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Theorem List for Metamath Proof Explorer - 7001-7100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremffnfv 7001* A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremffnfvf 7002 A function maps to a class to which all values belong. This version of ffnfv 7001 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝐹       (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremfnfvrnss 7003* An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)
((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → ran 𝐹𝐵)
 
Theoremfrnssb 7004* A function is a function into a subset of its codomain if all of its values are elements of this subset. (Contributed by AV, 7-Feb-2021.)
((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → (𝐹:𝐴𝑊𝐹:𝐴𝑉))
 
Theoremrnmptss 7005* The range of an operation given by the maps-to notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.)
𝐹 = (𝑥𝐴𝐵)       (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
 
Theoremfmpt2d 7006* Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑𝐹 = (𝑥𝐴𝐵))    &   ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ 𝐶)       (𝜑𝐹:𝐴𝐶)
 
Theoremffvresb 7007* A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)
(Fun 𝐹 → ((𝐹𝐴):𝐴𝐵 ↔ ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)))
 
Theoremf1oresrab 7008* Build a bijection between restricted abstract builders, given a bijection between the base classes, deduction version. (Contributed by Thierry Arnoux, 17-Aug-2018.)
𝐹 = (𝑥𝐴𝐶)    &   (𝜑𝐹:𝐴1-1-onto𝐵)    &   ((𝜑𝑥𝐴𝑦 = 𝐶) → (𝜒𝜓))       (𝜑 → (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
 
Theoremf1ossf1o 7009* Restricting a bijection, which is a mapping from a restricted class abstraction, to a subset is a bijection. (Contributed by AV, 7-Aug-2022.)
𝑋 = {𝑤𝐴 ∣ (𝜓𝜒)}    &   𝑌 = {𝑤𝐴𝜓}    &   𝐹 = (𝑥𝑋𝐵)    &   𝐺 = (𝑥𝑌𝐵)    &   (𝜑𝐺:𝑌1-1-onto𝐶)    &   ((𝜑𝑥𝑌𝑦 = 𝐵) → (𝜏 ↔ [𝑥 / 𝑤]𝜒))       (𝜑𝐹:𝑋1-1-onto→{𝑦𝐶𝜏})
 
Theoremfmptco 7010* Composition of two functions expressed as ordered-pair class abstractions. If 𝐹 has the equation (𝑥 + 2) and 𝐺 the equation (3∗𝑧) then (𝐺𝐹) has the equation (3∗(𝑥 + 2)). (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.)
((𝜑𝑥𝐴) → 𝑅𝐵)    &   (𝜑𝐹 = (𝑥𝐴𝑅))    &   (𝜑𝐺 = (𝑦𝐵𝑆))    &   (𝑦 = 𝑅𝑆 = 𝑇)       (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))
 
Theoremfmptcof 7011* Version of fmptco 7010 where 𝜑 needn't be distinct from 𝑥. (Contributed by NM, 27-Dec-2014.)
(𝜑 → ∀𝑥𝐴 𝑅𝐵)    &   (𝜑𝐹 = (𝑥𝐴𝑅))    &   (𝜑𝐺 = (𝑦𝐵𝑆))    &   (𝑦 = 𝑅𝑆 = 𝑇)       (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))
 
Theoremfmptcos 7012* Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝜑 → ∀𝑥𝐴 𝑅𝐵)    &   (𝜑𝐹 = (𝑥𝐴𝑅))    &   (𝜑𝐺 = (𝑦𝐵𝑆))       (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑅 / 𝑦𝑆))
 
Theoremcofmpt 7013* Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 29-Jun-2017.)
(𝜑𝐹:𝐶𝐷)    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → (𝐹 ∘ (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ (𝐹𝐵)))
 
Theoremfcompt 7014* Express composition of two functions as a maps-to applying both in sequence. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (𝐴‘(𝐵𝑥))))
 
Theoremfcoconst 7015 Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
((𝐹 Fn 𝑋𝑌𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹𝑌)}))
 
Theoremfsn 7016 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩})
 
Theoremfsn2 7017 A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.)
𝐴 ∈ V       (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
 
Theoremfsng 7018 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)
((𝐴𝐶𝐵𝐷) → (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩}))
 
Theoremfsn2g 7019 A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by Thierry Arnoux, 11-Jul-2020.)
(𝐴𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩})))
 
Theoremxpsng 7020 The Cartesian product of two singletons is the singleton consisting in the associated ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝐴𝑉𝐵𝑊) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
 
Theoremxpprsng 7021 The Cartesian product of an unordered pair and a singleton. (Contributed by AV, 20-May-2019.)
((𝐴𝑉𝐵𝑊𝐶𝑈) → ({𝐴, 𝐵} × {𝐶}) = {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐶⟩})
 
Theoremxpsn 7022 The Cartesian product of two singletons is the singleton consisting in the associated ordered pair. (Contributed by NM, 4-Nov-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩}
 
Theoremf1o2sn 7023 A singleton consisting in a nested ordered pair is a one-to-one function from the cartesian product of two singletons onto a singleton (case where the two singletons are equal). (Contributed by AV, 15-Aug-2019.)
((𝐸𝑉𝑋𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:({𝐸} × {𝐸})–1-1-onto→{𝑋})
 
Theoremresidpr 7024 Restriction of the identity to a pair. (Contributed by AV, 11-Dec-2018.)
((𝐴𝑉𝐵𝑊) → ( I ↾ {𝐴, 𝐵}) = {⟨𝐴, 𝐴⟩, ⟨𝐵, 𝐵⟩})
 
Theoremdfmpt 7025 Alternate definition for the maps-to notation df-mpt 5159 (although it requires that 𝐵 be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.)
𝐵 ∈ V       (𝑥𝐴𝐵) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
 
Theoremfnasrn 7026 A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
𝐵 ∈ V       (𝑥𝐴𝐵) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩)
 
Theoremidref 7027* 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 7028* 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 7029* 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 6492, as relsnopg 5715 is to relop 5762. (New usage is discouraged.)
𝑋 ∈ V    &   𝑌 ∈ V       ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
 
Theoremfunop 7030* 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 6492, as relsnopg 5715 is to relop 5762. (New usage is discouraged.)
𝑋 ∈ V    &   𝑌 ∈ V       (Fun ⟨𝑋, 𝑌⟩ ↔ ∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}))
 
Theoremfunopdmsn 7031 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 7032 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 7033 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 7034 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 7035 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 7036 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 7037 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 7038 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 7039 A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩})
 
Theoremfunressn 7040 A function restricted to a singleton. (Contributed by Mario Carneiro, 16-Nov-2014.)
(Fun 𝐹 → (𝐹 ↾ {𝐵}) ⊆ {⟨𝐵, (𝐹𝐵)⟩})
 
Theoremfressnfv 7041 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 7042 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 7043 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 7044 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 7045 The value of a constant function. (Contributed by NM, 30-May-1999.)
((𝐹:𝐴⟶{𝐵} ∧ 𝐶𝐴) → (𝐹𝐶) = 𝐵)
 
Theoremfnsnr 7046 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 7047 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 7048* 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 7049* Express a singleton function in maps-to notation. Version of fmptsn 7048 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 7050* Express a singleton function in maps-to notation. Deduction form of fmptsng 7049. (Contributed by AV, 4-Aug-2019.)
((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
 
Theoremfmptap 7051* 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 7052* Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.) (Revised by AV, 10-Aug-2024.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)    &   ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐵)       (𝜑 → ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶))
 
Theoremfmptpr 7053* 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 7054 The value of a restricted identity function. (Contributed by NM, 19-May-2004.)
(𝐵𝐴 → (( I ↾ 𝐴)‘𝐵) = 𝐵)
 
Theoremfninfp 7055* 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 7056 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 7057* 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 7058 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 7059 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 7060 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 7061 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 7062 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 7063 The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 7064. (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 7064 The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 7063. (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 7065 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 7066 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 7067 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 7068 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 7069 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 7070 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)
 
Theoremfvpr1g 7071 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 7072 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Proof shortened by BJ, 26-Sep-2024.)
((𝐵𝑉𝐷𝑊𝐴𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷)
 
Theoremfvpr2gOLD 7073 Obsolete version of fvpr2g 7072 as of 26-Sep-2024. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐵𝑉𝐷𝑊𝐴𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷)
 
Theoremfvpr1 7074 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by BJ, 26-Sep-2024.)
𝐴 ∈ V    &   𝐶 ∈ V       (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)
 
Theoremfvpr1OLD 7075 Obsolete version of fvpr1 7074 as of 26-Sep-2024. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ V    &   𝐶 ∈ V       (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)
 
Theoremfvpr2 7076 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by BJ, 26-Sep-2024.)
𝐵 ∈ V    &   𝐷 ∈ V       (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷)
 
Theoremfvpr2OLD 7077 Obsolete version of fvpr2 7076 as of 26-Sep-2024. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 ∈ V    &   𝐷 ∈ V       (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷)
 
Theoremfprb 7078* 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 7079 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 7080 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 7081 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 7082 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 7083 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 7084 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 7085 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 7086 The value of a constant function. (Contributed by NM, 20-Aug-2005.)
((𝐵𝐷𝐶𝐴) → ((𝐴 × {𝐵})‘𝐶) = 𝐵)
 
Theoremfconst2g 7087 A constant function expressed as a Cartesian product. (Contributed by NM, 27-Nov-2007.)
(𝐵𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))
 
Theoremfvconst2 7088 The value of a constant function. (Contributed by NM, 16-Apr-2005.)
𝐵 ∈ V       (𝐶𝐴 → ((𝐴 × {𝐵})‘𝐶) = 𝐵)
 
Theoremfconst2 7089 A constant function expressed as a Cartesian product. (Contributed by NM, 20-Aug-1999.)
𝐵 ∈ V       (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵}))
 
Theoremfconst5 7090 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 7091* 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 7092* Obsolete version of rnmptc 7091 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 7093 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 7094 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 7095 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 7096 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 7097* A constant function expressed in terms of its functionality, domain, and value. See also fconst2 7089. (Contributed by NM, 27-Aug-2004.) (Proof shortened by OpenAI, 25-Mar-2020.)
(𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵))
 
Theoremfconst3 7098 Two ways to express a constant function. (Contributed by NM, 15-Mar-2007.)
(𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))
 
Theoremfconst4 7099 Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
(𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (𝐹 “ {𝐵}) = 𝐴))
 
Theoremresfunexg 7100 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)
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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