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	csbfv 6801*	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 6802	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 6803	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 6804	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 6805	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 6806	Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))
Theorem	funopfvb 6807	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 6808	Version of fnbrfvb 6804 for functions on Cartesian products: function value expressed as a binary relation. See fnbrovb 7304 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 6809	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 6810*	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 6811*	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 6812*	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 6813*	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 6814*	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 6815*	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 6816*	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 6817*	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 6818*	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 6819*	Deduction form of dffn5 6810. (Contributed by Mario Carneiro, 8-Jan-2015.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
Theorem	feqresmpt 6820*	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 6821	Deduction form of dffn5f 6822. (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 6822*	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 6823*	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 6824*	Deduction form of fvelimab 6823. (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 6825	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 6826	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 6827	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 6828*	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 6829	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 6830	Obsolete version of fnsnfv 6829 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 6831*	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 6832*	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 6833*	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 6834	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 6835*	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 6836*	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 6837	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 6838*	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 6839	The value of a function. Version of funfv2 6838 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 6840	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 6841	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 6842	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 6843	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 6844	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 6845	Alternate definition of function value df-fv 6426 that doesn't require dummy variables. (Contributed by NM, 4-Aug-2010.)
⊢ (𝐹‘𝐴) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
Theorem	dmfco 6846	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 6847	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 6848	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 6849	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 6850	Value of a function composition. Deduction form of fvco3 6849. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶)))
Theorem	fvco4i 6851	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 6852*	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 6853*	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 6854*	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 7308. (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 6855*	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 6856*	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 6857*	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 6858	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 6859*	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 6860*	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 6861*	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 6862*	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 6863*	Deduction version of fvmptd 6864 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 6864*	Deduction version of fvmpt 6857. (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 6865*	Deduction version of fvmpt 6857 (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 6866*	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 6867*	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 6868*	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 6869*	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 6870*	Deduction version of fvmpt2 6868. (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 6871*	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 6769.) 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 6872*	Alternate deduction version of fvmpt 6857 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 6873*	Alternate deduction version of fvmpt 6857, 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 6874*	Alternate deduction version of fvmpt 6857, 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 6875*	Alternate deduction version of fvmpt 6857, 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 6876*	Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 6891. (Contributed by Mario Carneiro, 14-Nov-2014.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))
Theorem	fvmptt 6877*	Closed theorem form of fvmpt 6857. (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 6878*	Value of a function given by an ordered-pair class abstraction. This version of fvmptg 6855 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 6879*	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 6881 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 6880*	Deduction version of fvmpt 6857. (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 6881*	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 6855. (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 6882*	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
Theorem	elfvmptrab1w 6883*	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 6884 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by Gino Giotto, 26-Jan-2024.)
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})    &   ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)    ⇒   ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))
Theorem	elfvmptrab1 6884*	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 2372. Use the weaker elfvmptrab1w 6883 when possible. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (New usage is discouraged.)
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})    &   ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)    ⇒   ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))
Theorem	elfvmptrab 6885*	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 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑})    &   ⊢ (𝑋 ∈ 𝑉 → 𝑀 ∈ V)    ⇒   ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑀))
Theorem	fvopab4ndm 6886*	Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)
⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}    ⇒   ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐹‘𝐵) = ∅)
Theorem	fvmptndm 6887*	Value of a function given by the maps-to notation, outside of its domain. (Contributed by AV, 31-Dec-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (¬ 𝑋 ∈ 𝐴 → (𝐹‘𝑋) = ∅)
Theorem	fvmptrabfv 6888*	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 ∈ (𝐺‘𝑥) ∣ 𝜑})    &   ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓}
Theorem	fvopab5 6889*	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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = (℩𝑦𝜓))
Theorem	fvopab6 6890*	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 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅 ∧ 𝜓) → (𝐹‘𝐴) = 𝐶)
Theorem	eqfnfv 6891*	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 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
Theorem	eqfnfv2 6892*	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 (𝐹‘𝑥) = (𝐺‘𝑥))))
Theorem	eqfnfv3 6893*	Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ (𝐹‘𝑥) = (𝐺‘𝑥)))))
Theorem	eqfnfvd 6894*	Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥))    ⇒   ⊢ (𝜑 → 𝐹 = 𝐺)
Theorem	eqfnfv2f 6895*	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 6891 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐺    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
Theorem	eqfunfv 6896*	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(𝐹‘𝑥) = (𝐺‘𝑥))))
Theorem	fvreseq0 6897*	Equality of restricted functions is determined by their values (for functions with different domains). (Contributed by AV, 6-Jan-2019.)
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐶)) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
Theorem	fvreseq1 6898*	Equality of a function restricted to the domain of another function. (Contributed by AV, 6-Jan-2019.)
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ 𝐵 ⊆ 𝐴) → ((𝐹 ↾ 𝐵) = 𝐺 ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
Theorem	fvreseq 6899*	Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.) (Proof shortened by AV, 4-Mar-2019.)
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐵 ⊆ 𝐴) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
Theorem	fnmptfvd 6900*	A function with a given domain is a mapping defined by its function values. (Contributed by AV, 1-Mar-2019.)
⊢ (𝜑 → 𝑀 Fn 𝐴)    &   ⊢ (𝑖 = 𝑎 → 𝐷 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝐷 ∈ 𝑈)    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = 𝐷))