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Theorem List for Metamath Proof Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelfv 6101* Membership in a function value. (Contributed by NM, 30-Apr-2004.)
(𝐴 ∈ (𝐹𝐵) ↔ ∃𝑥(𝐴𝑥 ∧ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)))

Theoremfveq1 6102 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
(𝐹 = 𝐺 → (𝐹𝐴) = (𝐺𝐴))

Theoremfveq2 6103 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
(𝐴 = 𝐵 → (𝐹𝐴) = (𝐹𝐵))

Theoremfveq1i 6104 Equality inference for function value. (Contributed by NM, 2-Sep-2003.)
𝐹 = 𝐺       (𝐹𝐴) = (𝐺𝐴)

Theoremfveq1d 6105 Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)
(𝜑𝐹 = 𝐺)       (𝜑 → (𝐹𝐴) = (𝐺𝐴))

Theoremfveq2i 6106 Equality inference for function value. (Contributed by NM, 28-Jul-1999.)
𝐴 = 𝐵       (𝐹𝐴) = (𝐹𝐵)

Theoremfveq2d 6107 Equality deduction for function value. (Contributed by NM, 29-May-1999.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹𝐴) = (𝐹𝐵))

Theoremfveq12i 6108 Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)
𝐹 = 𝐺    &   𝐴 = 𝐵       (𝐹𝐴) = (𝐺𝐵)

Theoremfveq12d 6109 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹𝐴) = (𝐺𝐵))

Theoremnffv 6110 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 6111* Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝑥((𝑥𝐴𝐵)‘𝐶)

Theoremnffvd 6112 Deduction version of bound-variable hypothesis builder nffv 6110. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝑥𝐹)    &   (𝜑𝑥𝐴)       (𝜑𝑥(𝐹𝐴))

Theoremfvex 6113 The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by NM, 30-Dec-1996.)
(𝐹𝐴) ∈ V

Theoremfvif 6114 Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹𝐴), (𝐹𝐵))

Theoremiffv 6115 Move a conditional outside of a function. (Contributed by Thierry Arnoux, 28-Sep-2018.)
(if(𝜑, 𝐹, 𝐺)‘𝐴) = if(𝜑, (𝐹𝐴), (𝐺𝐴))

Theoremfv3 6116* 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(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)}

Theoremfvres 6117 The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
(𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))

Theoremfvresd 6118 The value of a restricted function, deduction version of fvres 6117. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴𝐵)       (𝜑 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))

Theoremfunssfv 6119 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 6120* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)

Theoremtz6.12 6121* 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 6122* 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 6123* Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
(∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))

Theoremtz6.12i 6124 Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by Mario Carneiro, 17-Nov-2014.)
(𝐵 ≠ ∅ → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))

Theoremfvbr0 6125 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 6126 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 6127 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 6128 The value of a class outside its domain is the empty set. (Contributed by NM, 24-Aug-1995.)
𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)

Theoremndmfvrcl 6129 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 6130 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 6131 If a function value has a member, the argument is a set. (Contributed by Mario Carneiro, 6-Nov-2015.)
(𝐴 ∈ (𝐹𝐵) → 𝐵 ∈ V)

Theoremelfvexd 6132 If a function value is nonempty, its argument is a set. Deduction form of elfvex 6131. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (𝐵𝐶))       (𝜑𝐶 ∈ V)

Theoremeliman0 6133 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 6134 The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)
𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)

Theoremnfunsn 6135 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 6136 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 6137 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
(∅‘𝐴) = ∅

Theoremfv2prc 6138 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 6139 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 6140 Equal values imply equal values in a restriction. (Contributed by NM, 13-Nov-1995.)
((𝐹𝐴) = (𝐺𝐴) → ((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴))

Theoremcsbfv12 6141 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 6142* Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)
(𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐹𝐴 / 𝑥𝐵))

Theoremcsbfv 6143* Substitution for a function value. (Contributed by NM, 1-Jan-2006.) (Revised by NM, 20-Aug-2018.)
𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴)

Theoremfunbrfv 6144 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 6145 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 6146 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 6147 Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))

Theoremfunbrfvb 6148 Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))

Theoremfunopfvb 6149 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 6150 Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)
(Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹𝐴) = 𝐵)))

Theoremdffn5 6151* 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 6152* 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 6153* 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 6154* 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 6155* 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 6156* 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 6157* 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 6158* 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 6159* Deduction form of dffn5 6151. (Contributed by Mario Carneiro, 8-Jan-2015.)
(𝜑𝐹:𝐴𝐵)       (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))

Theoremfeqresmpt 6160* Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ (𝐹𝑥)))

Theoremfeqmptdf 6161 Deduction form of dffn5f 6162. (Contributed by Mario Carneiro, 8-Jan-2015.) (Revised by Thierry Arnoux, 10-May-2017.)
𝑥𝐴    &   𝑥𝐹    &   (𝜑𝐹:𝐴𝐵)       (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))

Theoremdffn5f 6162* Representation of a function in terms of its values. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑥𝐹       (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))

Theoremfvelimab 6163* 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 6164* Deduction form of fvelimab 6163. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐵𝐴)       (𝜑 → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))

Theoremfvi 6165 The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐴𝑉 → ( I ‘𝐴) = 𝐴)

Theoremfviss 6166 The value of the identity function is a subset of the argument. (Contributed by Mario Carneiro, 27-Feb-2016.)
( I ‘𝐴) ⊆ 𝐴

Theoremfniinfv 6167* 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 6168 Singleton of function value. (Contributed by NM, 22-May-1998.)
((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))

Theoremopabiotafun 6169* 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 6170* 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 6171* 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 6172 The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011.)
((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹𝐵), (𝐹𝐶)})

Theoremssimaex 6173* The existence of a subimage. (Contributed by NM, 8-Apr-2007.)
𝐴 ∈ V       ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))

Theoremssimaexg 6174* The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
((𝐴𝐶 ∧ Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))

Theoremfunfv 6175 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 6176* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by NM, 22-May-1998.)
(Fun 𝐹 → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})

Theoremfunfv2f 6177 The value of a function. Version of funfv2 6176 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.)
𝑦𝐴    &   𝑦𝐹       (Fun 𝐹 → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})

Theoremfvun 6178 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 6179 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 6180 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 6181 Alternate definition of function value df-fv 5812 that doesn't require dummy variables. (Contributed by NM, 4-Aug-2010.)
(𝐹𝐴) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))

Theoremdmfco 6182 Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺𝐴) ∈ dom 𝐹))

Theoremfvco2 6183 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 6184 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 6185 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 6186 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 6187* 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 6188* 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 6189* 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 6190* 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 6191* 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 6192 Value of a function given by the "maps to" notation. (Contributed by Thierry Arnoux, 9-Mar-2017.)
𝑥𝐴       ((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)

Theoremfvtresfn 6193* Functionality of a tuple-restriction function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))       (𝑋𝐵 → (𝐹𝑋) = (𝑋𝑉))

Theoremfvmpts 6194* 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 6195* 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 6196* 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 6197* Deduction version of fvmpt 6191. (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 6198* 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 6199* Value of a function given by the "maps to" notation. (Contributed by Mario Carneiro, 23-Apr-2014.)
𝐹 = (𝑥𝐴𝐵)       (𝑥𝐴 → (𝐹𝑥) = ( I ‘𝐵))

Theoremfvmpt2 6200* Value of a function given by the "maps to" notation. (Contributed by FL, 21-Jun-2010.)
𝐹 = (𝑥𝐴𝐵)       ((𝑥𝐴𝐵𝐶) → (𝐹𝑥) = 𝐵)

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