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	rnresss 6001	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 6002	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 6003*	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 6004*	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 6005*	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 6006	Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → (𝐴 ↾ 𝐵) = 𝐴)
Theorem	dmressnsn 6007	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 6008	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 6009	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 6010	A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
Theorem	resexg 6011	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 6012	The restriction of a set is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ V)
Theorem	resex 6013	The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ↾ 𝐵) ∈ V
Theorem	resindm 6014	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 6015	Obsolete version of resindm 6014 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 6016	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 6017	Obsolete version of resdmdfsn 6016 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	reldisjun 6018	Split a relation into two disjoint parts based on its domain. (Contributed by Thierry Arnoux, 9-Oct-2023.)
⊢ ((Rel 𝑅 ∧ dom 𝑅 = (𝐴 ∪ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑅 = ((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵)))
Theorem	relresdm1 6019	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 6020*	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 6021	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 6022*	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 6023*	Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)
⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶))
Theorem	resmpt3 6024*	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 6025	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 6026*	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 6027*	Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
⊢ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}
Theorem	mptss 6028*	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 6029*	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 6030*	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 6031*	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 6032*	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 6033	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 6034) to cartesian product. (Revised by BJ, 23-Dec-2023.)
⊢ ( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴 ∩ 𝐵))
Theorem	idinxpresid 6034	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 6035	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 6036*	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 6037*	The restricted identity relation expressed in maps-to notation. (Contributed by FL, 25-Apr-2012.)
⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)
Theorem	dmresi 6038	The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
⊢ dom ( I ↾ 𝐴) = 𝐴
Theorem	restidsing 6039	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 6040	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 6041	Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶))
Theorem	imaeq2 6042	Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
⊢ (𝐴 = 𝐵 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵))
Theorem	imaeq1i 6043	Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 “ 𝐶) = (𝐵 “ 𝐶)
Theorem	imaeq2i 6044	Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 “ 𝐴) = (𝐶 “ 𝐵)
Theorem	imaeq1d 6045	Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶))
Theorem	imaeq2d 6046	Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵))
Theorem	imaeq12d 6047	Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐷))
Theorem	dfima2 6048*	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 6049*	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 6050*	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 6051*	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 6052*	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 6053*	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 6054	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 6055	Deduction version of bound-variable hypothesis builder nfima 6054. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(𝐴 “ 𝐵))
Theorem	imadmrn 6056	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 6057	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 6058*	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 6059*	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 6060	Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
⊢ ( I “ 𝐴) = 𝐴
Theorem	rnresi 6061	The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)
⊢ ran ( I ↾ 𝐴) = 𝐴
Theorem	resiima 6062	The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵)
Theorem	ima0 6063	Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
⊢ (𝐴 “ ∅) = ∅
Theorem	0ima 6064	Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
⊢ (∅ “ 𝐴) = ∅
Theorem	csbima12 6065	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 6066	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 6067	Deduction form of one negated side of imadisj 6066. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) ≠ ∅)    ⇒   ⊢ (𝜑 → (𝐴 “ 𝐵) ≠ ∅)
Theorem	cnvimass 6068	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 6069	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 6070*	The image of a singleton. (Contributed by NM, 8-May-2005.)
⊢ (𝐴 ∈ 𝐵 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})
Theorem	relimasn 6071*	The image of a singleton. (Contributed by NM, 20-May-1998.)
⊢ (Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})
Theorem	elrelimasn 6072	Elementhood in the image of a singleton. (Contributed by Mario Carneiro, 3-Nov-2015.)
⊢ (Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))
Theorem	elimasng1 6073	Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) Revise to use df-br 5100 and to prove elimasn1 6074 from it. (Revised by BJ, 16-Oct-2024.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
Theorem	elimasn1 6074	Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Use df-br 5100 and shorten proof. (Revised by BJ, 16-Oct-2024.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)
Theorem	elimasng 6075	Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) TODO: replace existing usages by usages of elimasng1 6073, remove, and relabel elimasng1 6073 to "elimasng".
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
Theorem	elimasn 6076	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 6074, remove, and relabel elimasn1 6074 to "elimasn".
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
Theorem	elimasni 6077	Membership in an image of a singleton. (Contributed by NM, 5-Aug-2010.)
⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵𝐴𝐶)
Theorem	args 6078*	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 6860 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 6079	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))
Theorem	eliniseg 6080	Membership in the inverse image of a singleton. An application is to express initial segments for an order relation. See for example Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
Theorem	epin 6081	Any set is equal to its preimage under the converse membership relation. (Contributed by Mario Carneiro, 9-Mar-2013.) Put in closed form. (Revised by BJ, 16-Oct-2024.)
⊢ (𝐴 ∈ 𝑉 → (◡ E “ {𝐴}) = 𝐴)
Theorem	epini 6082	Any set is equal to its preimage under the converse membership relation. (Contributed by Mario Carneiro, 9-Mar-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (◡ E “ {𝐴}) = 𝐴
Theorem	iniseg 6083*	An idiom that signifies an initial segment of an ordering, used, for example, in Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.)
⊢ (𝐵 ∈ 𝑉 → (◡𝐴 “ {𝐵}) = {𝑥 ∣ 𝑥𝐴𝐵})
Theorem	inisegn0 6084	Nonemptiness of an initial segment in terms of range. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ (𝐴 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝐴}) ≠ ∅)
Theorem	dffr3 6085*	Alternate definition of well-founded relation. Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 23-Jun-2015.)
⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅))
Theorem	dfse2 6086*	Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V)
Theorem	imass1 6087	Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶))
Theorem	imass2 6088	Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵))
Theorem	ndmima 6089	The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.) (Proof shortened by OpenAI, 3-Jul-2020.)
⊢ (¬ 𝐴 ∈ dom 𝐵 → (𝐵 “ {𝐴}) = ∅)
Theorem	relcnv 6090	A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)
⊢ Rel ◡𝐴
Theorem	relbrcnvg 6091	When 𝑅 is a relation, the sethood assumptions on brcnv 5852 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))
Theorem	eliniseg2 6092	Eliminate the class existence constraint in eliniseg 6080. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 17-Nov-2015.)
⊢ (Rel 𝐴 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
Theorem	relbrcnv 6093	When 𝑅 is a relation, the sethood assumptions on brcnv 5852 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
⊢ Rel 𝑅    ⇒   ⊢ (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)
Theorem	relco 6094	A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
⊢ Rel (𝐴 ∘ 𝐵)
Theorem	cotrg 6095*	Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 6096 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 6096. (Revised by Richard Penner, 24-Dec-2019.) (Proof shortened by SN, 19-Dec-2024.) Avoid ax-11 2190. (Revised by BTernaryTau, 29-Dec-2024.)
⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
Theorem	cotr 6096*	Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. Special instance of cotrg 6095. (Contributed by NM, 27-Dec-1996.)
⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Theorem	idrefALT 6097*	Alternate proof of idref 7124 not relying on definitions related to functions. Two ways to state that a relation is reflexive on a class. (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Revised by NM, 30-Mar-2016.) (Proof shortened by BJ, 28-Aug-2022.) The "proof modification is discouraged" tag is here only because this is an *ALT result. (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
Theorem	cnvsym 6098*	Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by SN, 23-Dec-2024.) Avoid ax-11 2190. (Revised by BTernaryTau, 29-Dec-2024.)
⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
Theorem	intasym 6099*	Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ((𝑅 ∩ ◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
Theorem	asymref 6100*	Two ways of saying a relation is antisymmetric and reflexive. ∪ ∪ 𝑅 is the field of a relation by relfld 6258. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ ∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦))