Home Intuitionistic Logic ExplorerTheorem List (p. 55 of 129) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 5401-5500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfvelrnb 5401* A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)
(𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵))

Theoremdfimafn 5402* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦})

Theoremdfimafn2 5403* Alternate definition of the image of a function as an indexed union of singletons of function values. (Contributed by Raph Levien, 20-Nov-2006.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹𝑥)})

Theoremfunimass4 5404* Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))

Theoremfvelima 5405* Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42. (Contributed by NM, 29-Apr-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → ∃𝑥𝐵 (𝐹𝑥) = 𝐴)

Theoremfeqmptd 5406* Deduction form of dffn5im 5399. (Contributed by Mario Carneiro, 8-Jan-2015.)
(𝜑𝐹:𝐴𝐵)       (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))

Theoremfeqresmpt 5407* Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ (𝐹𝑥)))

Theoremdffn5imf 5408* Representation of a function in terms of its values. (Contributed by Jim Kingdon, 31-Dec-2018.)
𝑥𝐹       (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))

Theoremfvelimab 5409* Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))

Theoremfvi 5410 The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐴𝑉 → ( I ‘𝐴) = 𝐴)

Theoremfniinfv 5411* The indexed intersection of a function's values is the intersection of its range. (Contributed by NM, 20-Oct-2005.)
(𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)

Theoremfnsnfv 5412 Singleton of function value. (Contributed by NM, 22-May-1998.)
((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))

Theoremfnimapr 5413 The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011.)
((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹𝐵), (𝐹𝐶)})

Theoremssimaex 5414* The existence of a subimage. (Contributed by NM, 8-Apr-2007.)
𝐴 ∈ V       ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))

Theoremssimaexg 5415* The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
((𝐴𝐶 ∧ Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))

Theoremfunfvdm 5416 A simplified expression for the value of a function when we know it's a function. (Contributed by Jim Kingdon, 1-Jan-2019.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = (𝐹 “ {𝐴}))

Theoremfunfvdm2 5417* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by Jim Kingdon, 1-Jan-2019.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})

Theoremfunfvdm2f 5418 The value of a function. Version of funfvdm2 5417 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 1-Jan-2019.)
𝑦𝐴    &   𝑦𝐹       ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})

Theoremfvun1 5419 The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))

Theoremfvun2 5420 The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐵)) → ((𝐹𝐺)‘𝑋) = (𝐺𝑋))

Theoremdmfco 5421 Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺𝐴) ∈ dom 𝐹))

Theoremfvco2 5422 Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (Contributed by NM, 9-Oct-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 16-Oct-2014.)
((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))

Theoremfvco 5423 Value of a function composition. Similar to Exercise 5 of [TakeutiZaring] p. 28. (Contributed by NM, 22-Apr-2006.) (Proof shortened by Mario Carneiro, 26-Dec-2014.)
((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))

Theoremfvco3 5424 Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.)
((𝐺:𝐴𝐵𝐶𝐴) → ((𝐹𝐺)‘𝐶) = (𝐹‘(𝐺𝐶)))

Theoremfvco4 5425 Value of a composition. (Contributed by BJ, 7-Jul-2022.)
(((𝐾:𝐴𝑋 ∧ (𝐻𝐾) = 𝐹) ∧ (𝑢𝐴𝑥 = (𝐾𝑢))) → (𝐻𝑥) = (𝐹𝑢))

Theoremfvopab3g 5426* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑥𝐶 → ∃!𝑦𝜑)    &   𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}       ((𝐴𝐶𝐵𝐷) → ((𝐹𝐴) = 𝐵𝜒))

Theoremfvopab3ig 5427* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑥𝐶 → ∃*𝑦𝜑)    &   𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}       ((𝐴𝐶𝐵𝐷) → (𝜒 → (𝐹𝐴) = 𝐵))

Theoremfvmptss2 5428* A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)
(𝑥 = 𝐷𝐵 = 𝐶)    &   𝐹 = (𝑥𝐴𝐵)       (𝐹𝐷) ⊆ 𝐶

Theoremfvmptg 5429* Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)       ((𝐴𝐷𝐶𝑅) → (𝐹𝐴) = 𝐶)

Theoremfvmpt 5430* Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)
(𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)    &   𝐶 ∈ V       (𝐴𝐷 → (𝐹𝐴) = 𝐶)

Theoremfvmpts 5431* Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐹 = (𝑥𝐶𝐵)       ((𝐴𝐶𝐴 / 𝑥𝐵𝑉) → (𝐹𝐴) = 𝐴 / 𝑥𝐵)

Theoremfvmpt3 5432* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)    &   (𝑥𝐷𝐵𝑉)       (𝐴𝐷 → (𝐹𝐴) = 𝐶)

Theoremfvmpt3i 5433* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)    &   𝐵 ∈ V       (𝐴𝐷 → (𝐹𝐴) = 𝐶)

Theoremfvmptd 5434* Deduction version of fvmpt 5430. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝜑𝐹 = (𝑥𝐷𝐵))    &   ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)    &   (𝜑𝐴𝐷)    &   (𝜑𝐶𝑉)       (𝜑 → (𝐹𝐴) = 𝐶)

Theoremmptrcl 5435* Reverse closure for a mapping: If the function value of a mapping has a member, the argument belongs to the base class of the mapping. (Contributed by AV, 4-Apr-2020.) (Revised by Jim Kingdon, 27-Mar-2023.)
𝐹 = (𝑥𝐴𝐵)       (𝐼 ∈ (𝐹𝑋) → 𝑋𝐴)

Theoremfvmpt2 5436* Value of a function given by the maps-to notation. (Contributed by FL, 21-Jun-2010.)
𝐹 = (𝑥𝐴𝐵)       ((𝑥𝐴𝐵𝐶) → (𝐹𝑥) = 𝐵)

Theoremfvmptssdm 5437* If all the values of the mapping are subsets of a class 𝐶, then so is any evaluation of the mapping at a value in the domain of the mapping. (Contributed by Jim Kingdon, 3-Jan-2018.)
𝐹 = (𝑥𝐴𝐵)       ((𝐷 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝐷) ⊆ 𝐶)

Theoremmptfvex 5438* Sufficient condition for a maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
𝐹 = (𝑥𝐴𝐵)       ((∀𝑥 𝐵𝑉𝐶𝑊) → (𝐹𝐶) ∈ V)

Theoremfvmpt2d 5439* Deduction version of fvmpt2 5436. (Contributed by Thierry Arnoux, 8-Dec-2016.)
(𝜑𝐹 = (𝑥𝐴𝐵))    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)

Theoremfvmptdf 5440* Alternate deduction version of fvmpt 5430, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))    &   𝑥𝐹    &   𝑥𝜓       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))

Theoremfvmptdv 5441* Alternate deduction version of fvmpt 5430, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))

Theoremfvmptdv2 5442* Alternate deduction version of fvmpt 5430, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = 𝐶))

Theoremmpteqb 5443* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5450. (Contributed by Mario Carneiro, 14-Nov-2014.)
(∀𝑥𝐴 𝐵𝑉 → ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ↔ ∀𝑥𝐴 𝐵 = 𝐶))

Theoremfvmptt 5444* Closed theorem form of fvmpt 5430. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)
((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → (𝐹𝐴) = 𝐶)

Theoremfvmptf 5445* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5429 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴    &   𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)       ((𝐴𝐷𝐶𝑉) → (𝐹𝐴) = 𝐶)

Theoremfvmptd3 5446* Deduction version of fvmpt 5430. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝐹 = (𝑥𝐷𝐵)    &   (𝑥 = 𝐴𝐵 = 𝐶)    &   (𝜑𝐴𝐷)    &   (𝜑𝐶𝑉)       (𝜑 → (𝐹𝐴) = 𝐶)

Theoremelfvmptrab1 5447* Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the argument of the function. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})    &   (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)       (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))

Theoremelfvmptrab 5448* Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝑀𝜑})    &   (𝑋𝑉𝑀 ∈ V)       (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑀))

Theoremfvopab6 5449* Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐵)}    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐴𝐵 = 𝐶)       ((𝐴𝐷𝐶𝑅𝜓) → (𝐹𝐴) = 𝐶)

Theoremeqfnfv 5450* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))

Theoremeqfnfv2 5451* Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))

Theoremeqfnfv3 5452* Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥)))))

Theoremeqfnfvd 5453* Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐴)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))       (𝜑𝐹 = 𝐺)

Theoremeqfnfv2f 5454* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 5450 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)
𝑥𝐹    &   𝑥𝐺       ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))

Theoremeqfunfv 5455* Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)
((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 = 𝐺 ↔ (dom 𝐹 = dom 𝐺 ∧ ∀𝑥 ∈ dom 𝐹(𝐹𝑥) = (𝐺𝑥))))

Theoremfvreseq 5456* Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)
(((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐵𝐴) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))

Theoremfndmdif 5457* Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})

Theoremfndmdifcom 5458 The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = dom (𝐺𝐹))

Theoremfndmin 5459* Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐺𝑥)})

Theoremfneqeql 5460 Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ dom (𝐹𝐺) = 𝐴))

Theoremfneqeql2 5461 Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺𝐴 ⊆ dom (𝐹𝐺)))

Theoremfnreseql 5462 Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = (𝐺𝑋) ↔ 𝑋 ⊆ dom (𝐹𝐺)))

Theoremchfnrn 5463* The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain. (Contributed by NM, 31-Aug-1999.)
((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → ran 𝐹 𝐴)

Theoremfunfvop 5464 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)

Theoremfunfvbrb 5465 Two ways to say that 𝐴 is in the domain of 𝐹. (Contributed by Mario Carneiro, 1-May-2014.)
(Fun 𝐹 → (𝐴 ∈ dom 𝐹𝐴𝐹(𝐹𝐴)))

Theoremfvimacnvi 5466 A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)
((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → (𝐹𝐴) ∈ 𝐵)

Theoremfvimacnv 5467 The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 5137 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))

Theoremfunimass3 5468 A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv 5467 would be the special case of 𝐴 being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))

Theoremfunimass5 5469* A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐴 ⊆ (𝐹𝐵) ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))

Theoremfunconstss 5470* Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∀𝑥𝐴 (𝐹𝑥) = 𝐵𝐴 ⊆ (𝐹 “ {𝐵})))

Theoremelpreima 5471 Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐹 Fn 𝐴 → (𝐵 ∈ (𝐹𝐶) ↔ (𝐵𝐴 ∧ (𝐹𝐵) ∈ 𝐶)))

Theoremfniniseg 5472 Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
(𝐹 Fn 𝐴 → (𝐶 ∈ (𝐹 “ {𝐵}) ↔ (𝐶𝐴 ∧ (𝐹𝐶) = 𝐵)))

Theoremfncnvima2 5473* Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐹 Fn 𝐴 → (𝐹𝐵) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ 𝐵})

Theoremfniniseg2 5474* Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) = {𝑥𝐴 ∣ (𝐹𝑥) = 𝐵})

Theoremfnniniseg2 5475* Support sets of functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐹 Fn 𝐴 → (𝐹 “ (V ∖ {𝐵})) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝐵})

Theoremrexsupp 5476* Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.)
(𝐹 Fn 𝐴 → (∃𝑥 ∈ (𝐹 “ (V ∖ {𝑍}))𝜑 ↔ ∃𝑥𝐴 ((𝐹𝑥) ≠ 𝑍𝜑)))

Theoremunpreima 5477 Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∪ (𝐹𝐵)))

Theoreminpreima 5478 Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))

Theoremdifpreima 5479 Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))

Theoremrespreima 5480 The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)
(Fun 𝐹 → ((𝐹𝐵) “ 𝐴) = ((𝐹𝐴) ∩ 𝐵))

Theoremfimacnv 5481 The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)
(𝐹:𝐴𝐵 → (𝐹𝐵) = 𝐴)

Theoremfnopfv 5482 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.)
((𝐹 Fn 𝐴𝐵𝐴) → ⟨𝐵, (𝐹𝐵)⟩ ∈ 𝐹)

Theoremfvelrn 5483 A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)

Theoremfnfvelrn 5484 A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) ∈ ran 𝐹)

Theoremffvelrn 5485 A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)
((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐵)

Theoremffvelrni 5486 A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)
𝐹:𝐴𝐵       (𝐶𝐴 → (𝐹𝐶) ∈ 𝐵)

Theoremffvelrnda 5487 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐹:𝐴𝐵)       ((𝜑𝐶𝐴) → (𝐹𝐶) ∈ 𝐵)

Theoremffvelrnd 5488 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → (𝐹𝐶) ∈ 𝐵)

Theoremrexrn 5489* Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
(𝑥 = (𝐹𝑦) → (𝜑𝜓))       (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦𝐴 𝜓))

Theoremralrn 5490* Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
(𝑥 = (𝐹𝑦) → (𝜑𝜓))       (𝐹 Fn 𝐴 → (∀𝑥 ∈ ran 𝐹𝜑 ↔ ∀𝑦𝐴 𝜓))

Theoremelrnrexdm 5491* For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
(Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))

Theoremelrnrexdmb 5492* For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
(Fun 𝐹 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))

Theoremeldmrexrn 5493* For any element in the domain of a function there is an element in the range of the function which is the function value for the element of the domain. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
(Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))

Theoremralrnmpt 5494* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑦 = 𝐵 → (𝜓𝜒))       (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 𝜒))

Theoremrexrnmpt 5495* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑦 = 𝐵 → (𝜓𝜒))       (∀𝑥𝐴 𝐵𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥𝐴 𝜒))

Theoremdff2 5496 Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)))

Theoremdff3im 5497* Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)
(𝐹:𝐴𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))

Theoremdff4im 5498* Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)
(𝐹:𝐴𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝐹𝑦))

Theoremdffo3 5499* An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)
(𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))

Theoremdffo4 5500* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
(𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))

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 12801-12832
 Copyright terms: Public domain < Previous  Next >