P. 69 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 6801-6900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fvbr0 6801	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(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅)
Theorem	fvrn0 6802	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.)
⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})
Theorem	fvssunirn 6803	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.)
⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹
Theorem	ndmfv 6804	The value of a class outside its domain is the empty set. (An artifact of our function value definition.) (Contributed by NM, 24-Aug-1995.)
⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
Theorem	ndmfvrcl 6805	Reverse closure law for function with the empty set not in its domain. (Contributed by NM, 26-Apr-1996.)
⊢ dom 𝐹 = 𝑆    &   ⊢ ¬ ∅ ∈ 𝑆    ⇒   ⊢ ((𝐹‘𝐴) ∈ 𝑆 → 𝐴 ∈ 𝑆)
Theorem	elfvdm 6806	If a function value has a member, then the argument belongs to the domain. (An artifact of our function value definition.) (Contributed by NM, 12-Feb-2007.) (Proof shortened by BJ, 22-Oct-2022.)
⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ dom 𝐹)
Theorem	elfvex 6807	If a function value has a member, then the argument is a set. (An artifact of our function value definition.) (Contributed by Mario Carneiro, 6-Nov-2015.)
⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ V)
Theorem	elfvexd 6808	If a function value has a member, then its argument is a set. Deduction form of elfvex 6807. (An artifact of our function value definition.) (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (𝐵‘𝐶))    ⇒   ⊢ (𝜑 → 𝐶 ∈ V)
Theorem	eliman0 6809	A nonempty function value is an element of the image of the function. (Contributed by Thierry Arnoux, 25-Jun-2019.)
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ (𝐹‘𝐴) = ∅) → (𝐹‘𝐴) ∈ (𝐹 “ 𝐵))
Theorem	nfvres 6810	The value of a non-member of a restriction is the empty set. (An artifact of our function value definition.) (Contributed by NM, 13-Nov-1995.)
⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)
Theorem	nfunsn 6811	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}) → (𝐹‘𝐴) = ∅)
Theorem	fvfundmfvn0 6812	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.)
⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
Theorem	0fv 6813	Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
⊢ (∅‘𝐴) = ∅
Theorem	fv2prc 6814	A function value of a function value at a proper class is the empty set. (Contributed by AV, 8-Apr-2021.)
⊢ (¬ 𝐴 ∈ V → ((𝐹‘𝐴)‘𝐵) = ∅)
Theorem	elfv2ex 6815	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 ∈ ((𝐹‘𝐵)‘𝐶) → 𝐵 ∈ V)
Theorem	fveqres 6816	Equal values imply equal values in a restriction. (Contributed by NM, 13-Nov-1995.)
⊢ ((𝐹‘𝐴) = (𝐺‘𝐴) → ((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴))
Theorem	csbfv12 6817	Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 20-Aug-2018.)
⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)
Theorem	csbfv2g 6818*	Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)
⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (𝐹‘⦋𝐴 / 𝑥⦌𝐵))
Theorem	csbfv 6819*	Substitution for a function value. (Contributed by NM, 1-Jan-2006.) (Revised by NM, 20-Aug-2018.)
⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴)
Theorem	funbrfv 6820	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 → (𝐹‘𝐴) = 𝐵))
Theorem	funopfv 6821	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 → (𝐹‘𝐴) = 𝐵))
Theorem	fnbrfvb 6822	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) → ((𝐹‘𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))
Theorem	fnopfvb 6823	Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))
Theorem	funbrfvb 6824	Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))
Theorem	funopfvb 6825	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) → ((𝐹‘𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
Theorem	fnbrfvb2 6826	Version of fnbrfvb 6822 for functions on Cartesian products: function value expressed as a binary relation. See fnbrovb 7324 for the form when 𝐹 is seen as a binary operation. (Contributed by BJ, 15-Feb-2022.)
⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((𝐹‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩𝐹𝐶))
Theorem	funbrfv2b 6827	Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹‘𝐴) = 𝐵)))
Theorem	dffn5 6828*	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 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
Theorem	fnrnfv 6829*	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 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
Theorem	fvelrnb 6830*	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 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
Theorem	foelrni 6831*	A member of a surjective function's codomain is a value of the function. (Contributed by Thierry Arnoux, 23-Jan-2020.)
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑌 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌)
Theorem	dfimafn 6832*	Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦})
Theorem	dfimafn2 6833*	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) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹‘𝑥)})
Theorem	funimass4 6834*	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) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
Theorem	fvelima 6835*	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 ∈ (𝐹 “ 𝐵)) → ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐴)
Theorem	fvelimad 6836*	Function value in an image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝐹    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐹 “ 𝐵))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑥) = 𝐶)
Theorem	feqmptd 6837*	Deduction form of dffn5 6828. (Contributed by Mario Carneiro, 8-Jan-2015.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
Theorem	feqresmpt 6838*	Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))
Theorem	feqmptdf 6839	Deduction form of dffn5f 6840. (Contributed by Mario Carneiro, 8-Jan-2015.) (Revised by Thierry Arnoux, 10-May-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
Theorem	dffn5f 6840*	Representation of a function in terms of its values. (Contributed by Mario Carneiro, 3-Jul-2015.)
⊢ Ⅎ𝑥𝐹    ⇒   ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
Theorem	fvelimab 6841*	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 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))
Theorem	fvelimabd 6842*	Deduction form of fvelimab 6841. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))
Theorem	unima 6843	Image of a union. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐹 “ (𝐵 ∪ 𝐶)) = ((𝐹 “ 𝐵) ∪ (𝐹 “ 𝐶)))
Theorem	fvi 6844	The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴)
Theorem	fviss 6845	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
Theorem	fniinfv 6846*	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 (𝐹‘𝑥) = ∩ ran 𝐹)
Theorem	fnsnfv 6847	Singleton of function value. (Contributed by NM, 22-May-1998.) (Proof shortened by Scott Fenton, 8-Aug-2024.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))
Theorem	fnsnfvOLD 6848	Obsolete version of fnsnfv 6847 as of 8-Aug-2024. (Contributed by NM, 22-May-1998.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))
Theorem	opabiotafun 6849*	Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 19-May-2015.)
⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}    ⇒   ⊢ Fun 𝐹
Theorem	opabiotadm 6850*	Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 16-Nov-2013.)
⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}    ⇒   ⊢ dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}
Theorem	opabiota 6851*	Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 16-Nov-2013.)
⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = (℩𝑦𝜓))
Theorem	fnimapr 6852	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}) = {(𝐹‘𝐵), (𝐹‘𝐶)})
Theorem	ssimaex 6853*	The existence of a subimage. (Contributed by NM, 8-Apr-2007.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))
Theorem	ssimaexg 6854*	The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
⊢ ((𝐴 ∈ 𝐶 ∧ Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))
Theorem	funfv 6855	A simplified expression for the value of a function when we know it is a function. (Contributed by NM, 22-May-1998.)
⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))
Theorem	funfv2 6856*	The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by NM, 22-May-1998.)
⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦})
Theorem	funfv2f 6857	The value of a function. Version of funfv2 6856 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐹    ⇒   ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦})
Theorem	fvun 6858	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) = ∅) → ((𝐹 ∪ 𝐺)‘𝐴) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴)))
Theorem	fvun1 6859	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 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋))
Theorem	fvun2 6860	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 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋))
Theorem	fvun1d 6861	The value of a union when the argument is in the first domain, a deduction version. (Contributed by metakunt, 28-May-2024.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋))
Theorem	fvun2d 6862	The value of a union when the argument is in the second domain, a deduction version. (Contributed by metakunt, 28-May-2024.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋))
Theorem	dffv2 6863	Alternate definition of function value df-fv 6441 that doesn't require dummy variables. (Contributed by NM, 4-Aug-2010.)
⊢ (𝐹‘𝐴) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
Theorem	dmfco 6864	Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺‘𝐴) ∈ dom 𝐹))
Theorem	fvco2 6865	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) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
Theorem	fvco 6866	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) → ((𝐹 ∘ 𝐺)‘𝐴) = (𝐹‘(𝐺‘𝐴)))
Theorem	fvco3 6867	Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.)
⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶)))
Theorem	fvco3d 6868	Value of a function composition. Deduction form of fvco3 6867. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶)))
Theorem	fvco4i 6869	Conditions for a composition to be expandable without conditions on the argument. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ ∅ = (𝐹‘∅)    &   ⊢ Fun 𝐺    ⇒   ⊢ ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))
Theorem	fvopab3g 6870*	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 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦𝜑)    &   ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐹‘𝐴) = 𝐵 ↔ 𝜒))
Theorem	fvopab3ig 6871*	Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 ∈ 𝐶 → ∃*𝑦𝜑)    &   ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝜒 → (𝐹‘𝐴) = 𝐵))
Theorem	brfvopabrbr 6872*	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 7329. (Contributed by AV, 29-Oct-2021.)
⊢ (𝐴‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐵‘𝑍)𝑦 ∧ 𝜑)}    &   ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓))    &   ⊢ Rel (𝐵‘𝑍)    ⇒   ⊢ (𝑋(𝐴‘𝑍)𝑌 ↔ (𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓))
Theorem	fvmptg 6873*	Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)    ⇒   ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝐹‘𝐴) = 𝐶)
Theorem	fvmpti 6874*	Value of a function given in maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)    ⇒   ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶))
Theorem	fvmpt 6875*	Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶)
Theorem	fvmpt2f 6876	Value of a function given by the maps-to notation. (Contributed by Thierry Arnoux, 9-Mar-2017.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
Theorem	fvtresfn 6877*	Functionality of a tuple-restriction function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉))    ⇒   ⊢ (𝑋 ∈ 𝐵 → (𝐹‘𝑋) = (𝑋 ↾ 𝑉))
Theorem	fvmpts 6878*	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 ∈ 𝐶 ↦ 𝐵)    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵)
Theorem	fvmpt3 6879*	Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)    &   ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶)
Theorem	fvmpt3i 6880*	Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶)
Theorem	fvmptdf 6881*	Deduction version of fvmptd 6882 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by AV, 29-Mar-2024.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐶    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)
Theorem	fvmptd 6882*	Deduction version of fvmpt 6875. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) (Proof shortened by AV, 29-Mar-2024.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)
Theorem	fvmptd2 6883*	Deduction version of fvmpt 6875 (where the definition of the mapping does not depend on the common antecedent 𝜑). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)
Theorem	mptrcl 6884*	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 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (𝐼 ∈ (𝐹‘𝑋) → 𝑋 ∈ 𝐴)
Theorem	fvmpt2i 6885*	Value of a function given by the maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = ( I ‘𝐵))
Theorem	fvmpt2 6886*	Value of a function given by the maps-to notation. (Contributed by FL, 21-Jun-2010.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐹‘𝑥) = 𝐵)
Theorem	fvmptss 6887*	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 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶)
Theorem	fvmpt2d 6888*	Deduction version of fvmpt2 6886. (Contributed by Thierry Arnoux, 8-Dec-2016.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
Theorem	fvmptex 6889*	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 6787.) 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 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵))    ⇒   ⊢ (𝐹‘𝐶) = (𝐺‘𝐶)
Theorem	fvmptd3f 6890*	Alternate deduction version of fvmpt 6875 with three nonfreeness hypotheses instead of distinct variable conditions. (Contributed by AV, 19-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓))    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓))
Theorem	fvmptd2f 6891*	Alternate deduction version of fvmpt 6875, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) (Proof shortened by AV, 19-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓))    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓))
Theorem	fvmptdv 6892*	Alternate deduction version of fvmpt 6875, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓))    ⇒   ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓))
Theorem	fvmptdv2 6893*	Alternate deduction version of fvmpt 6875, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = 𝐶))
Theorem	mpteqb 6894*	Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 6909. (Contributed by Mario Carneiro, 14-Nov-2014.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))
Theorem	fvmptt 6895*	Closed theorem form of fvmpt 6875. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)
⊢ ((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → (𝐹‘𝐴) = 𝐶)
Theorem	fvmptf 6896*	Value of a function given by an ordered-pair class abstraction. This version of fvmptg 6873 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)    ⇒   ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶)
Theorem	fvmptnf 6897*	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 6899 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)    ⇒   ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐴) = ∅)
Theorem	fvmptd3 6898*	Deduction version of fvmpt 6875. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)
Theorem	fvmptn 6899*	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 6873. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 9-Sep-2013.)
⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐷) = ∅)
Theorem	fvmptss2 6900*	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 = 𝐷 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (𝐹‘𝐷) ⊆ 𝐶