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Theorem List for Metamath Proof Explorer - 5801-5900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdmcnvcnv 5801 The domain of the double converse of a class is equal to its domain (even when that class in not a relation, in which case dfrel2 6043 gives another proof). (Contributed by NM, 8-Apr-2007.)
dom 𝐴 = dom 𝐴

Theoremrncnvcnv 5802 The range of the double converse of a class is equal to its range (even when that class in not a relation). (Contributed by NM, 8-Apr-2007.)
ran 𝐴 = ran 𝐴

Theoremelreldm 5803 The first member of an ordered pair in a relation belongs to the domain of the relation (see op1stb 5359). (Contributed by NM, 28-Jul-2004.)
((Rel 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐴)

Theoremrneq 5804 Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
(𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)

Theoremrneqi 5805 Equality inference for range. (Contributed by NM, 4-Mar-2004.)
𝐴 = 𝐵       ran 𝐴 = ran 𝐵

Theoremrneqd 5806 Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
(𝜑𝐴 = 𝐵)       (𝜑 → ran 𝐴 = ran 𝐵)

Theoremrnss 5807 Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
(𝐴𝐵 → ran 𝐴 ⊆ ran 𝐵)

Theoremrnssi 5808 Subclass inference for range. (Contributed by Peter Mazsa, 24-Sep-2022.)
𝐴𝐵       ran 𝐴 ⊆ ran 𝐵

Theorembrelrng 5809 The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.)
((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)

Theorembrelrn 5810 The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐶𝐵𝐵 ∈ ran 𝐶)

Theoremopelrn 5811 Membership of second member of an ordered pair in a range. (Contributed by NM, 23-Feb-1997.)
𝐴 ∈ V    &   𝐵 ∈ V       (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐵 ∈ ran 𝐶)

Theoremreleldm 5812 The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5671 and brv 5360. (Contributed by NM, 2-Jul-2008.)
((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Theoremrelelrn 5813 The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)
((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)

Theoremreleldmb 5814* Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
(Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))

Theoremrelelrnb 5815* Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
(Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))

Theoremreleldmi 5816 The first argument of a binary relation belongs to its domain. (Contributed by NM, 28-Apr-2015.)
Rel 𝑅       (𝐴𝑅𝐵𝐴 ∈ dom 𝑅)

Theoremrelelrni 5817 The second argument of a binary relation belongs to its range. (Contributed by NM, 28-Apr-2015.)
Rel 𝑅       (𝐴𝑅𝐵𝐵 ∈ ran 𝑅)

Theoremdfrnf 5818* Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴    &   𝑦𝐴       ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Theoremelrn2 5819* Membership in a range. (Contributed by NM, 10-Jul-1994.)
𝐴 ∈ V       (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)

Theoremelrn 5820* Membership in a range. (Contributed by NM, 2-Apr-2004.)
𝐴 ∈ V       (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)

Theoremnfdm 5821 Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥dom 𝐴

Theoremnfrn 5822 Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥ran 𝐴

Theoremdmiin 5823 Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
dom 𝑥𝐴 𝐵 𝑥𝐴 dom 𝐵

Theoremrnopab 5824* The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
ran {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑦 ∣ ∃𝑥𝜑}

Theoremrnmpt 5825* The range of a function in maps-to notation. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐹 = (𝑥𝐴𝐵)       ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}

Theoremelrnmpt 5826* The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015.)
𝐹 = (𝑥𝐴𝐵)       (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))

Theoremelrnmpt1s 5827* Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑥 = 𝐷𝐵 = 𝐶)       ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)

Theoremelrnmpt1 5828 Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐹 = (𝑥𝐴𝐵)       ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)

Theoremelrnmptg 5829* Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐹 = (𝑥𝐴𝐵)       (∀𝑥𝐴 𝐵𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))

Theoremelrnmpti 5830* Membership in the range of a function. (Contributed by NM, 30-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐹 = (𝑥𝐴𝐵)    &   𝐵 ∈ V       (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵)

Theoremelrnmptdv 5831* Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐶𝐴)    &   (𝜑𝐷𝑉)    &   ((𝜑𝑥 = 𝐶) → 𝐷 = 𝐵)       (𝜑𝐷 ∈ ran 𝐹)

Theoremelrnmpt2d 5832* Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐶 ∈ ran 𝐹)       (𝜑 → ∃𝑥𝐴 𝐶 = 𝐵)

Theoremdfiun3g 5833 Alternate definition of indexed union when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))

Theoremdfiin3g 5834 Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))

Theoremdfiun3 5835 Alternate definition of indexed union when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐵 ∈ V        𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)

Theoremdfiin3 5836 Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐵 ∈ V        𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)

Theoremriinint 5837* Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝑋𝑉 ∧ ∀𝑘𝐼 𝑆𝑋) → (𝑋 𝑘𝐼 𝑆) = ({𝑋} ∪ ran (𝑘𝐼𝑆)))

Theoremrelrn0 5838 A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)
(Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))

Theoremdmrnssfld 5839 The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.)
(dom 𝐴 ∪ ran 𝐴) ⊆ 𝐴

Theoremdmcoss 5840 Domain of a composition. Theorem 21 of [Suppes] p. 63. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
dom (𝐴𝐵) ⊆ dom 𝐵

Theoremrncoss 5841 Range of a composition. (Contributed by NM, 19-Mar-1998.)
ran (𝐴𝐵) ⊆ ran 𝐴

Theoremdmcosseq 5842 Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(ran 𝐵 ⊆ dom 𝐴 → dom (𝐴𝐵) = dom 𝐵)

Theoremdmcoeq 5843 Domain of a composition. (Contributed by NM, 19-Mar-1998.)
(dom 𝐴 = ran 𝐵 → dom (𝐴𝐵) = dom 𝐵)

Theoremrncoeq 5844 Range of a composition. (Contributed by NM, 19-Mar-1998.)
(dom 𝐴 = ran 𝐵 → ran (𝐴𝐵) = ran 𝐴)

Theoremreseq1 5845 Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremreseq2 5846 Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremreseq1i 5847 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)

Theoremreseq2i 5848 Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)

Theoremreseq12i 5849 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)

Theoremreseq1d 5850 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Theoremreseq2d 5851 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Theoremreseq12d 5852 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremnfres 5853 Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)

Theoremcsbres 5854 Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.) (Revised by NM, 23-Aug-2018.)
𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)

Theoremres0 5855 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
(𝐴 ↾ ∅) = ∅

Theoremdfres3 5856 Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝐴𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴))

Theoremopelres 5857 Ordered pair elementhood in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.) (Revised by BJ, 18-Feb-2022.) Commute the consequent. (Revised by Peter Mazsa, 24-Sep-2022.)
(𝐶𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴) ↔ (𝐵𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))

Theorembrres 5858 Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.) Commute the consequent. (Revised by Peter Mazsa, 24-Sep-2022.)
(𝐶𝑉 → (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))

Theoremopelresi 5859 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)
𝐶 ∈ V       (⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴) ↔ (𝐵𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅))

Theorembrresi 5860 Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
𝐶 ∈ V       (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶))

Theoremopres 5861 Ordered pair membership in a restriction when the first member belongs to the restricting class. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝐵 ∈ V       (𝐴𝐷 → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))

Theoremresieq 5862 A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)
((𝐵𝐴𝐶𝐴) → (𝐵( I ↾ 𝐴)𝐶𝐵 = 𝐶))

Theoremopelidres 5863 𝐴, 𝐴 belongs to a restriction of the identity class iff 𝐴 belongs to the restricting class. (Contributed by FL, 27-Oct-2008.) (Revised by NM, 30-Mar-2016.)
(𝐴𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ 𝐴𝐵))

Theoremresres 5864 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
((𝐴𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵𝐶))

Theoremresundi 5865 Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
(𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Theoremresundir 5866 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremresindi 5867 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
(𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Theoremresindir 5868 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Theoreminres 5869 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
(𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ↾ 𝐶)

Theoremresdifcom 5870 Commutative law for restriction and difference. (Contributed by AV, 7-Jun-2021.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ↾ 𝐵)

Theoremresiun1 5871* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.) (Proof shortened by JJ, 25-Aug-2021.)
( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)

Theoremresiun2 5872* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
(𝐶 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶𝐵)

Theoremdmres 5873 The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)
dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)

Theoremssdmres 5874 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
(𝐴 ⊆ dom 𝐵 ↔ dom (𝐵𝐴) = 𝐴)

Theoremdmresexg 5875 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
(𝐵𝑉 → dom (𝐴𝐵) ∈ V)

Theoremresss 5876 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
(𝐴𝐵) ⊆ 𝐴

Theoremrescom 5877 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ↾ 𝐵)

Theoremssres 5878 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremssres2 5879 Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

Theoremrelres 5880 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 5881 Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)
(𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))

Theoremresabs1d 5882 Absorption law for restriction, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐵𝐶)       (𝜑 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))

Theoremresabs2 5883 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
(𝐵𝐶 → ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐵))

Theoremresidm 5884 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
((𝐴𝐵) ↾ 𝐵) = (𝐴𝐵)

Theoremresima 5885 A restriction to an image. (Contributed by NM, 29-Sep-2004.)
((𝐴𝐵) “ 𝐵) = (𝐴𝐵)

Theoremresima2 5886 Image under a restricted class. (Contributed by FL, 31-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.)
(𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))

Theoremxpssres 5887 Restriction of a constant function (or other Cartesian product). (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐶𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))

Theoremelinxp 5888* Membership in an intersection with a Cartesian product. (Contributed by Peter Mazsa, 9-Sep-2022.)
(𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))

Theoremelres 5889* Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.) (Proof shortened by Peter Mazsa, 9-Sep-2022.)
(𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))

Theoremelsnres 5890* 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 5891 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
((Rel 𝐴 ∧ dom 𝐴𝐵) → (𝐴𝐵) = 𝐴)

Theoremdmressnsn 5892 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 5893 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 5894 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 5895 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
(Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)

Theoremresexg 5896 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 5897 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
𝐴 ∈ V       (𝐴𝐵) ∈ V

Theoremresindm 5898 When restricting a relation, intersecting with the domain of the relation has no effect. (Contributed by FL, 6-Oct-2008.)
(Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))

Theoremresdmdfsn 5899 Restricting a relation to its domain without a set is the same as restricting the relation to the universe without this set. (Contributed by AV, 2-Dec-2018.)
(Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋})))

Theoremresopab 5900* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}

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