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Theorem List for Metamath Proof Explorer - 6001-6100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfvresd 6001 The value of a restricted function, deduction version of fvres 6000. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴𝐵)       (𝜑 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
 
Theoremfunssfv 6002 The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.)
((Fun 𝐹𝐺𝐹𝐴 ∈ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
 
Theoremtz6.12-1 6003* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
 
Theoremtz6.12 6004* 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 6005* 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 6006* Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
(∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
 
Theoremtz6.12i 6007 Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by Mario Carneiro, 17-Nov-2014.)
(𝐵 ≠ ∅ → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))
 
Theoremfvbr0 6008 Two possibilities for the behavior of a function value. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
(𝑋𝐹(𝐹𝑋) ∨ (𝐹𝑋) = ∅)
 
Theoremfvrn0 6009 A function value is a member of the range plus null. (Contributed by Scott Fenton, 8-Jun-2011.) (Revised by Stefan O'Rear, 3-Jan-2015.)
(𝐹𝑋) ∈ (ran 𝐹 ∪ {∅})
 
Theoremfvssunirn 6010 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, 31-Aug-2015.)
(𝐹𝑋) ⊆ ran 𝐹
 
Theoremndmfv 6011 The value of a class outside its domain is the empty set. (Contributed by NM, 24-Aug-1995.)
𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
 
Theoremndmfvrcl 6012 Reverse closure law for function with the empty set not in its domain. (Contributed by NM, 26-Apr-1996.)
dom 𝐹 = 𝑆    &    ¬ ∅ ∈ 𝑆       ((𝐹𝐴) ∈ 𝑆𝐴𝑆)
 
Theoremelfvdm 6013 If a function value has a member, the argument belongs to the domain. (Contributed by NM, 12-Feb-2007.)
(𝐴 ∈ (𝐹𝐵) → 𝐵 ∈ dom 𝐹)
 
Theoremelfvex 6014 If a function value has a member, the argument is a set. (Contributed by Mario Carneiro, 6-Nov-2015.)
(𝐴 ∈ (𝐹𝐵) → 𝐵 ∈ V)
 
Theoremelfvexd 6015 If a function value is nonempty, its argument is a set. Deduction form of elfvex 6014. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (𝐵𝐶))       (𝜑𝐶 ∈ V)
 
Theoremeliman0 6016 A non-nul function value is an element of the image of the function. (Contributed by Thierry Arnoux, 25-Jun-2019.)
((𝐴𝐵 ∧ ¬ (𝐹𝐴) = ∅) → (𝐹𝐴) ∈ (𝐹𝐵))
 
Theoremnfvres 6017 The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)
𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)
 
Theoremnfunsn 6018 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) = ∅)
 
Theoremfvfundmfvn0 6019 If a class' value at an argument is not the empty set, the argument is contained in the domain of the class, and the class restricted to the argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐹𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
 
Theorem0fv 6020 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
(∅‘𝐴) = ∅
 
Theoremfv2prc 6021 A function's value at a function's value at a proper class is the empty set. (Contributed by AV, 8-Apr-2021.)
𝐴 ∈ V → ((𝐹𝐴)‘𝐵) = ∅)
 
Theoremelfv2ex 6022 If a function value of a function value has a member, the first argument is a set. (Contributed by AV, 8-Apr-2021.)
(𝐴 ∈ ((𝐹𝐵)‘𝐶) → 𝐵 ∈ V)
 
Theoremfveqres 6023 Equal values imply equal values in a restriction. (Contributed by NM, 13-Nov-1995.)
((𝐹𝐴) = (𝐺𝐴) → ((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴))
 
Theoremcsbfv12 6024 Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 20-Aug-2018.)
𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
 
Theoremcsbfv2g 6025* Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)
(𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐹𝐴 / 𝑥𝐵))
 
Theoremcsbfv 6026* Substitution for a function value. (Contributed by NM, 1-Jan-2006.) (Revised by NM, 20-Aug-2018.)
𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴)
 
Theoremfunbrfv 6027 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 6028 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 6029 Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶𝐵𝐹𝐶))
 
Theoremfnopfvb 6030 Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))
 
Theoremfunbrfvb 6031 Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))
 
Theoremfunopfvb 6032 Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
 
Theoremfunbrfv2b 6033 Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)
(Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹𝐴) = 𝐵)))
 
Theoremdffn5 6034* Representation of a function in terms of its values. (Contributed by FL, 14-Sep-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
(𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
 
Theoremfnrnfv 6035* 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 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
 
Theoremfvelrnb 6036* 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))
 
Theoremfoelrni 6037* A member of a surjective function's codomain is a value of the function. (Contributed by Thierry Arnoux, 23-Jan-2020.)
((𝐹:𝐴onto𝐵𝑌𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑌)
 
Theoremdfimafn 6038* 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 6039* 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 6040* 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 6041* 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 6042* Deduction form of dffn5 6034. (Contributed by Mario Carneiro, 8-Jan-2015.)
(𝜑𝐹:𝐴𝐵)       (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
 
Theoremfeqresmpt 6043* Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ (𝐹𝑥)))
 
Theoremfeqmptdf 6044 Deduction form of dffn5f 6045. (Contributed by Mario Carneiro, 8-Jan-2015.) (Revised by Thierry Arnoux, 10-May-2017.)
𝑥𝐴    &   𝑥𝐹    &   (𝜑𝐹:𝐴𝐵)       (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
 
Theoremdffn5f 6045* Representation of a function in terms of its values. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑥𝐹       (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
 
Theoremfvelimab 6046* 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))
 
Theoremfvelimabd 6047* Deduction form of fvelimab 6046. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐵𝐴)       (𝜑 → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
 
Theoremfvi 6048 The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐴𝑉 → ( I ‘𝐴) = 𝐴)
 
Theoremfviss 6049 The value of the identity function is a subset of the argument. (Contributed by Mario Carneiro, 27-Feb-2016.)
( I ‘𝐴) ⊆ 𝐴
 
Theoremfniinfv 6050* 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 6051 Singleton of function value. (Contributed by NM, 22-May-1998.)
((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))
 
Theoremopabiotafun 6052* Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 19-May-2015.)
𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}       Fun 𝐹
 
Theoremopabiotadm 6053* Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 16-Nov-2013.)
𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}       dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}
 
Theoremopabiota 6054* Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 16-Nov-2013.)
𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}    &   (𝑥 = 𝐵 → (𝜑𝜓))       (𝐵 ∈ dom 𝐹 → (𝐹𝐵) = (℩𝑦𝜓))
 
Theoremfnimapr 6055 The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011.)
((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹𝐵), (𝐹𝐶)})
 
Theoremssimaex 6056* The existence of a subimage. (Contributed by NM, 8-Apr-2007.)
𝐴 ∈ V       ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
 
Theoremssimaexg 6057* The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
((𝐴𝐶 ∧ Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
 
Theoremfunfv 6058 A simplified expression for the value of a function when we know it's a function. (Contributed by NM, 22-May-1998.)
(Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
 
Theoremfunfv2 6059* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by NM, 22-May-1998.)
(Fun 𝐹 → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
 
Theoremfunfv2f 6060 The value of a function. Version of funfv2 6059 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.)
𝑦𝐴    &   𝑦𝐹       (Fun 𝐹 → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
 
Theoremfvun 6061 Value of the union of two functions when the domains are separate. (Contributed by FL, 7-Nov-2011.)
(((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺)‘𝐴) = ((𝐹𝐴) ∪ (𝐺𝐴)))
 
Theoremfvun1 6062 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 6063 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))
 
Theoremdffv2 6064 Alternate definition of function value df-fv 5697 that doesn't require dummy variables. (Contributed by NM, 4-Aug-2010.)
(𝐹𝐴) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
 
Theoremdmfco 6065 Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺𝐴) ∈ dom 𝐹))
 
Theoremfvco2 6066 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 6067 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 6068 Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.)
((𝐺:𝐴𝐵𝐶𝐴) → ((𝐹𝐺)‘𝐶) = (𝐹‘(𝐺𝐶)))
 
Theoremfvco4i 6069 Conditions for a composition to be expandable without conditions on the argument. (Contributed by Stefan O'Rear, 31-Mar-2015.)
∅ = (𝐹‘∅)    &   Fun 𝐺       ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋))
 
Theoremfvopab3g 6070* 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 6071* 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))
 
Theoremfvmptg 6072* 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)
 
Theoremfvmpti 6073* Value of a function given in maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.)
(𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)       (𝐴𝐷 → (𝐹𝐴) = ( I ‘𝐶))
 
Theoremfvmpt 6074* 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)
 
Theoremfvmpt2f 6075 Value of a function given by the "maps to" notation. (Contributed by Thierry Arnoux, 9-Mar-2017.)
𝑥𝐴       ((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
 
Theoremfvtresfn 6076* Functionality of a tuple-restriction function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))       (𝑋𝐵 → (𝐹𝑋) = (𝑋𝑉))
 
Theoremfvmpts 6077* 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 6078* 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 6079* 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 6080* Deduction version of fvmpt 6074. (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 6081* 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.)
𝐹 = (𝑥𝐴𝐵)       (𝐼 ∈ (𝐹𝑋) → 𝑋𝐴)
 
Theoremfvmpt2i 6082* Value of a function given by the "maps to" notation. (Contributed by Mario Carneiro, 23-Apr-2014.)
𝐹 = (𝑥𝐴𝐵)       (𝑥𝐴 → (𝐹𝑥) = ( I ‘𝐵))
 
Theoremfvmpt2 6083* Value of a function given by the "maps to" notation. (Contributed by FL, 21-Jun-2010.)
𝐹 = (𝑥𝐴𝐵)       ((𝑥𝐴𝐵𝐶) → (𝐹𝑥) = 𝐵)
 
Theoremfvmptss 6084* If all the values of the mapping are subsets of a class 𝐶, then so is any evaluation of the mapping, even if 𝐷 is not in the base set 𝐴. (Contributed by Mario Carneiro, 13-Feb-2015.)
𝐹 = (𝑥𝐴𝐵)       (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)
 
Theoremfvmpt2d 6085* Deduction version of fvmpt2 6083. (Contributed by Thierry Arnoux, 8-Dec-2016.)
(𝜑𝐹 = (𝑥𝐴𝐵))    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
 
Theoremfvmptex 6086* Express a function 𝐹 whose value 𝐵 may not always be a set in terms of another function 𝐺 for which sethood is guaranteed. (Note that ( I ‘𝐵) is just shorthand for if(𝐵 ∈ V, 𝐵, ∅), and it is always a set by fvex 5996.) Note also that these functions are not the same; wherever 𝐵(𝐶) is not a set, 𝐶 is not in the domain of 𝐹 (so it evaluates to the empty set), but 𝐶 is in the domain of 𝐺, and 𝐺(𝐶) is defined to be the empty set. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴 ↦ ( I ‘𝐵))       (𝐹𝐶) = (𝐺𝐶)
 
Theoremfvmptdf 6087* Alternate deduction version of fvmpt 6074, 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 6088* Alternate deduction version of fvmpt 6074, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
 
Theoremfvmptdv2 6089* Alternate deduction version of fvmpt 6074, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = 𝐶))
 
Theoremmpteqb 6090* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 6102. (Contributed by Mario Carneiro, 14-Nov-2014.)
(∀𝑥𝐴 𝐵𝑉 → ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ↔ ∀𝑥𝐴 𝐵 = 𝐶))
 
Theoremfvmptt 6091* Closed theorem form of fvmpt 6074. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)
((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → (𝐹𝐴) = 𝐶)
 
Theoremfvmptf 6092* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 6072 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)
 
Theoremfvmptnf 6093* The value of a function given by an ordered-pair class abstraction is the empty set when the class it would otherwise map to is a proper class. This version of fvmptn 6094 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
𝑥𝐴    &   𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)       𝐶 ∈ V → (𝐹𝐴) = ∅)
 
Theoremfvmptn 6094* This somewhat non-intuitive theorem tells us the value of its function is the empty set when the class 𝐶 it would otherwise map to is a proper class. This is a technical lemma that can help eliminate redundant sethood antecedents otherwise required by fvmptg 6072. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 9-Sep-2013.)
(𝑥 = 𝐷𝐵 = 𝐶)    &   𝐹 = (𝑥𝐴𝐵)       𝐶 ∈ V → (𝐹𝐷) = ∅)
 
Theoremfvmptss2 6095* 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.)
(𝑥 = 𝐷𝐵 = 𝐶)    &   𝐹 = (𝑥𝐴𝐵)       (𝐹𝐷) ⊆ 𝐶
 
Theoremelfvmptrab1 6096* 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 6097* 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))
 
Theoremfvopab4ndm 6098* Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)
𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}       𝐵𝐴 → (𝐹𝐵) = ∅)
 
Theoremfvmptndm 6099* Value of a function given by the "maps to" notation, outside of its domain. (Contributed by AV, 31-Dec-2020.)
𝐹 = (𝑥𝐴𝐵)       𝑋𝐴 → (𝐹𝑋) = ∅)
 
Theoremfvopab5 6100* The value of a function that is expressed as an ordered pair abstraction. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 11-Sep-2015.)
𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (𝐹𝐴) = (℩𝑦𝜓))
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