P. 61 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 6001-6100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	inres 6001	Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
⊢ (𝐴 ∩ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ 𝐵) ↾ 𝐶)
Theorem	resdifcom 6002	Commutative law for restriction and difference. (Contributed by AV, 7-Jun-2021.)
⊢ ((𝐴 ↾ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ↾ 𝐵)
Theorem	resiun1 6003*	Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.) (Proof shortened by JJ, 25-Aug-2021.)
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶)
Theorem	resiun2 6004*	Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
⊢ (𝐶 ↾ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵)
Theorem	resss 6005	A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
Theorem	rescom 6006	Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ↾ 𝐵)
Theorem	ssres 6007	Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶))
Theorem	ssres2 6008	Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵))
Theorem	relres 6009	A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ Rel (𝐴 ↾ 𝐵)
Theorem	resabs1 6010	Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)
⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵))
Theorem	resabs1i 6011	Absorption law for restriction. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵)
Theorem	resabs1d 6012	Absorption law for restriction, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵))
Theorem	resabs2 6013	Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵))
Theorem	residm 6014	Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
⊢ ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵)
Theorem	dmresss 6015	The domain of a restriction is a subset of the original domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.) Proof shortened and axiom usage reduced. (Proof shortened by AV, 15-May-2025.)
⊢ dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴
Theorem	dmres 6016	The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)
⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
Theorem	ssdmres 6017	A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
⊢ (𝐴 ⊆ dom 𝐵 ↔ dom (𝐵 ↾ 𝐴) = 𝐴)
Theorem	dmresexg 6018	The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
⊢ (𝐵 ∈ 𝑉 → dom (𝐴 ↾ 𝐵) ∈ V)
Theorem	resima 6019	A restriction to an image. (Contributed by NM, 29-Sep-2004.)
⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = (𝐴 “ 𝐵)
Theorem	resima2 6020	Image under a restricted class. (Contributed by FL, 31-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.)
⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵))
Theorem	rnresss 6021	The range of a restriction is a subset of the whole range. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴
Theorem	xpssres 6022	Restriction of a constant function (or other Cartesian product). (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ (𝐶 ⊆ 𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))
Theorem	elinxp 6023*	Membership in an intersection with a Cartesian product. (Contributed by Peter Mazsa, 9-Sep-2022.)
⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
Theorem	elres 6024*	Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.) (Proof shortened by Peter Mazsa, 9-Sep-2022.)
⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
Theorem	elsnres 6025*	Membership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.)
⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑦(𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵))
Theorem	relssres 6026	Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → (𝐴 ↾ 𝐵) = 𝐴)
Theorem	dmressnsn 6027	The domain of a restriction to a singleton is a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
⊢ (𝐴 ∈ dom 𝐹 → dom (𝐹 ↾ {𝐴}) = {𝐴})
Theorem	eldmressnsn 6028	The element of the domain of a restriction to a singleton is the element of the singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ {𝐴}))
Theorem	eldmeldmressn 6029	An element of the domain (of a relation) is an element of the domain of the restriction (of the relation) to the singleton containing this element. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
⊢ (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ dom (𝐹 ↾ {𝑋}))
Theorem	resdm 6030	A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
Theorem	resexg 6031	The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V)
Theorem	resexd 6032	The restriction of a set is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ V)
Theorem	resex 6033	The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ↾ 𝐵) ∈ V
Theorem	resindm 6034	When restricting a class, intersecting with the domain of the class has no effect. (Contributed by FL, 6-Oct-2008.) Remove antecedent. (Revised by Eric Schmidt, 16-Jun-2026.)
⊢ (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵)
Theorem	resindmOLD 6035	Obsolete version of resindm 6034 as of 16-Jun-2026. (Contributed by FL, 6-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵))
Theorem	resdmdfsn 6036	Restricting a class to its domain without a set is the same as restricting the class to the universe without this set. (Contributed by AV, 2-Dec-2018.) Remove antecedent. (Revised by Eric Schmidt, 16-Jun-2026.)
⊢ (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))
Theorem	resdmdfsnOLD 6037	Obsolete version of resdmdfsn 6036 as of 16-Jun-2026. (Contributed by AV, 2-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋})))
Theorem	reldmun 6038	Split a relation into two parts based on its domain. (Contributed by Thierry Arnoux, 9-Oct-2023.) Remove requirement that 𝐴 and 𝐵 are disjoint. (Revised by Eric Schmidt, 20-Jun-2026.)
⊢ ((Rel 𝑅 ∧ dom 𝑅 = (𝐴 ∪ 𝐵)) → 𝑅 = ((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵)))
Theorem	reldisjunOLD 6039	Obsolete version of reldmun 6038 as of 20-Jun-2026. (Contributed by Thierry Arnoux, 9-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((Rel 𝑅 ∧ dom 𝑅 = (𝐴 ∪ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑅 = ((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵)))
Theorem	relresdm1 6040	Restriction of a disjoint union to the domain of the first term. (Contributed by Thierry Arnoux, 9-Dec-2021.)
⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ∪ 𝐵) ↾ dom 𝐴) = 𝐴)
Theorem	resopab 6041*	Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
Theorem	iss 6042	A subclass of the identity function is the identity function restricted to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝐴 ⊆ I ↔ 𝐴 = ( I ↾ dom 𝐴))
Theorem	resopab2 6043*	Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)
⊢ (𝐴 ⊆ 𝐵 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
Theorem	resmpt 6044*	Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)
⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶))
Theorem	resmpt3 6045*	Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)
⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶)
Theorem	resmptf 6046	Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶))
Theorem	resmptd 6047*	Restriction of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶))
Theorem	dfres2 6048*	Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
⊢ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}
Theorem	mptss 6049*	Sufficient condition for inclusion among two functions in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶))
Theorem	elimampt 6050*	Membership in the image of a mapping. (Contributed by Thierry Arnoux, 3-Jan-2022.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐷 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐷) ↔ ∃𝑥 ∈ 𝐷 𝐶 = 𝐵))
Theorem	elidinxp 6051*	Characterization of the elements of the intersection of the identity relation with a Cartesian product. (Contributed by Peter Mazsa, 9-Sep-2022.)
⊢ (𝐶 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝐶 = ⟨𝑥, 𝑥⟩)
Theorem	elidinxpid 6052*	Characterization of the elements of the intersection of the identity relation with a Cartesian square. (Contributed by Peter Mazsa, 9-Sep-2022.)
⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ 𝐴 𝐵 = ⟨𝑥, 𝑥⟩)
Theorem	elrid 6053*	Characterization of the elements of a restricted identity relation. (Contributed by BJ, 28-Aug-2022.) (Proof shortened by Peter Mazsa, 9-Sep-2022.)
⊢ (𝐴 ∈ ( I ↾ 𝑋) ↔ ∃𝑥 ∈ 𝑋 𝐴 = ⟨𝑥, 𝑥⟩)
Theorem	idinxpres 6054	The intersection of the identity relation with a cartesian product is the restriction of the identity relation to the intersection of the factors. (Contributed by FL, 2-Aug-2009.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) Generalize statement from cartesian square (now idinxpresid 6055) to cartesian product. (Revised by BJ, 23-Dec-2023.)
⊢ ( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴 ∩ 𝐵))
Theorem	idinxpresid 6055	The intersection of the identity relation with the cartesian square of a class is the restriction of the identity relation to that class. (Contributed by FL, 2-Aug-2009.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) (Proof shortened by BJ, 23-Dec-2023.)
⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
Theorem	idssxp 6056	A diagonal set as a subset of a Cartesian square. (Contributed by Thierry Arnoux, 29-Dec-2019.) (Proof shortened by BJ, 9-Sep-2022.)
⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
Theorem	opabresid 6057*	The restricted identity relation expressed as an ordered-pair class abstraction. (Contributed by FL, 25-Apr-2012.)
⊢ ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)}
Theorem	mptresid 6058*	The restricted identity relation expressed in maps-to notation. (Contributed by FL, 25-Apr-2012.)
⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)
Theorem	dmresi 6059	The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
⊢ dom ( I ↾ 𝐴) = 𝐴
Theorem	restidsing 6060	Restriction of the identity to a singleton. (Contributed by FL, 2-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.) (Proof shortened by Peter Mazsa, 6-Oct-2022.)
⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴})
Theorem	iresn0n0 6061	The identity function restricted to a class 𝐴 is empty iff the class 𝐴 is empty. (Contributed by AV, 30-Jan-2024.)
⊢ (𝐴 = ∅ ↔ ( I ↾ 𝐴) = ∅)
Theorem	imaeq1 6062	Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶))
Theorem	imaeq2 6063	Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
⊢ (𝐴 = 𝐵 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵))
Theorem	imaeq1i 6064	Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 “ 𝐶) = (𝐵 “ 𝐶)
Theorem	imaeq2i 6065	Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 “ 𝐴) = (𝐶 “ 𝐵)
Theorem	imaeq1d 6066	Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶))
Theorem	imaeq2d 6067	Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵))
Theorem	imaeq12d 6068	Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐷))
Theorem	dfima2 6069*	Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦}
Theorem	dfima3 6070*	Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
Theorem	elimag 6071*	Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴))
Theorem	elima 6072*	Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴)
Theorem	elima2 6073*	Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝐴))
Theorem	elima3 6074*	Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
Theorem	nfima 6075	Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 “ 𝐵)
Theorem	nfimad 6076	Deduction version of bound-variable hypothesis builder nfima 6075. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(𝐴 “ 𝐵))
Theorem	imadmrn 6077	The image of the domain of a class is the range of the class. (Contributed by NM, 14-Aug-1994.)
⊢ (𝐴 “ dom 𝐴) = ran 𝐴
Theorem	imassrn 6078	The image of a class is a subset of its range. Theorem 3.16(xi) of [Monk1] p. 39. (Contributed by NM, 31-Mar-1995.)
⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴
Theorem	mptima 6079*	Image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
Theorem	mptimass 6080*	Image of a function in maps-to notation for a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐶 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ 𝐶 ↦ 𝐵))
Theorem	imai 6081	Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
⊢ ( I “ 𝐴) = 𝐴
Theorem	rnresi 6082	The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)
⊢ ran ( I ↾ 𝐴) = 𝐴
Theorem	resiima 6083	The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵)
Theorem	ima0 6084	Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
⊢ (𝐴 “ ∅) = ∅
Theorem	0ima 6085	Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
⊢ (∅ “ 𝐴) = ∅
Theorem	csbima12 6086	Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Revised by NM, 20-Aug-2018.)
⊢ ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵)
Theorem	imadisj 6087	A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)
⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅)
Theorem	imadisjlnd 6088	Deduction form of one negated side of imadisj 6087. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) ≠ ∅)    ⇒   ⊢ (𝜑 → (𝐴 “ 𝐵) ≠ ∅)
Theorem	cnvimass 6089	A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.)
⊢ (◡𝐴 “ 𝐵) ⊆ dom 𝐴
Theorem	cnvimarndm 6090	The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.)
⊢ (◡𝐴 “ ran 𝐴) = dom 𝐴
Theorem	imasng 6091*	The image of a singleton. (Contributed by NM, 8-May-2005.)
⊢ (𝐴 ∈ 𝐵 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})
Theorem	relimasn 6092*	The image of a singleton. (Contributed by NM, 20-May-1998.)
⊢ (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})
Theorem	elrelimasn 6093	Elementhood in the image of a singleton. (Contributed by Mario Carneiro, 3-Nov-2015.)
⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))
Theorem	elimasng1 6094	Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) Revise to use df-br 5115 and to prove elimasn1 6095 from it. (Revised by BJ, 16-Oct-2024.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
Theorem	elimasn1 6095	Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Use df-br 5115 and shorten proof. (Revised by BJ, 16-Oct-2024.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)
Theorem	elimasng 6096	Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) TODO: replace existing usages by usages of elimasng1 6094, remove, and relabel elimasng1 6094 to "elimasng".
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
Theorem	elimasn 6097	Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by BJ, 16-Oct-2024.) TODO: replace existing usages by usages of elimasn1 6095, remove, and relabel elimasn1 6095 to "elimasn".
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
Theorem	elimasni 6098	Membership in an image of a singleton. (Contributed by NM, 5-Aug-2010.)
⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵𝐴𝐶)
Theorem	args 6099*	Two ways to express the class of unique-valued arguments of 𝐹, which is the same as the domain of 𝐹 whenever 𝐹 is a function. The left-hand side of the equality is from Definition 10.2 of [Quine] p. 65. Quine uses the notation "arg 𝐹 " for this class (for which we have no separate notation). Observe the resemblance to the alternate definition dffv4 6885 of function value, which is based on the idea in Quine's definition. (Contributed by NM, 8-May-2005.)
⊢ {𝑥 ∣ ∃𝑦(𝐹 “ {𝑥}) = {𝑦}} = {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
Theorem	elinisegg 6100	Membership in the inverse image of a singleton. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Put in closed form and shorten proof. (Revised by BJ, 16-Oct-2024.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))