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Theorem List for Intuitionistic Logic Explorer - 5101-5200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfveq1d 5101 Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)
(φ𝐹 = 𝐺)       (φ → (𝐹A) = (𝐺A))

Theoremfveq2i 5102 Equality inference for function value. (Contributed by NM, 28-Jul-1999.)
A = B       (𝐹A) = (𝐹B)

Theoremfveq2d 5103 Equality deduction for function value. (Contributed by NM, 29-May-1999.)
(φA = B)       (φ → (𝐹A) = (𝐹B))

Theoremfveq12i 5104 Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)
𝐹 = 𝐺    &   A = B       (𝐹A) = (𝐺B)

Theoremfveq12d 5105 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
(φ𝐹 = 𝐺)    &   (φA = B)       (φ → (𝐹A) = (𝐺B))

Theoremnffv 5106 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 5107* Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)
x((x AB)‘𝐶)

Theoremnffvd 5108 Deduction version of bound-variable hypothesis builder nffv 5106. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
(φx𝐹)    &   (φxA)       (φx(𝐹A))

Theoremfunfveu 5109* 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 5110* 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 5111 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 5112 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 5113 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 5114 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 5115 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 5116 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 5117 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 5118* 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 5119 The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
(A B → ((𝐹B)‘A) = (𝐹A))

Theoremfunssfv 5120 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 5121* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
((A𝐹y ∃!y A𝐹y) → (𝐹A) = y)

Theoremtz6.12 5122* 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 5123* 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 5124* Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
(∃!y A𝐹y → ((𝐹A) = yA𝐹y))

Theoremndmfvg 5125 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 5126 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 5127 The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)
A B → ((𝐹B)‘A) = ∅)

Theoremnfunsn 5128 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 5129 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
(∅‘A) = ∅

Theoremcsbfv12g 5130 Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)
(A 𝐶A / x(𝐹B) = (A / x𝐹A / xB))

Theoremcsbfv2g 5131* Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)
(A 𝐶A / x(𝐹B) = (𝐹A / xB))

Theoremcsbfvg 5132* Substitution for a function value. (Contributed by NM, 1-Jan-2006.)
(A 𝐶A / x(𝐹x) = (𝐹A))

Theoremfunbrfv 5133 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 5134 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 5135 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 5136 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 5137 Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)
((Fun 𝐹 A dom 𝐹) → ((𝐹A) = BA𝐹B))

Theoremfunopfvb 5138 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 5139 Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)
(Fun 𝐹 → (A𝐹B ↔ (A dom 𝐹 (𝐹A) = B)))

Theoremdffn5im 5140* 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 4860 and dmmptss 4740. (Contributed by Jim Kingdon, 31-Dec-2018.)
(𝐹 Fn A𝐹 = (x A ↦ (𝐹x)))

Theoremfnrnfv 5141* 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 5142* 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 5143* 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 5144* 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 5145* 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 5146* 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 5147* Deduction form of dffn5im 5140. (Contributed by Mario Carneiro, 8-Jan-2015.)
(φ𝐹:AB)       (φ𝐹 = (x A ↦ (𝐹x)))

Theoremfeqresmpt 5148* Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)
(φ𝐹:AB)    &   (φ𝐶A)       (φ → (𝐹𝐶) = (x 𝐶 ↦ (𝐹x)))

Theoremdffn5imf 5149* Representation of a function in terms of its values. (Contributed by Jim Kingdon, 31-Dec-2018.)
x𝐹       (𝐹 Fn A𝐹 = (x A ↦ (𝐹x)))

Theoremfvelimab 5150* 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 5151 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 5152* 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 5153 Singleton of function value. (Contributed by NM, 22-May-1998.)
((𝐹 Fn A B A) → {(𝐹B)} = (𝐹 “ {B}))

Theoremfnimapr 5154 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 5155* The existence of a subimage. (Contributed by NM, 8-Apr-2007.)
A V       ((Fun 𝐹 B ⊆ (𝐹A)) → x(xA B = (𝐹x)))

Theoremssimaexg 5156* The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
((A 𝐶 Fun 𝐹 B ⊆ (𝐹A)) → x(xA B = (𝐹x)))

Theoremfunfvdm 5157 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 5158* 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 5159 The value of a function. Version of funfvdm2 5158 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 5160 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 5161 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 5162 Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
((Fun 𝐺 A dom 𝐺) → (A dom (𝐹𝐺) ↔ (𝐺A) dom 𝐹))

Theoremfvco2 5163 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 5164 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 5165 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 5166* 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 5167* 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 5168* 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 5169* 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 5170* 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 5171* 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 5172* 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 5173* 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 5174* Deduction version of fvmpt 5170. (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 5175* 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 5176* 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 5177* 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 5178* Deduction version of fvmpt2 5175. (Contributed by Thierry Arnoux, 8-Dec-2016.)
(φ𝐹 = (x AB))    &   ((φ x A) → B 𝑉)       ((φ x A) → (𝐹x) = B)

Theoremfvmptdf 5179* Alternate deduction version of fvmpt 5170, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(φA 𝐷)    &   ((φ x = A) → B 𝑉)    &   ((φ x = A) → ((𝐹A) = Bψ))    &   x𝐹    &   xψ       (φ → (𝐹 = (x 𝐷B) → ψ))

Theoremfvmptdv 5180* Alternate deduction version of fvmpt 5170, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(φA 𝐷)    &   ((φ x = A) → B 𝑉)    &   ((φ x = A) → ((𝐹A) = Bψ))       (φ → (𝐹 = (x 𝐷B) → ψ))

Theoremfvmptdv2 5181* Alternate deduction version of fvmpt 5170, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(φA 𝐷)    &   ((φ x = A) → B 𝑉)    &   ((φ x = A) → B = 𝐶)       (φ → (𝐹 = (x 𝐷B) → (𝐹A) = 𝐶))

Theoremmpteqb 5182* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5186. (Contributed by Mario Carneiro, 14-Nov-2014.)
(x A B 𝑉 → ((x AB) = (x A𝐶) ↔ x A B = 𝐶))

Theoremfvmptt 5183* Closed theorem form of fvmpt 5170. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)
((x(x = AB = 𝐶) 𝐹 = (x 𝐷B) (A 𝐷 𝐶 𝑉)) → (𝐹A) = 𝐶)

Theoremfvmptf 5184* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5169 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 = (x 𝐷B)       ((A 𝐷 𝐶 𝑉) → (𝐹A) = 𝐶)

Theoremfvopab6 5185* 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⟩ ∣ (φ y = B)}    &   (x = A → (φψ))    &   (x = AB = 𝐶)       ((A 𝐷 𝐶 𝑅 ψ) → (𝐹A) = 𝐶)

Theoremeqfnfv 5186* 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 A 𝐺 Fn A) → (𝐹 = 𝐺x A (𝐹x) = (𝐺x)))

Theoremeqfnfv2 5187* 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 A 𝐺 Fn B) → (𝐹 = 𝐺 ↔ (A = B x A (𝐹x) = (𝐺x))))

Theoremeqfnfv3 5188* Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐹 Fn A 𝐺 Fn B) → (𝐹 = 𝐺 ↔ (BA x A (x B (𝐹x) = (𝐺x)))))

Theoremeqfnfvd 5189* Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)
(φ𝐹 Fn A)    &   (φ𝐺 Fn A)    &   ((φ x A) → (𝐹x) = (𝐺x))       (φ𝐹 = 𝐺)

Theoremeqfnfv2f 5190* 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 5186 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)
x𝐹    &   x𝐺       ((𝐹 Fn A 𝐺 Fn A) → (𝐹 = 𝐺x A (𝐹x) = (𝐺x)))

Theoremeqfunfv 5191* 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 5192* Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)
(((𝐹 Fn A 𝐺 Fn A) BA) → ((𝐹B) = (𝐺B) ↔ x B (𝐹x) = (𝐺x)))

Theoremfndmdif 5193* Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
((𝐹 Fn A 𝐺 Fn A) → dom (𝐹𝐺) = {x A ∣ (𝐹x) ≠ (𝐺x)})

Theoremfndmdifcom 5194 The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)
((𝐹 Fn A 𝐺 Fn A) → dom (𝐹𝐺) = dom (𝐺𝐹))

Theoremfndmin 5195* Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
((𝐹 Fn A 𝐺 Fn A) → dom (𝐹𝐺) = {x A ∣ (𝐹x) = (𝐺x)})

Theoremfneqeql 5196 Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)
((𝐹 Fn A 𝐺 Fn A) → (𝐹 = 𝐺 ↔ dom (𝐹𝐺) = A))

Theoremfneqeql2 5197 Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)
((𝐹 Fn A 𝐺 Fn A) → (𝐹 = 𝐺A ⊆ dom (𝐹𝐺)))

Theoremfnreseql 5198 Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)
((𝐹 Fn A 𝐺 Fn A 𝑋A) → ((𝐹𝑋) = (𝐺𝑋) ↔ 𝑋 ⊆ dom (𝐹𝐺)))

Theoremchfnrn 5199* 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 x A (𝐹x) x) → ran 𝐹 A)

Theoremfunfvop 5200 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)
((Fun 𝐹 A dom 𝐹) → ⟨A, (𝐹A)⟩ 𝐹)

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