)

Theoremfexd 6983 If the domain of a mapping is a set, the function is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝐶)       (𝜑𝐹 ∈ V)

Theoremmptfvmpt 6984* A function in maps-to notation as the value of another function in maps-to notation. (Contributed by AV, 20-Aug-2022.)
(𝑦 = 𝑌𝑀 = (𝑥𝑉𝐴))    &   𝐺 = (𝑦𝑊𝑀)    &   𝑉 = (𝐹𝑋)       (𝑌𝑊 → (𝐺𝑌) = (𝑥𝑉𝐴))

Theoremeufnfv 6985* A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       ∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵)

Theoremfunfvima 6986 A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)
((Fun 𝐹𝐵 ∈ dom 𝐹) → (𝐵𝐴 → (𝐹𝐵) ∈ (𝐹𝐴)))

Theoremfunfvima2 6987 A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐵𝐴 → (𝐹𝐵) ∈ (𝐹𝐴)))

Theoremfunfvima2d 6988 A function's value in a preimage belongs to the image. (Contributed by Stanislas Polu, 9-Mar-2020.) (Revised by AV, 23-Mar-2024.)
(𝜑𝐹:𝐴𝐵)       ((𝜑𝑋𝐴) → (𝐹𝑋) ∈ (𝐹𝐴))

Theoremfnfvima 6989 The function value of an operand in a set is contained in the image of that set, using the Fn abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.)
((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → (𝐹𝑋) ∈ (𝐹𝑆))

Theoremfnfvimad 6990 A function's value belongs to the image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐵𝐴)    &   (𝜑𝐵𝐶)       (𝜑 → (𝐹𝐵) ∈ (𝐹𝐶))

Theoremresfvresima 6991 The value of the function value of a restriction for a function restricted to the image of the restricting subset. (Contributed by AV, 6-Mar-2021.)
(𝜑 → Fun 𝐹)    &   (𝜑𝑆 ⊆ dom 𝐹)    &   (𝜑𝑋𝑆)       (𝜑 → ((𝐻 ↾ (𝐹𝑆))‘((𝐹𝑆)‘𝑋)) = (𝐻‘(𝐹𝑋)))

Theoremfunfvima3 6992 A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.)
((Fun 𝐹𝐹𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ (𝐺 “ {𝐴})))

Theoremrexima 6993* Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
(𝑥 = (𝐹𝑦) → (𝜑𝜓))       ((𝐹 Fn 𝐴𝐵𝐴) → (∃𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∃𝑦𝐵 𝜓))

Theoremralima 6994* Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
(𝑥 = (𝐹𝑦) → (𝜑𝜓))       ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∀𝑦𝐵 𝜓))

Theoremfvclss 6995* Upper bound for the class of values of a class. (Contributed by NM, 9-Nov-1995.)
{𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ⊆ (ran 𝐹 ∪ {∅})

Theoremelabrex 6996* Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)
𝐵 ∈ V       (𝑥𝐴𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})

Theoremabrexco 6997* Composition of two image maps 𝐶(𝑦) and 𝐵(𝑤). (Contributed by NM, 27-May-2013.)
𝐵 ∈ V    &   (𝑦 = 𝐵𝐶 = 𝐷)       {𝑥 ∣ ∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶} = {𝑥 ∣ ∃𝑤𝐴 𝑥 = 𝐷}

Theoremimaiun 6998* The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝐴 𝑥𝐵 𝐶) = 𝑥𝐵 (𝐴𝐶)

Theoremimauni 6999* The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
(𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥)

Theoremfniunfv 7000* The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.)
(𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)

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