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Theorem List for Metamath Proof Explorer - 5701-5800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremididg 5701 A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝑉𝐴 I 𝐴)

Theoremissetid 5702 Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴 ∈ V ↔ 𝐴 I 𝐴)

Theoremcoss1 5703 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremcoss2 5704 Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

Theoremcoeq1 5705 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremcoeq2 5706 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremcoeq1i 5707 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)

Theoremcoeq2i 5708 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)

Theoremcoeq1d 5709 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Theoremcoeq2d 5710 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Theoremcoeq12i 5711 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)

Theoremcoeq12d 5712 Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremnfco 5713 Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)

Theorembrcog 5714* Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
((𝐴𝑉𝐵𝑊) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))

Theoremopelco2g 5715* Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)
((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐷 ∧ ⟨𝑥, 𝐵⟩ ∈ 𝐶)))

Theorembrcogw 5716 Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)
(((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → 𝐴(𝐶𝐷)𝐵)

Theoremeqbrrdva 5717* Deduction from extensionality principle for relations, given an equivalence only on the relation domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.)
(𝜑𝐴 ⊆ (𝐶 × 𝐷))    &   (𝜑𝐵 ⊆ (𝐶 × 𝐷))    &   ((𝜑𝑥𝐶𝑦𝐷) → (𝑥𝐴𝑦𝑥𝐵𝑦))       (𝜑𝐴 = 𝐵)

Theorembrco 5718* Binary relation on a composition. (Contributed by NM, 21-Sep-2004.) (Revised by Mario Carneiro, 24-Feb-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))

Theoremopelco 5719* Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))

Theoremcnvss 5720 Subset theorem for converse. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Kyle Wyonch, 27-Apr-2021.)
(𝐴𝐵𝐴𝐵)

Theoremcnveq 5721 Equality theorem for converse relation. (Contributed by NM, 13-Aug-1995.)
(𝐴 = 𝐵𝐴 = 𝐵)

Theoremcnveqi 5722 Equality inference for converse relation. (Contributed by NM, 23-Dec-2008.)
𝐴 = 𝐵       𝐴 = 𝐵

Theoremcnveqd 5723 Equality deduction for converse relation. (Contributed by NM, 6-Dec-2013.)
(𝜑𝐴 = 𝐵)       (𝜑𝐴 = 𝐵)

Theoremelcnv 5724* Membership in a converse relation. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)
(𝐴𝑅 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥))

Theoremelcnv2 5725* Membership in a converse relation. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)
(𝐴𝑅 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))

Theoremnfcnv 5726 Bound-variable hypothesis builder for converse relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥𝐴

Theorembrcnvg 5727 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)
((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝐵𝑅𝐴))

Theoremopelcnvg 5728 Ordered-pair membership in converse relation. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))

Theoremopelcnv 5729 Ordered-pair membership in converse relation. (Contributed by NM, 13-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V       (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)

Theorembrcnv 5730 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 13-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝑅𝐵𝐵𝑅𝐴)

Theoremcsbcnv 5731 Move class substitution in and out of the converse of a relation. Version of csbcnvgALT 5732 without a sethood antecedent but depending on more axioms. (Contributed by Thierry Arnoux, 8-Feb-2017.) (Revised by NM, 23-Aug-2018.)
𝐴 / 𝑥𝐹 = 𝐴 / 𝑥𝐹

TheoremcsbcnvgALT 5732 Move class substitution in and out of the converse of a relation. Version of csbcnv 5731 with a sethood antecedent but depending on fewer axioms. (Contributed by Thierry Arnoux, 8-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉𝐴 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)

Theoremcnvco 5733 Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝐵) = (𝐵𝐴)

Theoremcnvuni 5734* The converse of a class union is the (indexed) union of the converses of its members. (Contributed by NM, 11-Aug-2004.)
𝐴 = 𝑥𝐴 𝑥

Theoremdfdm3 5735* Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
dom 𝐴 = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴}

Theoremdfrn2 5736* Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)
ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Theoremdfrn3 5737* 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 5738* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝐴𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵))

Theoremelrng 5739* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝐴𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴))

Theoremssrelrn 5740* 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 5741 Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
dom 𝐴 = ran 𝐴

Theoremdfdmf 5742* 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 5743 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 5744* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))

Theoremeldm2g 5745* 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 5746* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
𝐴 ∈ V       (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)

Theoremeldm2 5747* 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 5748 Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
(𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)

Theoremdmeq 5749 Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
(𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)

Theoremdmeqi 5750 Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
𝐴 = 𝐵       dom 𝐴 = dom 𝐵

Theoremdmeqd 5751 Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
(𝜑𝐴 = 𝐵)       (𝜑 → dom 𝐴 = dom 𝐵)

Theoremopeldmd 5752 Membership of first of an ordered pair in a domain. Deduction version of opeldm 5753. (Contributed by AV, 11-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 ∈ dom 𝐶))

Theoremopeldm 5753 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 5754 Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝑅𝐵𝐴 ∈ dom 𝑅)

Theorembreldmg 5755 Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)
((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Theoremdmun 5756 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 5757 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 5758 Membership of first of a binary relation in a domain. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)    &   (𝜑𝐴𝑅𝐵)       (𝜑𝐴 ∈ dom 𝑅)

Theoremdmiun 5759 The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)
dom 𝑥𝐴 𝐵 = 𝑥𝐴 dom 𝐵

Theoremdmuni 5760* 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 5761* 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 5762* 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 5763* 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 5764* Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ⊆ 𝐴

Theoremdmopab3 5765* The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
(∀𝑥𝐴𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴)

Theoremopabssxpd 5766* An ordered-pair class abstraction is a subset of a Cartesian product. Formerly part of proof for opabex2 7741. (Contributed by AV, 26-Nov-2021.)
((𝜑𝜓) → 𝑥𝐴)    &   ((𝜑𝜓) → 𝑦𝐵)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))

Theoremdm0 5767 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 5768 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 5769 The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)
dom V = V

Theoremdmep 5770 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 5771 Obsolete proof of dmep 5770 as of 26-Dec-2023. (Contributed by Scott Fenton, 27-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
dom E = V

Theoremdm0rn0 5772 An empty domain is equivalent to an empty range. (Contributed by NM, 21-May-1998.)
(dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)

Theoremrn0 5773 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 5774 The range of the membership relation is the universal class minus the empty set. (Contributed by BJ, 26-Dec-2023.)
ran E = (V ∖ {∅})

Theoremreldm0 5775 A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.)
(Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))

Theoremdmxp 5776 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 5777 The domain of a Cartesian square. (Contributed by NM, 28-Jul-1995.)
dom (𝐴 × 𝐴) = 𝐴

Theoremdmxpin 5778 The domain of the intersection of two Cartesian squares. Unlike in dmin 5757, equality holds. (Contributed by NM, 29-Jan-2008.)
dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴𝐵)

Theoremxpid11 5779 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 5780 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 6024 gives another proof). (Contributed by NM, 8-Apr-2007.)
dom 𝐴 = dom 𝐴

Theoremrncnvcnv 5781 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 5782 The first member of an ordered pair in a relation belongs to the domain of the relation (see op1stb 5340). (Contributed by NM, 28-Jul-2004.)
((Rel 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐴)

Theoremrneq 5783 Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
(𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)

Theoremrneqi 5784 Equality inference for range. (Contributed by NM, 4-Mar-2004.)
𝐴 = 𝐵       ran 𝐴 = ran 𝐵

Theoremrneqd 5785 Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
(𝜑𝐴 = 𝐵)       (𝜑 → ran 𝐴 = ran 𝐵)

Theoremrnss 5786 Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
(𝐴𝐵 → ran 𝐴 ⊆ ran 𝐵)

Theoremrnssi 5787 Subclass inference for range. (Contributed by Peter Mazsa, 24-Sep-2022.)
𝐴𝐵       ran 𝐴 ⊆ ran 𝐵

Theorembrelrng 5788 The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.)
((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)

Theorembrelrn 5789 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 5790 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 5791 The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5650 and brv 5341. (Contributed by NM, 2-Jul-2008.)
((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Theoremrelelrn 5792 The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)
((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)

Theoremreleldmb 5793* Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
(Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))

Theoremrelelrnb 5794* Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
(Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))

Theoremreleldmi 5795 The first argument of a binary relation belongs to its domain. (Contributed by NM, 28-Apr-2015.)
Rel 𝑅       (𝐴𝑅𝐵𝐴 ∈ dom 𝑅)

Theoremrelelrni 5796 The second argument of a binary relation belongs to its range. (Contributed by NM, 28-Apr-2015.)
Rel 𝑅       (𝐴𝑅𝐵𝐵 ∈ ran 𝑅)

Theoremdfrnf 5797* 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 5798* Membership in a range. (Contributed by NM, 10-Jul-1994.)
𝐴 ∈ V       (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)

Theoremelrn 5799* Membership in a range. (Contributed by NM, 2-Apr-2004.)
𝐴 ∈ V       (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)

Theoremnfdm 5800 Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥dom 𝐴

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