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Theorem List for Intuitionistic Logic Explorer - 4601-4700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelrn2 4601* Membership in a range. (Contributed by NM, 10-Jul-1994.)
𝐴 ∈ V       (𝐴 ∈ ran 𝐵 ↔ ∃𝑥𝑥, 𝐴⟩ ∈ 𝐵)
 
Theoremelrn 4602* Membership in a range. (Contributed by NM, 2-Apr-2004.)
𝐴 ∈ V       (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)
 
Theoremnfdm 4603 Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥dom 𝐴
 
Theoremnfrn 4604 Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥ran 𝐴
 
Theoremdmiin 4605 Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
dom 𝑥𝐴 𝐵 𝑥𝐴 dom 𝐵
 
Theoremrnopab 4606* 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 4607* 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 4608* 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 4609* Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑥 = 𝐷𝐵 = 𝐶)       ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)
 
Theoremelrnmpt1 4610 Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐹 = (𝑥𝐴𝐵)       ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)
 
Theoremelrnmptg 4611* 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 4612* 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 4613 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 4614 Alternate definition of indexed union when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
 
Theoremdfiin3g 4615 Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
 
Theoremdfiun3 4616 Alternate definition of indexed union when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐵 ∈ V        𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)
 
Theoremdfiin3 4617 Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐵 ∈ V        𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)
 
Theoremriinint 4618* 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 4619 A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)
(Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))
 
Theoremdmrnssfld 4620 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 4621 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 4622 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 4623 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 4624 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 4625 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
 
Theoremdmcoss 4626 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 4627 Range of a composition. (Contributed by NM, 19-Mar-1998.)
ran (𝐴𝐵) ⊆ ran 𝐴
 
Theoremdmcosseq 4628 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 4629 Domain of a composition. (Contributed by NM, 19-Mar-1998.)
(dom 𝐴 = ran 𝐵 → dom (𝐴𝐵) = dom 𝐵)
 
Theoremrncoeq 4630 Range of a composition. (Contributed by NM, 19-Mar-1998.)
(dom 𝐴 = ran 𝐵 → ran (𝐴𝐵) = ran 𝐴)
 
Theoremreseq1 4631 Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremreseq2 4632 Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremreseq1i 4633 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)
 
Theoremreseq2i 4634 Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)
 
Theoremreseq12i 4635 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)
 
Theoremreseq1d 4636 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremreseq2d 4637 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremreseq12d 4638 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))
 
Theoremnfres 4639 Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)
 
Theoremcsbresg 4640 Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
 
Theoremres0 4641 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
(𝐴 ↾ ∅) = ∅
 
Theoremopelres 4642 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 4643 Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
𝐵 ∈ V       (𝐴(𝐶𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐴𝐷))
 
Theoremopelresg 4644 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 4645 Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)
(𝐵𝑉 → (𝐴(𝐶𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐴𝐷)))
 
Theoremopres 4646 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 4647 A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)
((𝐵𝐴𝐶𝐴) → (𝐵( I ↾ 𝐴)𝐶𝐵 = 𝐶))
 
Theoremopelresi 4648 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 4649 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
((𝐴𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵𝐶))
 
Theoremresundi 4650 Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
(𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
 
Theoremresundir 4651 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
 
Theoremresindi 4652 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
(𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
 
Theoremresindir 4653 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
 
Theoreminres 4654 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
(𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ↾ 𝐶)
 
Theoremresiun1 4655* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
 
Theoremresiun2 4656* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
(𝐶 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶𝐵)
 
Theoremdmres 4657 The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)
dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
 
Theoremssdmres 4658 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
(𝐴 ⊆ dom 𝐵 ↔ dom (𝐵𝐴) = 𝐴)
 
Theoremdmresexg 4659 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
(𝐵𝑉 → dom (𝐴𝐵) ∈ V)
 
Theoremresss 4660 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
(𝐴𝐵) ⊆ 𝐴
 
Theoremrescom 4661 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ↾ 𝐵)
 
Theoremssres 4662 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
 
Theoremssres2 4663 Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
 
Theoremrelres 4664 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 4665 Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)
(𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
 
Theoremresabs2 4666 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
(𝐵𝐶 → ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐵))
 
Theoremresidm 4667 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
((𝐴𝐵) ↾ 𝐵) = (𝐴𝐵)
 
Theoremresima 4668 A restriction to an image. (Contributed by NM, 29-Sep-2004.)
((𝐴𝐵) “ 𝐵) = (𝐴𝐵)
 
Theoremresima2 4669 Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
(𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))
 
Theoremxpssres 4670 Restriction of a constant function (or other cross product). (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐶𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))
 
Theoremelres 4671* Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)
(𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
 
Theoremelsnres 4672* 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 4673 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
((Rel 𝐴 ∧ dom 𝐴𝐵) → (𝐴𝐵) = 𝐴)
 
Theoremresdm 4674 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
(Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
 
Theoremresexg 4675 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 4676 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
𝐴 ∈ V       (𝐴𝐵) ∈ V
 
Theoremresopab 4677* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
 
Theoremresiexg 4678 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 4679 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 4680* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)
(𝐴𝐵 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
 
Theoremresmpt 4681* Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)
(𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
 
Theoremresmpt3 4682* 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 4683* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
(𝑅𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
 
Theoremopabresid 4684* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)
{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ( I ↾ 𝐴)
 
Theoremmptresid 4685* The restricted identity expressed with the "maps to" notation. (Contributed by FL, 25-Apr-2012.)
(𝑥𝐴𝑥) = ( I ↾ 𝐴)
 
Theoremdmresi 4686 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
dom ( I ↾ 𝐴) = 𝐴
 
Theoremresid 4687 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)
(Rel 𝐴 → (𝐴 ↾ V) = 𝐴)
 
Theoremimaeq1 4688 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremimaeq2 4689 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremimaeq1i 4690 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)
 
Theoremimaeq2i 4691 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)
 
Theoremimaeq1d 4692 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremimaeq2d 4693 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremimaeq12d 4694 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))
 
Theoremdfima2 4695* 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 4696* 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 4697* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)
(𝐴𝑉 → (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴))
 
Theoremelima 4698* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)
𝐴 ∈ V       (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴)
 
Theoremelima2 4699* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)
𝐴 ∈ V       (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐶𝑥𝐵𝐴))
 
Theoremelima3 4700* 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))
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