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Theorem List for Intuitionistic Logic Explorer - 4601-4700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdmcoss 4601 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 4602 Range of a composition. (Contributed by NM, 19-Mar-1998.)
ran (𝐴𝐵) ⊆ ran 𝐴

Theoremdmcosseq 4603 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 4604 Domain of a composition. (Contributed by NM, 19-Mar-1998.)
(dom 𝐴 = ran 𝐵 → dom (𝐴𝐵) = dom 𝐵)

Theoremrncoeq 4605 Range of a composition. (Contributed by NM, 19-Mar-1998.)
(dom 𝐴 = ran 𝐵 → ran (𝐴𝐵) = ran 𝐴)

Theoremreseq1 4606 Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremreseq2 4607 Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremreseq1i 4608 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)

Theoremreseq2i 4609 Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)

Theoremreseq12i 4610 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)

Theoremreseq1d 4611 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Theoremreseq2d 4612 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Theoremreseq12d 4613 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremnfres 4614 Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)

Theoremcsbresg 4615 Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Theoremres0 4616 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
(𝐴 ↾ ∅) = ∅

Theoremopelres 4617 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)
𝐵 ∈ V       (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷))

Theorembrres 4618 Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
𝐵 ∈ V       (𝐴(𝐶𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐴𝐷))

Theoremopelresg 4619 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.)
(𝐵𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷)))

Theorembrresg 4620 Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)
(𝐵𝑉 → (𝐴(𝐶𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐴𝐷)))

Theoremopres 4621 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 4622 A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)
((𝐵𝐴𝐶𝐴) → (𝐵( I ↾ 𝐴)𝐶𝐵 = 𝐶))

Theoremopelresi 4623 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 4624 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
((𝐴𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵𝐶))

Theoremresundi 4625 Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
(𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Theoremresundir 4626 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremresindi 4627 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
(𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Theoremresindir 4628 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Theoreminres 4629 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
(𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ↾ 𝐶)

Theoremresiun1 4630* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)

Theoremresiun2 4631* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
(𝐶 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶𝐵)

Theoremdmres 4632 The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)
dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)

Theoremssdmres 4633 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
(𝐴 ⊆ dom 𝐵 ↔ dom (𝐵𝐴) = 𝐴)

Theoremdmresexg 4634 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
(𝐵𝑉 → dom (𝐴𝐵) ∈ V)

Theoremresss 4635 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
(𝐴𝐵) ⊆ 𝐴

Theoremrescom 4636 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ↾ 𝐵)

Theoremssres 4637 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremssres2 4638 Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

Theoremrelres 4639 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 4640 Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)
(𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))

Theoremresabs2 4641 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
(𝐵𝐶 → ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐵))

Theoremresidm 4642 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
((𝐴𝐵) ↾ 𝐵) = (𝐴𝐵)

Theoremresima 4643 A restriction to an image. (Contributed by NM, 29-Sep-2004.)
((𝐴𝐵) “ 𝐵) = (𝐴𝐵)

Theoremresima2 4644 Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
(𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))

Theoremxpssres 4645 Restriction of a constant function (or other cross product). (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐶𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))

Theoremelres 4646* Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)
(𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))

Theoremelsnres 4647* Memebership 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 4648 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
((Rel 𝐴 ∧ dom 𝐴𝐵) → (𝐴𝐵) = 𝐴)

Theoremresdm 4649 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
(Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)

Theoremresexg 4650 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 4651 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
𝐴 ∈ V       (𝐴𝐵) ∈ V

Theoremresopab 4652* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}

Theoremresiexg 4653 The existence of a restricted identity function, proved without using the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.)
(𝐴𝑉 → ( I ↾ 𝐴) ∈ V)

Theoremiss 4654 A subclass of the identity function is the identity function restricted to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴 ⊆ I ↔ 𝐴 = ( I ↾ dom 𝐴))

Theoremresopab2 4655* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)
(𝐴𝐵 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})

Theoremresmpt 4656* Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)
(𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))

Theoremresmpt3 4657* Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)
((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴𝐵) ↦ 𝐶)

Theoremdfres2 4658* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
(𝑅𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}

Theoremopabresid 4659* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)
{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ( I ↾ 𝐴)

Theoremmptresid 4660* The restricted identity expressed with the "maps to" notation. (Contributed by FL, 25-Apr-2012.)
(𝑥𝐴𝑥) = ( I ↾ 𝐴)

Theoremdmresi 4661 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
dom ( I ↾ 𝐴) = 𝐴

Theoremresid 4662 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)
(Rel 𝐴 → (𝐴 ↾ V) = 𝐴)

Theoremimaeq1 4663 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremimaeq2 4664 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremimaeq1i 4665 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)

Theoremimaeq2i 4666 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)

Theoremimaeq1d 4667 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Theoremimaeq2d 4668 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Theoremimaeq12d 4669 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremdfima2 4670* Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}

Theoremdfima3 4671* Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝐵) = {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}

Theoremelimag 4672* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)
(𝐴𝑉 → (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴))

Theoremelima 4673* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)
𝐴 ∈ V       (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴)

Theoremelima2 4674* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)
𝐴 ∈ V       (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐶𝑥𝐵𝐴))

Theoremelima3 4675* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)
𝐴 ∈ V       (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))

Theoremnfima 4676 Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)

Theoremnfimad 4677 Deduction version of bound-variable hypothesis builder nfima 4676. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥(𝐴𝐵))

Theoremimadmrn 4678 The image of the domain of a class is the range of the class. (Contributed by NM, 14-Aug-1994.)
(𝐴 “ dom 𝐴) = ran 𝐴

Theoremimassrn 4679 The image of a class is a subset of its range. Theorem 3.16(xi) of [Monk1] p. 39. (Contributed by NM, 31-Mar-1995.)
(𝐴𝐵) ⊆ ran 𝐴

Theoremimaexg 4680 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)

Theoremimai 4681 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
( I “ 𝐴) = 𝐴

Theoremrnresi 4682 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)
ran ( I ↾ 𝐴) = 𝐴

Theoremresiima 4683 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
(𝐵𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵)

Theoremima0 4684 Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
(𝐴 “ ∅) = ∅

Theorem0ima 4685 Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
(∅ “ 𝐴) = ∅

Theoremcsbima12g 4686 Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
(𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))

Theoremimadisj 4687 A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)
((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)

Theoremcnvimass 4688 A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.)
(𝐴𝐵) ⊆ dom 𝐴

Theoremcnvimarndm 4689 The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.)
(𝐴 “ ran 𝐴) = dom 𝐴

Theoremimasng 4690* The image of a singleton. (Contributed by NM, 8-May-2005.)
(𝐴𝐵 → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})

Theoremelreimasng 4691 Elementhood in the image of a singleton. (Contributed by Jim Kingdon, 10-Dec-2018.)
((Rel 𝑅𝐴𝑉) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))

Theoremelimasn 4692 Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝐵 ∈ V    &   𝐶 ∈ V       (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)

Theoremelimasng 4693 Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)
((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))

Theoremargs 4694* Two ways to express the class of unique-valued arguments of 𝐹, which is the same as the domain of 𝐹 whenever 𝐹 is a function. The left-hand side of the equality is from Definition 10.2 of [Quine] p. 65. Quine uses the notation "arg 𝐹 " for this class (for which we have no separate notation). (Contributed by NM, 8-May-2005.)
{𝑥 ∣ ∃𝑦(𝐹 “ {𝑥}) = {𝑦}} = {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}

Theoremeliniseg 4695 Membership in an initial segment. The idiom (𝐴 “ {𝐵}), meaning {𝑥𝑥𝐴𝐵}, is used to specify an initial segment in (for example) Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝐶 ∈ V       (𝐵𝑉 → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))

Theoremepini 4696 Any set is equal to its preimage under the converse epsilon relation. (Contributed by Mario Carneiro, 9-Mar-2013.)
𝐴 ∈ V       ( E “ {𝐴}) = 𝐴

Theoreminiseg 4697* An idiom that signifies an initial segment of an ordering, used, for example, in Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.)
(𝐵𝑉 → (𝐴 “ {𝐵}) = {𝑥𝑥𝐴𝐵})

Theoremdfse2 4698* Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
(𝑅 Se 𝐴 ↔ ∀𝑥𝐴 (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)

Theoremexse2 4699 Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
(𝑅𝑉𝑅 Se 𝐴)

Theoremimass1 4700 Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

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