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Theorem List for Metamath Proof Explorer - 6701-6800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnfunsn 6701 If the restriction of a class to a singleton is not a function, then its value is the empty set. (An artifact of our function value definition.) (Contributed by NM, 8-Aug-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ∅)

Theoremfvfundmfvn0 6702 If the "value of a class" at an argument is not the empty set, then the argument is in the domain of the class and the class restricted to the singleton formed on that argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.) (Proof shortened by BJ, 13-Aug-2022.)
((𝐹𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

Theorem0fv 6703 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
(∅‘𝐴) = ∅

Theoremfv2prc 6704 A function value of a function value at a proper class is the empty set. (Contributed by AV, 8-Apr-2021.)
𝐴 ∈ V → ((𝐹𝐴)‘𝐵) = ∅)

Theoremelfv2ex 6705 If a function value of a function value has a member, then the first argument is a set. (Contributed by AV, 8-Apr-2021.)
(𝐴 ∈ ((𝐹𝐵)‘𝐶) → 𝐵 ∈ V)

Theoremfveqres 6706 Equal values imply equal values in a restriction. (Contributed by NM, 13-Nov-1995.)
((𝐹𝐴) = (𝐺𝐴) → ((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴))

Theoremcsbfv12 6707 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 6708* Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)
(𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐹𝐴 / 𝑥𝐵))

Theoremcsbfv 6709* Substitution for a function value. (Contributed by NM, 1-Jan-2006.) (Revised by NM, 20-Aug-2018.)
𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴)

Theoremfunbrfv 6710 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 6711 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 6712 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 6713 Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))

Theoremfunbrfvb 6714 Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))

Theoremfunopfvb 6715 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))

Theoremfnbrfvb2 6716 Version of fnbrfvb 6712 for functions on Cartesian products: function value expressed as a binary relation. See fnbrovb 7199 for the form when 𝐹 is seen as a binary operation. (Contributed by BJ, 15-Feb-2022.)
((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴𝑉𝐵𝑊)) → ((𝐹‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵𝐹𝐶))

Theoremfunbrfv2b 6717 Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)
(Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹𝐴) = 𝐵)))

Theoremdffn5 6718* 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 6719* 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 6720* 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 6721* 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 6722* 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 6723* 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 6724* 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 6725* 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)

Theoremfvelimad 6726* Function value in an image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐹    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐶 ∈ (𝐹𝐵))       (𝜑 → ∃𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = 𝐶)

Theoremfeqmptd 6727* Deduction form of dffn5 6718. (Contributed by Mario Carneiro, 8-Jan-2015.)
(𝜑𝐹:𝐴𝐵)       (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))

Theoremfeqresmpt 6728* Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ (𝐹𝑥)))

Theoremfeqmptdf 6729 Deduction form of dffn5f 6730. (Contributed by Mario Carneiro, 8-Jan-2015.) (Revised by Thierry Arnoux, 10-May-2017.)
𝑥𝐴    &   𝑥𝐹    &   (𝜑𝐹:𝐴𝐵)       (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))

Theoremdffn5f 6730* Representation of a function in terms of its values. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑥𝐹       (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))

Theoremfvelimab 6731* 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 6732* Deduction form of fvelimab 6731. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐵𝐴)       (𝜑 → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))

Theoremunima 6733 Image of a union. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ (𝐵𝐶)) = ((𝐹𝐵) ∪ (𝐹𝐶)))

Theoremfvi 6734 The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐴𝑉 → ( I ‘𝐴) = 𝐴)

Theoremfviss 6735 The value of the identity function is a subset of the argument. (An artifact of our function value definition.) (Contributed by Mario Carneiro, 27-Feb-2016.)
( I ‘𝐴) ⊆ 𝐴

Theoremfniinfv 6736* 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 6737 Singleton of function value. (Contributed by NM, 22-May-1998.)
((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))

Theoremopabiotafun 6738* 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 6739* 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 6740* 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 6741 The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011.)
((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹𝐵), (𝐹𝐶)})

Theoremssimaex 6742* The existence of a subimage. (Contributed by NM, 8-Apr-2007.)
𝐴 ∈ V       ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))

Theoremssimaexg 6743* The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
((𝐴𝐶 ∧ Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))

Theoremfunfv 6744 A simplified expression for the value of a function when we know it is a function. (Contributed by NM, 22-May-1998.)
(Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))

Theoremfunfv2 6745* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by NM, 22-May-1998.)
(Fun 𝐹 → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})

Theoremfunfv2f 6746 The value of a function. Version of funfv2 6745 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.)
𝑦𝐴    &   𝑦𝐹       (Fun 𝐹 → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})

Theoremfvun 6747 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 6748 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 6749 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 6750 Alternate definition of function value df-fv 6357 that doesn't require dummy variables. (Contributed by NM, 4-Aug-2010.)
(𝐹𝐴) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))

Theoremdmfco 6751 Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺𝐴) ∈ dom 𝐹))

Theoremfvco2 6752 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 6753 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 6754 Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.)
((𝐺:𝐴𝐵𝐶𝐴) → ((𝐹𝐺)‘𝐶) = (𝐹‘(𝐺𝐶)))

Theoremfvco3d 6755 Value of a function composition. Deduction form of fvco3 6754. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐺:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → ((𝐹𝐺)‘𝐶) = (𝐹‘(𝐺𝐶)))

Theoremfvco4i 6756 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 6757* 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 6758* 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))

Theorembrfvopabrbr 6759* The binary relation of a function value which is an ordered-pair class abstraction of a restricted binary relation is the restricted binary relation. The first hypothesis can often be obtained by using fvmptopab 7203. (Contributed by AV, 29-Oct-2021.)
(𝐴𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐵𝑍)𝑦𝜑)}    &   ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑𝜓))    &   Rel (𝐵𝑍)       (𝑋(𝐴𝑍)𝑌 ↔ (𝑋(𝐵𝑍)𝑌𝜓))

Theoremfvmptg 6760* 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 6761* 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 6762* 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 6763 Value of a function given by the maps-to notation. (Contributed by Thierry Arnoux, 9-Mar-2017.)
𝑥𝐴       ((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)

Theoremfvtresfn 6764* Functionality of a tuple-restriction function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))       (𝑋𝐵 → (𝐹𝑋) = (𝑋𝑉))

Theoremfvmpts 6765* 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 6766* 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 6767* 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)

Theoremfvmptdf 6768* Deduction version of fvmptd 6769 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by AV, 29-Mar-2024.)
(𝜑𝐹 = (𝑥𝐷𝐵))    &   ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)    &   (𝜑𝐴𝐷)    &   (𝜑𝐶𝑉)    &   𝑥𝜑    &   𝑥𝐴    &   𝑥𝐶       (𝜑 → (𝐹𝐴) = 𝐶)

Theoremfvmptd 6769* Deduction version of fvmpt 6762. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) (Proof shortened by AV, 29-Mar-2024.)
(𝜑𝐹 = (𝑥𝐷𝐵))    &   ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)    &   (𝜑𝐴𝐷)    &   (𝜑𝐶𝑉)       (𝜑 → (𝐹𝐴) = 𝐶)

Theoremfvmptd2 6770* Deduction version of fvmpt 6762 (where the definition of the mapping does not depend on the common antecedent 𝜑). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝐹 = (𝑥𝐷𝐵)    &   ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)    &   (𝜑𝐴𝐷)    &   (𝜑𝐶𝑉)       (𝜑 → (𝐹𝐴) = 𝐶)

Theoremmptrcl 6771* 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 6772* Value of a function given by the maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.)
𝐹 = (𝑥𝐴𝐵)       (𝑥𝐴 → (𝐹𝑥) = ( I ‘𝐵))

Theoremfvmpt2 6773* Value of a function given by the maps-to notation. (Contributed by FL, 21-Jun-2010.)
𝐹 = (𝑥𝐴𝐵)       ((𝑥𝐴𝐵𝐶) → (𝐹𝑥) = 𝐵)

Theoremfvmptss 6774* 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 6775* Deduction version of fvmpt2 6773. (Contributed by Thierry Arnoux, 8-Dec-2016.)
(𝜑𝐹 = (𝑥𝐴𝐵))    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)

Theoremfvmptex 6776* 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 6677.) 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)

Theoremfvmptd3f 6777* Alternate deduction version of fvmpt 6762 with three non-freeness hypotheses instead of distinct variable conditions. (Contributed by AV, 19-Jan-2022.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))    &   𝑥𝐹    &   𝑥𝜓    &   𝑥𝜑       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))

Theoremfvmptd2f 6778* Alternate deduction version of fvmpt 6762, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) (Proof shortened by AV, 19-Jan-2022.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))    &   𝑥𝐹    &   𝑥𝜓       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))

Theoremfvmptdv 6779* Alternate deduction version of fvmpt 6762, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))

Theoremfvmptdv2 6780* Alternate deduction version of fvmpt 6762, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = 𝐶))

Theoremmpteqb 6781* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 6796. (Contributed by Mario Carneiro, 14-Nov-2014.)
(∀𝑥𝐴 𝐵𝑉 → ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ↔ ∀𝑥𝐴 𝐵 = 𝐶))

Theoremfvmptt 6782* Closed theorem form of fvmpt 6762. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)
((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → (𝐹𝐴) = 𝐶)

Theoremfvmptf 6783* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 6760 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 6784* 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 6786 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) = ∅)

Theoremfvmptd3 6785* Deduction version of fvmpt 6762. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝐹 = (𝑥𝐷𝐵)    &   (𝑥 = 𝐴𝐵 = 𝐶)    &   (𝜑𝐴𝐷)    &   (𝜑𝐶𝑉)       (𝜑 → (𝐹𝐴) = 𝐶)

Theoremfvmptn 6786* 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 6760. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 9-Sep-2013.)
(𝑥 = 𝐷𝐵 = 𝐶)    &   𝐹 = (𝑥𝐴𝐵)       𝐶 ∈ V → (𝐹𝐷) = ∅)

Theoremfvmptss2 6787* 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

Theoremelfvmptrab1w 6788* 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. Version of elfvmptrab1 6789 with a disjoint variable condition, which does not require ax-13 2386. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by Gino Giotto, 26-Jan-2024.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})    &   (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)       (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))

Theoremelfvmptrab1 6789* 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. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker elfvmptrab1w 6788 when possible. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (New usage is discouraged.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})    &   (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)       (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))

Theoremelfvmptrab 6790* 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 6791* 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 6792* Value of a function given by the maps-to notation, outside of its domain. (Contributed by AV, 31-Dec-2020.)
𝐹 = (𝑥𝐴𝐵)       𝑋𝐴 → (𝐹𝑋) = ∅)

Theoremfvmptrabfv 6793* Value of a function mapping a set to a class abstraction restricting the value of another function. (Contributed by AV, 18-Feb-2022.)
𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ (𝐺𝑥) ∣ 𝜑})    &   (𝑥 = 𝑋 → (𝜑𝜓))       (𝐹𝑋) = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓}

Theoremfvopab5 6794* 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))

Theoremfvopab6 6795* 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 6796* 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 6797* 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 6798* 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 6799* Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐴)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))       (𝜑𝐹 = 𝐺)

Theoremeqfnfv2f 6800* 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 6796 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)))

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