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Theorem List for Intuitionistic Logic Explorer - 4501-4600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnfco 4501 Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)

Theorembrcog 4502* Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
((𝐴𝑉𝐵𝑊) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))

Theoremopelco2g 4503* 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 4504 Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)
(((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → 𝐴(𝐶𝐷)𝐵)

Theoremeqbrrdva 4505* 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 4506* 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 4507* 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 4508 Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
(𝐴𝐵𝐴𝐵)

Theoremcnveq 4509 Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
(𝐴 = 𝐵𝐴 = 𝐵)

Theoremcnveqi 4510 Equality inference for converse. (Contributed by NM, 23-Dec-2008.)
𝐴 = 𝐵       𝐴 = 𝐵

Theoremcnveqd 4511 Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)
(𝜑𝐴 = 𝐵)       (𝜑𝐴 = 𝐵)

Theoremelcnv 4512* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)
(𝐴𝑅 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥))

Theoremelcnv2 4513* 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 4514 Bound-variable hypothesis builder for converse. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥𝐴

Theoremopelcnvg 4515 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 4516 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)
((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝐵𝑅𝐴))

Theoremopelcnv 4517 Ordered-pair membership in converse. (Contributed by NM, 13-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V       (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)

Theorembrcnv 4518 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)

Theoremcsbcnvg 4519 Move class substitution in and out of the converse of a function. (Contributed by Thierry Arnoux, 8-Feb-2017.)
(𝐴𝑉𝐴 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)

Theoremcnvco 4520 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 4521* 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 4522* 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 4523* Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)
ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Theoremdfrn3 4524* 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 4525* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝐴𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵))

Theoremelrng 4526* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝐴𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴))

Theoremdfdm4 4527 Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
dom 𝐴 = ran 𝐴

Theoremdfdmf 4528* 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}

Theoremcsbdmg 4529 Distribute proper substitution through the domain of a class. (Contributed by Jim Kingdon, 8-Dec-2018.)
(𝐴𝑉𝐴 / 𝑥dom 𝐵 = dom 𝐴 / 𝑥𝐵)

Theoremeldmg 4530* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝐴𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))

Theoremeldm2g 4531* 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 4532* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
𝐴 ∈ V       (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)

Theoremeldm2 4533* 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 4534 Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
(𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)

Theoremdmeq 4535 Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
(𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)

Theoremdmeqi 4536 Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
𝐴 = 𝐵       dom 𝐴 = dom 𝐵

Theoremdmeqd 4537 Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
(𝜑𝐴 = 𝐵)       (𝜑 → dom 𝐴 = dom 𝐵)

Theoremopeldm 4538 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 4539 Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝑅𝐵𝐴 ∈ dom 𝑅)

Theoremopeldmg 4540 Membership of first of an ordered pair in a domain. (Contributed by Jim Kingdon, 9-Jul-2019.)
((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 ∈ dom 𝐶))

Theorembreldmg 4541 Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)
((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Theoremdmun 4542 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 4543 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 4544 The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)
dom 𝑥𝐴 𝐵 = 𝑥𝐴 dom 𝐵

Theoremdmuni 4545* 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 4546* 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 4547* Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ⊆ 𝐴

Theoremdmopab3 4548* The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
(∀𝑥𝐴𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴)

Theoremdm0 4549 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 4550 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 4551 The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)
dom V = V

Theoremdm0rn0 4552 An empty domain implies an empty range. (Contributed by NM, 21-May-1998.)
(dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)

Theoremreldm0 4553 A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.)
(Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))

Theoremdmmrnm 4554* A domain is inhabited if and only if the range is inhabited. (Contributed by Jim Kingdon, 15-Dec-2018.)
(∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴)

Theoremdmxpm 4555* The domain of a cross 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.)
(∃𝑥 𝑥𝐵 → dom (𝐴 × 𝐵) = 𝐴)

Theoremdmxpinm 4556* The domain of the intersection of two square cross products. Unlike dmin 4543, equality holds. (Contributed by NM, 29-Jan-2008.)
(∃𝑥 𝑥 ∈ (𝐴𝐵) → dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴𝐵))

Theoremxpid11m 4557* The cross product of a class with itself is one-to-one. (Contributed by Jim Kingdon, 8-Dec-2018.)
((∃𝑥 𝑥𝐴 ∧ ∃𝑥 𝑥𝐵) → ((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵))

Theoremdmcnvcnv 4558 The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 4771). (Contributed by NM, 8-Apr-2007.)
dom 𝐴 = dom 𝐴

Theoremrncnvcnv 4559 The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
ran 𝐴 = ran 𝐴

Theoremelreldm 4560 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 4561 Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
(𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)

Theoremrneqi 4562 Equality inference for range. (Contributed by NM, 4-Mar-2004.)
𝐴 = 𝐵       ran 𝐴 = ran 𝐵

Theoremrneqd 4563 Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
(𝜑𝐴 = 𝐵)       (𝜑 → ran 𝐴 = ran 𝐵)

Theoremrnss 4564 Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
(𝐴𝐵 → ran 𝐴 ⊆ ran 𝐵)

Theorembrelrng 4565 The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.)
((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)

Theoremopelrng 4566 Membership of second member of an ordered pair in a range. (Contributed by Jim Kingdon, 26-Jan-2019.)
((𝐴𝐹𝐵𝐺 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) → 𝐵 ∈ ran 𝐶)

Theorembrelrn 4567 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 4568 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 4569 The first argument of a binary relation belongs to its domain. (Contributed by NM, 2-Jul-2008.)
((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Theoremrelelrn 4570 The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)
((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)

Theoremreleldmb 4571* Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
(Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))

Theoremrelelrnb 4572* Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
(Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))

Theoremreleldmi 4573 The first argument of a binary relation belongs to its domain. (Contributed by NM, 28-Apr-2015.)
Rel 𝑅       (𝐴𝑅𝐵𝐴 ∈ dom 𝑅)

Theoremrelelrni 4574 The second argument of a binary relation belongs to its range. (Contributed by NM, 28-Apr-2015.)
Rel 𝑅       (𝐴𝑅𝐵𝐵 ∈ ran 𝑅)

Theoremdfrnf 4575* 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 4576* Membership in a range. (Contributed by NM, 10-Jul-1994.)
𝐴 ∈ V       (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)

Theoremelrn 4577* Membership in a range. (Contributed by NM, 2-Apr-2004.)
𝐴 ∈ V       (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)

Theoremnfdm 4578 Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥dom 𝐴

Theoremnfrn 4579 Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥ran 𝐴

Theoremdmiin 4580 Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
dom 𝑥𝐴 𝐵 𝑥𝐴 dom 𝐵

Theoremrnopab 4581* 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 4582* 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 4583* 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 4584* Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑥 = 𝐷𝐵 = 𝐶)       ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)

Theoremelrnmpt1 4585 Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐹 = (𝑥𝐴𝐵)       ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)

Theoremelrnmptg 4586* 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 4587* 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 4588 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 4589 Alternate definition of indexed union when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))

Theoremdfiin3g 4590 Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))

Theoremdfiun3 4591 Alternate definition of indexed union when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐵 ∈ V        𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)

Theoremdfiin3 4592 Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐵 ∈ V        𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)

Theoremriinint 4593* 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 4594 A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)
(Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))

Theoremdmrnssfld 4595 The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.)
(dom 𝐴 ∪ ran 𝐴) ⊆ 𝐴

Theoremdmexg 4596 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Apr-1995.)
(𝐴𝑉 → dom 𝐴 ∈ V)

Theoremrnexg 4597 The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.)
(𝐴𝑉 → ran 𝐴 ∈ V)

Theoremdmex 4598 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Jul-2008.)
𝐴 ∈ V       dom 𝐴 ∈ V

Theoremrnex 4599 The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 7-Jul-2008.)
𝐴 ∈ V       ran 𝐴 ∈ V

Theoremiprc 4600 The identity function is a proper class. This means, for example, that we cannot use it as a member of the class of continuous functions unless it is restricted to a set. (Contributed by NM, 1-Jan-2007.)
¬ I ∈ V

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