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Theorem List for Metamath Proof Explorer - 5201-5300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcoeq1 5201 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremcoeq2 5202 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremcoeq1i 5203 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)
 
Theoremcoeq2i 5204 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)
 
Theoremcoeq1d 5205 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremcoeq2d 5206 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremcoeq12i 5207 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)
 
Theoremcoeq12d 5208 Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))
 
Theoremnfco 5209 Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)
 
Theorembrcog 5210* Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
((𝐴𝑉𝐵𝑊) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
 
Theoremopelco2g 5211* 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 5212 Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)
(((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → 𝐴(𝐶𝐷)𝐵)
 
Theoremeqbrrdva 5213* Deduction from extensionality principle for relations, given an equivalence only on the relation's domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.)
(𝜑𝐴 ⊆ (𝐶 × 𝐷))    &   (𝜑𝐵 ⊆ (𝐶 × 𝐷))    &   ((𝜑𝑥𝐶𝑦𝐷) → (𝑥𝐴𝑦𝑥𝐵𝑦))       (𝜑𝐴 = 𝐵)
 
Theorembrco 5214* 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 5215* 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 5216 Subset theorem for converse. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Kyle Wyonch, 27-Apr-2021.)
(𝐴𝐵𝐴𝐵)
 
TheoremcnvssOLD 5217 Obsolete proof of cnvss 5216 as of 27-Apr-2021. Subset theorem for converse. (Contributed by NM, 22-Mar-1998.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝐵𝐴𝐵)
 
Theoremcnveq 5218 Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
(𝐴 = 𝐵𝐴 = 𝐵)
 
Theoremcnveqi 5219 Equality inference for converse. (Contributed by NM, 23-Dec-2008.)
𝐴 = 𝐵       𝐴 = 𝐵
 
Theoremcnveqd 5220 Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)
(𝜑𝐴 = 𝐵)       (𝜑𝐴 = 𝐵)
 
Theoremelcnv 5221* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)
(𝐴𝑅 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥))
 
Theoremelcnv2 5222* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)
(𝐴𝑅 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
 
Theoremnfcnv 5223 Bound-variable hypothesis builder for converse. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥𝐴
 
Theoremopelcnvg 5224 Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
 
Theorembrcnvg 5225 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)
((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝐵𝑅𝐴))
 
Theoremopelcnv 5226 Ordered-pair membership in converse. (Contributed by NM, 13-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V       (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)
 
Theorembrcnv 5227 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 5228 Move class substitution in and out of the converse of a function. Version of csbcnvgALT 5229 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 5229 Move class substitution in and out of the converse of a function. Version of csbcnv 5228 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 5230 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 5231* 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 5232* 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 5233* Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)
ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
 
Theoremdfrn3 5234* 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 5235* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝐴𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵))
 
Theoremelrng 5236* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝐴𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴))
 
Theoremssrelrn 5237* 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 5238 Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
dom 𝐴 = ran 𝐴
 
Theoremdfdmf 5239* 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 5240 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 5241* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
 
Theoremeldm2g 5242* 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 5243* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
𝐴 ∈ V       (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)
 
Theoremeldm2 5244* 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 5245 Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
(𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
 
Theoremdmeq 5246 Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
(𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
 
Theoremdmeqi 5247 Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
𝐴 = 𝐵       dom 𝐴 = dom 𝐵
 
Theoremdmeqd 5248 Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
(𝜑𝐴 = 𝐵)       (𝜑 → dom 𝐴 = dom 𝐵)
 
Theoremopeldmd 5249 Membership of first of an ordered pair in a domain. Deduction version of opeldm 5250. (Contributed by AV, 11-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 ∈ dom 𝐶))
 
Theoremopeldm 5250 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 5251 Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝑅𝐵𝐴 ∈ dom 𝑅)
 
Theorembreldmg 5252 Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)
((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
 
Theoremdmun 5253 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 5254 The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)
dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
 
Theoremdmiun 5255 The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)
dom 𝑥𝐴 𝐵 = 𝑥𝐴 dom 𝐵
 
Theoremdmuni 5256* 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 5257* 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}
 
Theoremdmopabss 5258* Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
 
Theoremdmopab3 5259* The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
(∀𝑥𝐴𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴)
 
Theoremdm0 5260 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 5261 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 5262 The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)
dom V = V
 
Theoremdm0rn0 5263 An empty domain implies an empty range. (Contributed by NM, 21-May-1998.)
(dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)
 
Theoremreldm0 5264 A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.)
(Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
 
Theoremdmxp 5265 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 5266 The domain of a square Cartesian product. (Contributed by NM, 28-Jul-1995.)
dom (𝐴 × 𝐴) = 𝐴
 
Theoremdmxpin 5267 The domain of the intersection of two square Cartesian products. Unlike dmin 5254, equality holds. (Contributed by NM, 29-Jan-2008.)
dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴𝐵)
 
Theoremxpid11 5268 The Cartesian product of a class with itself is one-to-one. (Contributed by NM, 5-Nov-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵)
 
Theoremdmcnvcnv 5269 The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 5502). (Contributed by NM, 8-Apr-2007.)
dom 𝐴 = dom 𝐴
 
Theoremrncnvcnv 5270 The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
ran 𝐴 = ran 𝐴
 
Theoremelreldm 5271 The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.)
((Rel 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐴)
 
Theoremrneq 5272 Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
(𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
 
Theoremrneqi 5273 Equality inference for range. (Contributed by NM, 4-Mar-2004.)
𝐴 = 𝐵       ran 𝐴 = ran 𝐵
 
Theoremrneqd 5274 Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
(𝜑𝐴 = 𝐵)       (𝜑 → ran 𝐴 = ran 𝐵)
 
Theoremrnss 5275 Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
(𝐴𝐵 → ran 𝐴 ⊆ ran 𝐵)
 
Theorembrelrng 5276 The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.)
((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)
 
Theorembrelrn 5277 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 5278 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 5279 The first argument of a binary relation belongs to its domain. (Contributed by NM, 2-Jul-2008.)
((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
 
Theoremrelelrn 5280 The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)
((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)
 
Theoremreleldmb 5281* Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
(Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
 
Theoremrelelrnb 5282* Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
(Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))
 
Theoremreleldmi 5283 The first argument of a binary relation belongs to its domain. (Contributed by NM, 28-Apr-2015.)
Rel 𝑅       (𝐴𝑅𝐵𝐴 ∈ dom 𝑅)
 
Theoremrelelrni 5284 The second argument of a binary relation belongs to its range. (Contributed by NM, 28-Apr-2015.)
Rel 𝑅       (𝐴𝑅𝐵𝐵 ∈ ran 𝑅)
 
Theoremdfrnf 5285* 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 5286* Membership in a range. (Contributed by NM, 10-Jul-1994.)
𝐴 ∈ V       (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)
 
Theoremelrn 5287* Membership in a range. (Contributed by NM, 2-Apr-2004.)
𝐴 ∈ V       (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)
 
Theoremnfdm 5288 Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥dom 𝐴
 
Theoremnfrn 5289 Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥ran 𝐴
 
Theoremdmiin 5290 Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
dom 𝑥𝐴 𝐵 𝑥𝐴 dom 𝐵
 
Theoremrnopab 5291* 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 5292* 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 5293* 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 5294* Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑥 = 𝐷𝐵 = 𝐶)       ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)
 
Theoremelrnmpt1 5295 Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐹 = (𝑥𝐴𝐵)       ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)
 
Theoremelrnmptg 5296* 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 5297* 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)
 
Theoremrn0 5298 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 ∅ = ∅
 
Theoremdfiun3g 5299 Alternate definition of indexed union when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
 
Theoremdfiin3g 5300 Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
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