Home Metamath Proof ExplorerTheorem List (p. 64 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27159) Hilbert Space Explorer (27160-28684) Users' Mathboxes (28685-42360)

Theorem List for Metamath Proof Explorer - 6301-6400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremffvresb 6301* 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 6302* 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})

Theoremfmptco 6303* 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 6304* Version of fmptco 6303 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 6305* 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))

Theoremfcompt 6306* 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 6307 Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
((𝐹 Fn 𝑋𝑌𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹𝑌)}))

Theoremfsn 6308 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 6309 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 6310 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 6311 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 6312 The Cartesian product of two singletons. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝐴𝑉𝐵𝑊) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})

Theoremxpsn 6313 The Cartesian product of two singletons. (Contributed by NM, 4-Nov-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩}

Theoremf1o2sn 6314 A singleton with a nested ordered pair is a 1-1 function of the cartesian product of two singleton onto a singleton. (Contributed by AV, 15-Aug-2019.)
((𝐸𝑉𝑋𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:({𝐸} × {𝐸})–1-1-onto→{𝑋})

Theoremresidpr 6315 Restriction of the identity to a pair. (Contributed by AV, 11-Dec-2018.)
((𝐴𝑉𝐵𝑊) → ( I ↾ {𝐴, 𝐵}) = {⟨𝐴, 𝐴⟩, ⟨𝐵, 𝐵⟩})

Theoremdfmpt 6316 Alternate definition for the "maps to" notation df-mpt 4645 (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 6317 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⟩)

Theoremfuniun 6318* 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 6319* If a function is an ordered pair then it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.)
𝑋 ∈ V    &   𝑌 ∈ V       ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))

Theoremfunop 6320* An ordered pair is a function iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.)
𝑋 ∈ V    &   𝑌 ∈ V       (Fun ⟨𝑋, 𝑌⟩ ↔ ∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}))

Theoremfunsndifnop 6321 A singleton of an ordered pair is not an ordered pair if the components are different. (Contributed by AV, 23-Sep-2020.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐺 = {⟨𝐴, 𝐵⟩}       (𝐴𝐵 → ¬ 𝐺 ∈ (V × V))

Theoremfunsneqopsn 6322 A singleton of an ordered pair is an ordered pair of equal singletons if the components are equal. (Contributed by AV, 24-Sep-2020.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐺 = {⟨𝐴, 𝐵⟩}       (𝐴 = 𝐵𝐺 = ⟨{𝐴}, {𝐴}⟩)

Theoremfunsneqop 6323 A singleton of an ordered pair is an ordered pair if the components are equal. (Contributed by AV, 24-Sep-2020.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐺 = {⟨𝐴, 𝐵⟩}       (𝐴 = 𝐵𝐺 ∈ (V × V))

Theoremfunsneqopb 6324 A singleton of an ordered pair is an ordered pair iff the components are equal. (Contributed by AV, 24-Sep-2020.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐺 = {⟨𝐴, 𝐵⟩}       (𝐴 = 𝐵𝐺 ∈ (V × V))

Theoremressnop0 6325 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 6326 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 6327 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 6328 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 6329 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 6330 A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩})

Theoremfunressn 6331 A function restricted to a singleton. (Contributed by Mario Carneiro, 16-Nov-2014.)
(Fun 𝐹 → (𝐹 ↾ {𝐵}) ⊆ {⟨𝐵, (𝐹𝐵)⟩})

Theoremfressnfv 6332 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 6333 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 6334 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 6335 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 6336 The value of a constant function. (Contributed by NM, 30-May-1999.)
((𝐹:𝐴⟶{𝐵} ∧ 𝐶𝐴) → (𝐹𝐶) = 𝐵)

Theoremfnsnb 6337 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 6338* 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 6339* Express a singleton function in maps-to notation. Version of fmptsn 6338 allowing the mapping value to depend on the mapping variable (usual case). (Contributed by AV, 27-Feb-2019.)
(𝑥 = 𝐴𝐵 = 𝐶)       ((𝐴𝑉𝐶𝑊) → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))

Theoremfmptsnd 6340* Express a singleton function in maps-to notation. Deduction form of fmptsng 6339. (Contributed by AV, 4-Aug-2019.)
((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))

Theoremfmptap 6341* 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 6342* 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 6343* 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 6344 The value of a restricted identity function. (Contributed by NM, 19-May-2004.)
(𝐵𝐴 → (( I ↾ 𝐴)‘𝐵) = 𝐵)

Theoremfninfp 6345* 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 6346 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 6347* 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 6348 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 6349 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 6350 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))

Theoremfvsn 6351 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.)
𝐴 ∈ V    &   𝐵 ∈ V       ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵

Theoremfvsng 6352 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.)
((𝐴𝑉𝐵𝑊) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)

Theoremfvsnun1 6353 The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 6354. (Contributed by NM, 23-Sep-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))       (𝐺𝐴) = 𝐵

Theoremfvsnun2 6354 The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 6353. (Contributed by NM, 23-Sep-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))       (𝐷 ∈ (𝐶 ∖ {𝐴}) → (𝐺𝐷) = (𝐹𝐷))

Theoremfnsnsplit 6355 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 6356 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 6357 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 6358 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 6359 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 6360 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 6361 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 6362 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 6363 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 6364 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)

Theoremfvtp1 6365 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 6366 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 6367 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 6368 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 6369 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 6370 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 6371 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 6372 The value of a constant function. (Contributed by NM, 20-Aug-2005.)
((𝐵𝐷𝐶𝐴) → ((𝐴 × {𝐵})‘𝐶) = 𝐵)

Theoremfconst2g 6373 A constant function expressed as a Cartesian product. (Contributed by NM, 27-Nov-2007.)
(𝐵𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))

Theoremfvconst2 6374 The value of a constant function. (Contributed by NM, 16-Apr-2005.)
𝐵 ∈ V       (𝐶𝐴 → ((𝐴 × {𝐵})‘𝐶) = 𝐵)

Theoremfconst2 6375 A constant function expressed as a Cartesian product. (Contributed by NM, 20-Aug-1999.)
𝐵 ∈ V       (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵}))

Theoremfconst5 6376 Two ways to express that a function is constant. (Contributed by NM, 27-Nov-2007.)
((𝐹 Fn 𝐴𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))

Theoremfnprb 6377 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.) Revised to eliminate unnecessary antecedent 𝐴𝐵. (Revised by NM, 29-Dec-2018.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})

Theoremfntpb 6378 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 6379 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 6380 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 6381* A constant function expressed in terms of its functionality, domain, and value. See also fconst2 6375. (Contributed by NM, 27-Aug-2004.) (Proof shortened by OpenAI, 25-Mar-2020.)
(𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵))

Theoremfconst3 6382 Two ways to express a constant function. (Contributed by NM, 15-Mar-2007.)
(𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))

Theoremfconst4 6383 Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
(𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (𝐹 “ {𝐵}) = 𝐴))

Theoremresfunexg 6384 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 6385 The restriction of the identity relation to a set is a set. (Contributed by AV, 15-Feb-2020.)
(𝜑𝐵𝑉)       (𝜑 → ( I ↾ 𝐵) ∈ V)

Theoremfnex 6386 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 6384. See fnexALT 7025 for alternate proof. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)

Theoremfunex 6387 If the domain of a function exists, so 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 6386. (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 6388* 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 6389* 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)

Theoremmptex 6390* If the domain of a function given by maps-to notation is a set, the function is a set. Inference version of mptexg 6389. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.)
𝐴 ∈ V       (𝑥𝐴𝐵) ∈ V

Theoremmptexd 6391* If the domain of a function given by maps-to notation is a set, the function is a set. Deduction version of mptexg 6389. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴𝑉)       (𝜑 → (𝑥𝐴𝐵) ∈ V)

Theoremmptrabex 6392* 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

TheoremmptrabexOLD 6393* Obsolete version of mptrabex 6392 as of 26-Mar-2021. (Contributed by AV, 16-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴𝑉       (𝑥 ∈ {𝑦𝐴𝜑} ↦ 𝐵) ∈ V

Theoremfex 6394 If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)
((𝐹:𝐴𝐵𝐴𝐶) → 𝐹 ∈ V)

Theoremeufnfv 6395* 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 6396 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 6397 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)))

Theoremresfvresima 6398 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 6399 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})))

Theoremfnfvima 6400 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))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
 Copyright terms: Public domain < Previous  Next >