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Theorem List for Metamath Proof Explorer - 5801-5900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdfrn3 5801* Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
ran 𝐴 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
 
Theoremelrn2g 5802* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝐴𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵))
 
Theoremelrng 5803* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝐴𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴))
 
Theoremelrn2 5804* Membership in a range. (Contributed by NM, 10-Jul-1994.)
𝐴 ∈ V       (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)
 
Theoremelrn 5805* Membership in a range. (Contributed by NM, 2-Apr-2004.)
𝐴 ∈ V       (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)
 
Theoremssrelrn 5806* If a relation is a subset of a cartesian product, then for each element of the range of the relation there is an element of the first set of the cartesian product which is related to the element of the range by the relation. (Contributed by AV, 24-Oct-2020.)
((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎𝐴 𝑎𝑅𝑌)
 
Theoremdfdm4 5807 Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
dom 𝐴 = ran 𝐴
 
Theoremdfdmf 5808* Definition of domain, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴    &   𝑦𝐴       dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
 
Theoremcsbdm 5809 Distribute proper substitution through the domain of a class. (Contributed by Alexander van der Vekens, 23-Jul-2017.) (Revised by NM, 24-Aug-2018.)
𝐴 / 𝑥dom 𝐵 = dom 𝐴 / 𝑥𝐵
 
Theoremeldmg 5810* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
 
Theoremeldm2g 5811* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)
(𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵))
 
Theoremeldm 5812* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
𝐴 ∈ V       (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)
 
Theoremeldm2 5813* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 1-Aug-1994.)
𝐴 ∈ V       (𝐴 ∈ dom 𝐵 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵)
 
Theoremdmss 5814 Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
(𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
 
Theoremdmeq 5815 Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
(𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
 
Theoremdmeqi 5816 Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
𝐴 = 𝐵       dom 𝐴 = dom 𝐵
 
Theoremdmeqd 5817 Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
(𝜑𝐴 = 𝐵)       (𝜑 → dom 𝐴 = dom 𝐵)
 
Theoremopeldmd 5818 Membership of first of an ordered pair in a domain. Deduction version of opeldm 5819. (Contributed by AV, 11-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 ∈ dom 𝐶))
 
Theoremopeldm 5819 Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.)
𝐴 ∈ V    &   𝐵 ∈ V       (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 ∈ dom 𝐶)
 
Theorembreldm 5820 Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝑅𝐵𝐴 ∈ dom 𝑅)
 
Theorembreldmg 5821 Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)
((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
 
Theoremdmun 5822 The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
dom (𝐴𝐵) = (dom 𝐴 ∪ dom 𝐵)
 
Theoremdmin 5823 The domain of an intersection is included in the intersection of the domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)
dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
 
Theorembreldmd 5824 Membership of first of a binary relation in a domain. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)    &   (𝜑𝐴𝑅𝐵)       (𝜑𝐴 ∈ dom 𝑅)
 
Theoremdmiun 5825 The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)
dom 𝑥𝐴 𝐵 = 𝑥𝐴 dom 𝐵
 
Theoremdmuni 5826* The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 3-Feb-2004.)
dom 𝐴 = 𝑥𝐴 dom 𝑥
 
Theoremdmopab 5827* The domain of a class of ordered pairs. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑}
 
Theoremdmopabelb 5828* A set is an element of the domain of a ordered pair class abstraction iff there is a second set so that both sets fulfil the wff of the class abstraction. (Contributed by AV, 19-Oct-2023.)
(𝑥 = 𝑋 → (𝜑𝜓))       (𝑋𝑉 → (𝑋 ∈ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑦𝜓))
 
Theoremdmopab2rex 5829* The domain of an ordered pair class abstraction with two nested restricted existential quantifiers. (Contributed by AV, 23-Oct-2023.)
(∀𝑢𝑈 (∀𝑣𝑉 𝐵𝑋 ∧ ∀𝑖𝐼 𝐷𝑊) → dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝑈 (∃𝑣𝑉 (𝑥 = 𝐴𝑦 = 𝐵) ∨ ∃𝑖𝐼 (𝑥 = 𝐶𝑦 = 𝐷))} = {𝑥 ∣ ∃𝑢𝑈 (∃𝑣𝑉 𝑥 = 𝐴 ∨ ∃𝑖𝐼 𝑥 = 𝐶)})
 
Theoremdmopabss 5830* Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
 
Theoremdmopab3 5831* The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
(∀𝑥𝐴𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴)
 
Theoremdm0 5832 The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
dom ∅ = ∅
 
Theoremdmi 5833 The domain of the identity relation is the universe. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
dom I = V
 
Theoremdmv 5834 The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)
dom V = V
 
Theoremdmep 5835 The domain of the membership relation is the universal class. (Contributed by Scott Fenton, 27-Oct-2010.) (Proof shortened by BJ, 26-Dec-2023.)
dom E = V
 
TheoremdomepOLD 5836 Obsolete proof of dmep 5835 as of 26-Dec-2023. (Contributed by Scott Fenton, 27-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
dom E = V
 
Theoremdm0rn0 5837 An empty domain is equivalent to an empty range. (Contributed by NM, 21-May-1998.)
(dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)
 
Theoremrn0 5838 The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.)
ran ∅ = ∅
 
Theoremrnep 5839 The range of the membership relation is the universal class minus the empty set. (Contributed by BJ, 26-Dec-2023.)
ran E = (V ∖ {∅})
 
Theoremreldm0 5840 A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.)
(Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
 
Theoremdmxp 5841 The domain of a Cartesian product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
 
Theoremdmxpid 5842 The domain of a Cartesian square. (Contributed by NM, 28-Jul-1995.)
dom (𝐴 × 𝐴) = 𝐴
 
Theoremdmxpin 5843 The domain of the intersection of two Cartesian squares. Unlike in dmin 5823, equality holds. (Contributed by NM, 29-Jan-2008.)
dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴𝐵)
 
Theoremxpid11 5844 The Cartesian square is a one-to-one construction. (Contributed by NM, 5-Nov-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵)
 
Theoremdmcnvcnv 5845 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 6097 gives another proof). (Contributed by NM, 8-Apr-2007.)
dom 𝐴 = dom 𝐴
 
Theoremrncnvcnv 5846 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 5847 The first member of an ordered pair in a relation belongs to the domain of the relation (see op1stb 5387). (Contributed by NM, 28-Jul-2004.)
((Rel 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐴)
 
Theoremrneq 5848 Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
(𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
 
Theoremrneqi 5849 Equality inference for range. (Contributed by NM, 4-Mar-2004.)
𝐴 = 𝐵       ran 𝐴 = ran 𝐵
 
Theoremrneqd 5850 Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
(𝜑𝐴 = 𝐵)       (𝜑 → ran 𝐴 = ran 𝐵)
 
Theoremrnss 5851 Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
(𝐴𝐵 → ran 𝐴 ⊆ ran 𝐵)
 
Theoremrnssi 5852 Subclass inference for range. (Contributed by Peter Mazsa, 24-Sep-2022.)
𝐴𝐵       ran 𝐴 ⊆ ran 𝐵
 
Theorembrelrng 5853 The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.)
((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)
 
Theorembrelrn 5854 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 5855 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 5856 The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5712 and brv 5388. (Contributed by NM, 2-Jul-2008.)
((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
 
Theoremrelelrn 5857 The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)
((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)
 
Theoremreleldmb 5858* Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
(Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
 
Theoremrelelrnb 5859* Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
(Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))
 
Theoremreleldmi 5860 The first argument of a binary relation belongs to its domain. (Contributed by NM, 28-Apr-2015.)
Rel 𝑅       (𝐴𝑅𝐵𝐴 ∈ dom 𝑅)
 
Theoremrelelrni 5861 The second argument of a binary relation belongs to its range. (Contributed by NM, 28-Apr-2015.)
Rel 𝑅       (𝐴𝑅𝐵𝐵 ∈ ran 𝑅)
 
Theoremdfrnf 5862* 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}
 
Theoremnfdm 5863 Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥dom 𝐴
 
Theoremnfrn 5864 Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥ran 𝐴
 
Theoremdmiin 5865 Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
dom 𝑥𝐴 𝐵 𝑥𝐴 dom 𝐵
 
Theoremrnopab 5866* 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 5867* 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 5868* 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 5869* Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑥 = 𝐷𝐵 = 𝐶)       ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)
 
Theoremelrnmpt1 5870 Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐹 = (𝑥𝐴𝐵)       ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)
 
Theoremelrnmptg 5871* 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 5872* 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)
 
Theoremelrnmptd 5873* The range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑 → ∃𝑥𝐴 𝐶 = 𝐵)    &   (𝜑𝐶𝑉)       (𝜑𝐶 ∈ ran 𝐹)
 
Theoremelrnmptdv 5874* 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 5875* 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 5876 Alternate definition of indexed union when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
 
Theoremdfiin3g 5877 Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
 
Theoremdfiun3 5878 Alternate definition of indexed union when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐵 ∈ V        𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)
 
Theoremdfiin3 5879 Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐵 ∈ V        𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)
 
Theoremriinint 5880* 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 5881 A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)
(Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))
 
Theoremdmrnssfld 5882 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 5883 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 5884 Range of a composition. (Contributed by NM, 19-Mar-1998.)
ran (𝐴𝐵) ⊆ ran 𝐴
 
Theoremdmcosseq 5885 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 5886 Domain of a composition. (Contributed by NM, 19-Mar-1998.)
(dom 𝐴 = ran 𝐵 → dom (𝐴𝐵) = dom 𝐵)
 
Theoremrncoeq 5887 Range of a composition. (Contributed by NM, 19-Mar-1998.)
(dom 𝐴 = ran 𝐵 → ran (𝐴𝐵) = ran 𝐴)
 
Theoremreseq1 5888 Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremreseq2 5889 Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremreseq1i 5890 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)
 
Theoremreseq2i 5891 Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)
 
Theoremreseq12i 5892 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)
 
Theoremreseq1d 5893 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremreseq2d 5894 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremreseq12d 5895 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))
 
Theoremnfres 5896 Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)
 
Theoremcsbres 5897 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 5898 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
(𝐴 ↾ ∅) = ∅
 
Theoremdfres3 5899 Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝐴𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴))
 
Theoremopelres 5900 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)))
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