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Theorem List for Intuitionistic Logic Explorer - 5101-5200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremffoss 5101* Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)
𝐹 V       (𝐹:ABx(𝐹:Aontox xB))
 
Theoremf11o 5102* Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)
𝐹 V       (𝐹:A1-1Bx(𝐹:A1-1-ontox xB))
 
Theoremf10 5103 The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
∅:∅–1-1A
 
Theoremf1o00 5104 One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
(𝐹:∅–1-1-ontoA ↔ (𝐹 = ∅ A = ∅))
 
Theoremfo00 5105 Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
(𝐹:∅–ontoA ↔ (𝐹 = ∅ A = ∅))
 
Theoremf1o0 5106 One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
∅:∅–1-1-onto→∅
 
Theoremf1oi 5107 A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
( I ↾ A):A1-1-ontoA
 
Theoremf1ovi 5108 The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.)
I :V–1-1-onto→V
 
Theoremf1osn 5109 A singleton of an ordered pair is one-to-one onto function. (Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
A V    &   B V       {⟨A, B⟩}:{A}–1-1-onto→{B}
 
Theoremf1osng 5110 A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)
((A 𝑉 B 𝑊) → {⟨A, B⟩}:{A}–1-1-onto→{B})
 
Theoremf1oprg 5111 An unordered pair of ordered pairs with different elements is a one-to-one onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
(((A 𝑉 B 𝑊) (𝐶 𝑋 𝐷 𝑌)) → ((A𝐶 B𝐷) → {⟨A, B⟩, ⟨𝐶, 𝐷⟩}:{A, 𝐶}–1-1-onto→{B, 𝐷}))
 
Theoremtz6.12-2 5112* Function value when 𝐹 is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
∃!x A𝐹x → (𝐹A) = ∅)
 
Theoremfveu 5113* The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)
(∃!x A𝐹x → (𝐹A) = {xA𝐹x})
 
Theorembrprcneu 5114* If A is a proper class, then there is no unique binary relationship with A as the first element. (Contributed by Scott Fenton, 7-Oct-2017.)
A V → ¬ ∃!x A𝐹x)
 
Theoremfvprc 5115 A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)
A V → (𝐹A) = ∅)
 
Theoremfv2 5116* Alternate definition of function value. Definition 10.11 of [Quine] p. 68. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝐹A) = {xy(A𝐹yy = x)}
 
Theoremdffv3g 5117* A definition of function value in terms of iota. (Contributed by Jim Kingdon, 29-Dec-2018.)
(A 𝑉 → (𝐹A) = (℩xx (𝐹 “ {A})))
 
Theoremdffv4g 5118* The previous definition of function value, from before the operator was introduced. Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 4637), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
(A 𝑉 → (𝐹A) = {x ∣ (𝐹 “ {A}) = {x}})
 
Theoremelfv 5119* Membership in a function value. (Contributed by NM, 30-Apr-2004.)
(A (𝐹B) ↔ x(A x y(B𝐹yy = x)))
 
Theoremfveq1 5120 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
(𝐹 = 𝐺 → (𝐹A) = (𝐺A))
 
Theoremfveq2 5121 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
(A = B → (𝐹A) = (𝐹B))
 
Theoremfveq1i 5122 Equality inference for function value. (Contributed by NM, 2-Sep-2003.)
𝐹 = 𝐺       (𝐹A) = (𝐺A)
 
Theoremfveq1d 5123 Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)
(φ𝐹 = 𝐺)       (φ → (𝐹A) = (𝐺A))
 
Theoremfveq2i 5124 Equality inference for function value. (Contributed by NM, 28-Jul-1999.)
A = B       (𝐹A) = (𝐹B)
 
Theoremfveq2d 5125 Equality deduction for function value. (Contributed by NM, 29-May-1999.)
(φA = B)       (φ → (𝐹A) = (𝐹B))
 
Theoremfveq12i 5126 Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)
𝐹 = 𝐺    &   A = B       (𝐹A) = (𝐺B)
 
Theoremfveq12d 5127 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
(φ𝐹 = 𝐺)    &   (φA = B)       (φ → (𝐹A) = (𝐺B))
 
Theoremnffv 5128 Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
x𝐹    &   xA       x(𝐹A)
 
Theoremnffvmpt1 5129* Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)
x((x AB)‘𝐶)
 
Theoremnffvd 5130 Deduction version of bound-variable hypothesis builder nffv 5128. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
(φx𝐹)    &   (φxA)       (φx(𝐹A))
 
Theoremfunfveu 5131* A function has one value given an argument in its domain. (Contributed by Jim Kingdon, 29-Dec-2018.)
((Fun 𝐹 A dom 𝐹) → ∃!y A𝐹y)
 
Theoremfvss 5132* The value of a function is a subset of B if every element that could be a candidate for the value is a subset of B. (Contributed by Mario Carneiro, 24-May-2019.)
(x(A𝐹xxB) → (𝐹A) ⊆ B)
 
Theoremfvssunirng 5133 The result of a function value is always a subset of the union of the range, if the input is a set. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)
(A V → (𝐹A) ⊆ ran 𝐹)
 
Theoremrelfvssunirn 5134 The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)
(Rel 𝐹 → (𝐹A) ⊆ ran 𝐹)
 
Theoremfunfvex 5135 The value of a function exists. A special case of Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by Jim Kingdon, 29-Dec-2018.)
((Fun 𝐹 A dom 𝐹) → (𝐹A) V)
 
Theoremrelrnfvex 5136 If a function has a set range, then the function value exists unconditional on the domain. (Contributed by Mario Carneiro, 24-May-2019.)
((Rel 𝐹 ran 𝐹 V) → (𝐹A) V)
 
Theoremfvexg 5137 Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)
((𝐹 𝑉 A 𝑊) → (𝐹A) V)
 
Theoremfvex 5138 Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)
𝐹 𝑉    &   A 𝑊       (𝐹A) V
 
Theoremsefvex 5139 If a function is set-like, then the function value exists if the input does. (Contributed by Mario Carneiro, 24-May-2019.)
((𝐹 Se V A V) → (𝐹A) V)
 
Theoremfv3 5140* Alternate definition of the value of a function. Definition 6.11 of [TakeutiZaring] p. 26. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝐹A) = {x ∣ (y(x y A𝐹y) ∃!y A𝐹y)}
 
Theoremfvres 5141 The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
(A B → ((𝐹B)‘A) = (𝐹A))
 
Theoremfunssfv 5142 The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.)
((Fun 𝐹 𝐺𝐹 A dom 𝐺) → (𝐹A) = (𝐺A))
 
Theoremtz6.12-1 5143* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
((A𝐹y ∃!y A𝐹y) → (𝐹A) = y)
 
Theoremtz6.12 5144* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.)
((⟨A, y 𝐹 ∃!yA, y 𝐹) → (𝐹A) = y)
 
Theoremtz6.12f 5145* Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)
y𝐹       ((⟨A, y 𝐹 ∃!yA, y 𝐹) → (𝐹A) = y)
 
Theoremtz6.12c 5146* Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
(∃!y A𝐹y → ((𝐹A) = yA𝐹y))
 
Theoremndmfvg 5147 The value of a class outside its domain is the empty set. (Contributed by Jim Kingdon, 15-Jan-2019.)
((A V ¬ A dom 𝐹) → (𝐹A) = ∅)
 
Theoremrelelfvdm 5148 If a function value has a member, the argument belongs to the domain. (Contributed by Jim Kingdon, 22-Jan-2019.)
((Rel 𝐹 A (𝐹B)) → B dom 𝐹)
 
Theoremnfvres 5149 The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)
A B → ((𝐹B)‘A) = ∅)
 
Theoremnfunsn 5150 If the restriction of a class to a singleton is not a function, its value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(¬ Fun (𝐹 ↾ {A}) → (𝐹A) = ∅)
 
Theorem0fv 5151 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
(∅‘A) = ∅
 
Theoremcsbfv12g 5152 Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)
(A 𝐶A / x(𝐹B) = (A / x𝐹A / xB))
 
Theoremcsbfv2g 5153* Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)
(A 𝐶A / x(𝐹B) = (𝐹A / xB))
 
Theoremcsbfvg 5154* Substitution for a function value. (Contributed by NM, 1-Jan-2006.)
(A 𝐶A / x(𝐹x) = (𝐹A))
 
Theoremfunbrfv 5155 The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
(Fun 𝐹 → (A𝐹B → (𝐹A) = B))
 
Theoremfunopfv 5156 The second element in an ordered pair member of a function is the function's value. (Contributed by NM, 19-Jul-1996.)
(Fun 𝐹 → (⟨A, B 𝐹 → (𝐹A) = B))
 
Theoremfnbrfvb 5157 Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝐹 Fn A B A) → ((𝐹B) = 𝐶B𝐹𝐶))
 
Theoremfnopfvb 5158 Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.)
((𝐹 Fn A B A) → ((𝐹B) = 𝐶 ↔ ⟨B, 𝐶 𝐹))
 
Theoremfunbrfvb 5159 Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)
((Fun 𝐹 A dom 𝐹) → ((𝐹A) = BA𝐹B))
 
Theoremfunopfvb 5160 Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.)
((Fun 𝐹 A dom 𝐹) → ((𝐹A) = B ↔ ⟨A, B 𝐹))
 
Theoremfunbrfv2b 5161 Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)
(Fun 𝐹 → (A𝐹B ↔ (A dom 𝐹 (𝐹A) = B)))
 
Theoremdffn5im 5162* Representation of a function in terms of its values. The converse holds given the law of the excluded middle; as it is we have most of the converse via funmpt 4881 and dmmptss 4760. (Contributed by Jim Kingdon, 31-Dec-2018.)
(𝐹 Fn A𝐹 = (x A ↦ (𝐹x)))
 
Theoremfnrnfv 5163* The range of a function expressed as a collection of the function's values. (Contributed by NM, 20-Oct-2005.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
(𝐹 Fn A → ran 𝐹 = {yx A y = (𝐹x)})
 
Theoremfvelrnb 5164* A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)
(𝐹 Fn A → (B ran 𝐹x A (𝐹x) = B))
 
Theoremdfimafn 5165* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)
((Fun 𝐹 A ⊆ dom 𝐹) → (𝐹A) = {yx A (𝐹x) = y})
 
Theoremdfimafn2 5166* 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 𝐹 A ⊆ dom 𝐹) → (𝐹A) = x A {(𝐹x)})
 
Theoremfunimass4 5167* Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.)
((Fun 𝐹 A ⊆ dom 𝐹) → ((𝐹A) ⊆ Bx A (𝐹x) B))
 
Theoremfvelima 5168* 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 𝐹 A (𝐹B)) → x B (𝐹x) = A)
 
Theoremfeqmptd 5169* Deduction form of dffn5im 5162. (Contributed by Mario Carneiro, 8-Jan-2015.)
(φ𝐹:AB)       (φ𝐹 = (x A ↦ (𝐹x)))
 
Theoremfeqresmpt 5170* Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)
(φ𝐹:AB)    &   (φ𝐶A)       (φ → (𝐹𝐶) = (x 𝐶 ↦ (𝐹x)))
 
Theoremdffn5imf 5171* Representation of a function in terms of its values. (Contributed by Jim Kingdon, 31-Dec-2018.)
x𝐹       (𝐹 Fn A𝐹 = (x A ↦ (𝐹x)))
 
Theoremfvelimab 5172* 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 A BA) → (𝐶 (𝐹B) ↔ x B (𝐹x) = 𝐶))
 
Theoremfvi 5173 The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
(A 𝑉 → ( I ‘A) = A)
 
Theoremfniinfv 5174* The indexed intersection of a function's values is the intersection of its range. (Contributed by NM, 20-Oct-2005.)
(𝐹 Fn A x A (𝐹x) = ran 𝐹)
 
Theoremfnsnfv 5175 Singleton of function value. (Contributed by NM, 22-May-1998.)
((𝐹 Fn A B A) → {(𝐹B)} = (𝐹 “ {B}))
 
Theoremfnimapr 5176 The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011.)
((𝐹 Fn A B A 𝐶 A) → (𝐹 “ {B, 𝐶}) = {(𝐹B), (𝐹𝐶)})
 
Theoremssimaex 5177* The existence of a subimage. (Contributed by NM, 8-Apr-2007.)
A V       ((Fun 𝐹 B ⊆ (𝐹A)) → x(xA B = (𝐹x)))
 
Theoremssimaexg 5178* The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
((A 𝐶 Fun 𝐹 B ⊆ (𝐹A)) → x(xA B = (𝐹x)))
 
Theoremfunfvdm 5179 A simplified expression for the value of a function when we know it's a function. (Contributed by Jim Kingdon, 1-Jan-2019.)
((Fun 𝐹 A dom 𝐹) → (𝐹A) = (𝐹 “ {A}))
 
Theoremfunfvdm2 5180* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by Jim Kingdon, 1-Jan-2019.)
((Fun 𝐹 A dom 𝐹) → (𝐹A) = {yA𝐹y})
 
Theoremfunfvdm2f 5181 The value of a function. Version of funfvdm2 5180 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 1-Jan-2019.)
yA    &   y𝐹       ((Fun 𝐹 A dom 𝐹) → (𝐹A) = {yA𝐹y})
 
Theoremfvun1 5182 The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)
((𝐹 Fn A 𝐺 Fn B ((AB) = ∅ 𝑋 A)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))
 
Theoremfvun2 5183 The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)
((𝐹 Fn A 𝐺 Fn B ((AB) = ∅ 𝑋 B)) → ((𝐹𝐺)‘𝑋) = (𝐺𝑋))
 
Theoremdmfco 5184 Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
((Fun 𝐺 A dom 𝐺) → (A dom (𝐹𝐺) ↔ (𝐺A) dom 𝐹))
 
Theoremfvco2 5185 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 A 𝑋 A) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
 
Theoremfvco 5186 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 𝐺 A dom 𝐺) → ((𝐹𝐺)‘A) = (𝐹‘(𝐺A)))
 
Theoremfvco3 5187 Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.)
((𝐺:AB 𝐶 A) → ((𝐹𝐺)‘𝐶) = (𝐹‘(𝐺𝐶)))
 
Theoremfvopab3g 5188* 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))    &   (x 𝐶∃!yφ)    &   𝐹 = {⟨x, y⟩ ∣ (x 𝐶 φ)}       ((A 𝐶 B 𝐷) → ((𝐹A) = Bχ))
 
Theoremfvopab3ig 5189* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)
(x = A → (φψ))    &   (y = B → (ψχ))    &   (x 𝐶∃*yφ)    &   𝐹 = {⟨x, y⟩ ∣ (x 𝐶 φ)}       ((A 𝐶 B 𝐷) → (χ → (𝐹A) = B))
 
Theoremfvmptss2 5190* 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 = (x AB)       (𝐹𝐷) ⊆ 𝐶
 
Theoremfvmptg 5191* 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 = (x 𝐷B)       ((A 𝐷 𝐶 𝑅) → (𝐹A) = 𝐶)
 
Theoremfvmpt 5192* Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)
(x = AB = 𝐶)    &   𝐹 = (x 𝐷B)    &   𝐶 V       (A 𝐷 → (𝐹A) = 𝐶)
 
Theoremfvmpts 5193* 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 = (x 𝐶B)       ((A 𝐶 A / xB 𝑉) → (𝐹A) = A / xB)
 
Theoremfvmpt3 5194* 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 = (x 𝐷B)    &   (x 𝐷B 𝑉)       (A 𝐷 → (𝐹A) = 𝐶)
 
Theoremfvmpt3i 5195* 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 = (x 𝐷B)    &   B V       (A 𝐷 → (𝐹A) = 𝐶)
 
Theoremfvmptd 5196* Deduction version of fvmpt 5192. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
(φ𝐹 = (x 𝐷B))    &   ((φ x = A) → B = 𝐶)    &   (φA 𝐷)    &   (φ𝐶 𝑉)       (φ → (𝐹A) = 𝐶)
 
Theoremfvmpt2 5197* Value of a function given by the "maps to" notation. (Contributed by FL, 21-Jun-2010.)
𝐹 = (x AB)       ((x A B 𝐶) → (𝐹x) = B)
 
Theoremfvmptssdm 5198* 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 = (x AB)       ((𝐷 dom 𝐹 x A B𝐶) → (𝐹𝐷) ⊆ 𝐶)
 
Theoremmptfvex 5199* Sufficient condition for a maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
𝐹 = (x AB)       ((x B 𝑉 𝐶 𝑊) → (𝐹𝐶) V)
 
Theoremfvmpt2d 5200* Deduction version of fvmpt2 5197. (Contributed by Thierry Arnoux, 8-Dec-2016.)
(φ𝐹 = (x AB))    &   ((φ x A) → B 𝑉)       ((φ x A) → (𝐹x) = B)
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