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Theorem List for Metamath Proof Explorer - 5401-5500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremresindir 5401 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
 
Theoreminres 5402 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
(𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ↾ 𝐶)
 
Theoremresdifcom 5403 Commutative law for restriction and difference. (Contributed by AV, 7-Jun-2021.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ↾ 𝐵)
 
Theoremresiun1 5404* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.) (Proof shortened by JJ, 25-Aug-2021.)
( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
 
Theoremresiun1OLD 5405* Obsolete proof of resiun1 5404 as of 25-Aug-2021. (Contributed by Mario Carneiro, 29-May-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
 
Theoremresiun2 5406* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
(𝐶 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶𝐵)
 
Theoremdmres 5407 The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)
dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
 
Theoremssdmres 5408 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
(𝐴 ⊆ dom 𝐵 ↔ dom (𝐵𝐴) = 𝐴)
 
Theoremdmresexg 5409 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
(𝐵𝑉 → dom (𝐴𝐵) ∈ V)
 
Theoremresss 5410 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
(𝐴𝐵) ⊆ 𝐴
 
Theoremrescom 5411 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ↾ 𝐵)
 
Theoremssres 5412 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
 
Theoremssres2 5413 Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
 
Theoremrelres 5414 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 (𝐴𝐵)
 
Theoremresabs1 5415 Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)
(𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
 
Theoremresabs1d 5416 Absorption law for restriction, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐵𝐶)       (𝜑 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
 
Theoremresabs2 5417 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
(𝐵𝐶 → ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐵))
 
Theoremresidm 5418 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
((𝐴𝐵) ↾ 𝐵) = (𝐴𝐵)
 
Theoremresima 5419 A restriction to an image. (Contributed by NM, 29-Sep-2004.)
((𝐴𝐵) “ 𝐵) = (𝐴𝐵)
 
Theoremresima2 5420 Image under a restricted class. (Contributed by FL, 31-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.)
(𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))
 
Theoremresima2OLD 5421 Obsolete proof of resima2 5420 as of 25-Aug-2021. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))
 
Theoremxpssres 5422 Restriction of a constant function (or other Cartesian product). (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐶𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))
 
Theoremelres 5423* Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)
(𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
 
Theoremelsnres 5424* 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))
 
Theoremrelssres 5425 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
((Rel 𝐴 ∧ dom 𝐴𝐵) → (𝐴𝐵) = 𝐴)
 
Theoremdmressnsn 5426 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})
 
Theoremeldmressnsn 5427 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 𝐹𝐴 ∈ dom (𝐹 ↾ {𝐴}))
 
Theoremeldmeldmressn 5428 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 𝐹𝑋 ∈ dom (𝐹 ↾ {𝑋}))
 
Theoremresdm 5429 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
(Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
 
Theoremresexg 5430 The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)
 
Theoremresex 5431 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
𝐴 ∈ V       (𝐴𝐵) ∈ V
 
Theoremresindm 5432 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
(Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))
 
Theoremresdmdfsn 5433 Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself. (Contributed by AV, 2-Dec-2018.)
(Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋})))
 
Theoremresopab 5434* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
 
Theoremiss 5435 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))
 
Theoremresopab2 5436* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)
(𝐴𝐵 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
 
Theoremresmpt 5437* Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)
(𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
 
Theoremresmpt3 5438* Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)
((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴𝐵) ↦ 𝐶)
 
Theoremresmptf 5439 Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
 
Theoremresmptd 5440* Restriction of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐵𝐴)       (𝜑 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
 
Theoremdfres2 5441* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
(𝑅𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
 
Theoremmptss 5442* Sufficient condition for inclusion in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))
 
Theoremopabresid 5443* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)
{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ( I ↾ 𝐴)
 
Theoremmptresid 5444* The restricted identity expressed with the "maps to" notation. (Contributed by FL, 25-Apr-2012.)
(𝑥𝐴𝑥) = ( I ↾ 𝐴)
 
Theoremdmresi 5445 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
dom ( I ↾ 𝐴) = 𝐴
 
Theoremrestidsing 5446 Restriction of the identity to a singleton. (Contributed by FL, 2-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.)
( I ↾ {𝐴}) = ({𝐴} × {𝐴})
 
TheoremrestidsingOLD 5447 Obsolete proof of restidsing 5446 as of 25-Aug-2021. (Contributed by FL, 2-Aug-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
( I ↾ {𝐴}) = ({𝐴} × {𝐴})
 
Theoremresid 5448 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)
(Rel 𝐴 → (𝐴 ↾ V) = 𝐴)
 
Theoremimaeq1 5449 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremimaeq2 5450 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremimaeq1i 5451 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)
 
Theoremimaeq2i 5452 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)
 
Theoremimaeq1d 5453 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremimaeq2d 5454 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremimaeq12d 5455 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))
 
Theoremdfima2 5456* 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.)
(𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}
 
Theoremdfima3 5457* 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.)
(𝐴𝐵) = {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
 
Theoremelimag 5458* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)
(𝐴𝑉 → (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴))
 
Theoremelima 5459* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)
𝐴 ∈ V       (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴)
 
Theoremelima2 5460* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)
𝐴 ∈ V       (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐶𝑥𝐵𝐴))
 
Theoremelima3 5461* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)
𝐴 ∈ V       (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
 
Theoremnfima 5462 Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)
 
Theoremnfimad 5463 Deduction version of bound-variable hypothesis builder nfima 5462. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥(𝐴𝐵))
 
Theoremimadmrn 5464 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
 
Theoremimassrn 5465 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.)
(𝐴𝐵) ⊆ ran 𝐴
 
Theoremimai 5466 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
( I “ 𝐴) = 𝐴
 
Theoremrnresi 5467 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)
ran ( I ↾ 𝐴) = 𝐴
 
Theoremresiima 5468 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
(𝐵𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵)
 
Theoremima0 5469 Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
(𝐴 “ ∅) = ∅
 
Theorem0ima 5470 Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
(∅ “ 𝐴) = ∅
 
Theoremcsbima12 5471 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(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
 
Theoremimadisj 5472 A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)
((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
 
Theoremcnvimass 5473 A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.)
(𝐴𝐵) ⊆ dom 𝐴
 
Theoremcnvimarndm 5474 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
 
Theoremimasng 5475* The image of a singleton. (Contributed by NM, 8-May-2005.)
(𝐴𝐵 → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
 
Theoremrelimasn 5476* The image of a singleton. (Contributed by NM, 20-May-1998.)
(Rel 𝑅 → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
 
Theoremelrelimasn 5477 Elementhood in the image of a singleton. (Contributed by Mario Carneiro, 3-Nov-2015.)
(Rel 𝑅 → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))
 
Theoremelimasn 5478 Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝐵 ∈ V    &   𝐶 ∈ V       (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
 
Theoremelimasng 5479 Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)
((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
 
Theoremelimasni 5480 Membership in an image of a singleton. (Contributed by NM, 5-Aug-2010.)
(𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵𝐴𝐶)
 
Theoremargs 5481* 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 6175 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}
 
Theoremeliniseg 5482 Membership in an initial segment. The idiom (𝐴 “ {𝐵}), meaning {𝑥𝑥𝐴𝐵}, is used to specify an initial segment in (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       (𝐵𝑉 → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
 
Theoremepini 5483 Any set is equal to its preimage under the converse epsilon relation. (Contributed by Mario Carneiro, 9-Mar-2013.)
𝐴 ∈ V       ( E “ {𝐴}) = 𝐴
 
Theoreminiseg 5484* 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.)
(𝐵𝑉 → (𝐴 “ {𝐵}) = {𝑥𝑥𝐴𝐵})
 
Theoreminisegn0 5485 Nonemptiness of an initial segment in terms of range. (Contributed by Stefan O'Rear, 18-Jan-2015.)
(𝐴 ∈ ran 𝐹 ↔ (𝐹 “ {𝐴}) ≠ ∅)
 
Theoremdffr3 5486* 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((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))
 
Theoremdfse2 5487* Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
(𝑅 Se 𝐴 ↔ ∀𝑥𝐴 (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
 
Theoremimass1 5488 Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
 
Theoremimass2 5489 Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
 
Theoremndmima 5490 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}) = ∅)
 
Theoremrelcnv 5491 A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)
Rel 𝐴
 
Theoremrelbrcnvg 5492 When 𝑅 is a relation, the sethood assumptions on brcnv 5294 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
(Rel 𝑅 → (𝐴𝑅𝐵𝐵𝑅𝐴))
 
Theoremeliniseg2 5493 Eliminate the class existence constraint in eliniseg 5482. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 17-Nov-2015.)
(Rel 𝐴 → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
 
Theoremrelbrcnv 5494 When 𝑅 is a relation, the sethood assumptions on brcnv 5294 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
Rel 𝑅       (𝐴𝑅𝐵𝐵𝑅𝐴)
 
Theoremcotrg 5495* Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 5496 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 5496. (Revised by Richard Penner, 24-Dec-2019.)
((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
 
Theoremcotr 5496* Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. Special instance of cotrg 5495. (Contributed by NM, 27-Dec-1996.)
((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
 
Theoremissref 5497* Two ways to state a relation is reflexive. Adapted from Tarski. (Contributed by FL, 15-Jan-2012.) (Revised by NM, 30-Mar-2016.)
(( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
 
Theoremcnvsym 5498* 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.)
(𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
 
Theoremintasym 5499* 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.)
((𝑅𝑅) ⊆ I ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
 
Theoremasymref 5500* Two ways of saying a relation is antisymmetric and reflexive. 𝑅 is the field of a relation by relfld 5649. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝑅𝑅) = ( I ↾ 𝑅) ↔ ∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦))
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