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

Theoremdfres2 4601* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
(𝑅A) = {⟨x, y⟩ ∣ (x A x𝑅y)}

Theoremopabresid 4602* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)
{⟨x, y⟩ ∣ (x A y = x)} = ( I ↾ A)

Theoremmptresid 4603* The restricted identity expressed with the "maps to" notation. (Contributed by FL, 25-Apr-2012.)
(x Ax) = ( I ↾ A)

Theoremdmresi 4604 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
dom ( I ↾ A) = A

Theoremresid 4605 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)
(Rel A → (A ↾ V) = A)

Theoremimaeq1 4606 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
(A = B → (A𝐶) = (B𝐶))

Theoremimaeq2 4607 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
(A = B → (𝐶A) = (𝐶B))

Theoremimaeq1i 4608 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
A = B       (A𝐶) = (B𝐶)

Theoremimaeq2i 4609 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
A = B       (𝐶A) = (𝐶B)

Theoremimaeq1d 4610 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
(φA = B)       (φ → (A𝐶) = (B𝐶))

Theoremimaeq2d 4611 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
(φA = B)       (φ → (𝐶A) = (𝐶B))

Theoremimaeq12d 4612 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝐶) = (B𝐷))

Theoremdfima2 4613* 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) = {yx B xAy}

Theoremdfima3 4614* 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) = {yx(x B x, y A)}

Theoremelimag 4615* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)
(A 𝑉 → (A (B𝐶) ↔ x 𝐶 xBA))

Theoremelima 4616* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)
A V       (A (B𝐶) ↔ x 𝐶 xBA)

Theoremelima2 4617* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)
A V       (A (B𝐶) ↔ x(x 𝐶 xBA))

Theoremelima3 4618* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)
A V       (A (B𝐶) ↔ x(x 𝐶 x, A B))

Theoremnfima 4619 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 4620 Deduction version of bound-variable hypothesis builder nfima 4619. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
(φxA)    &   (φxB)       (φx(AB))

Theoremimadmrn 4621 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 4622 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 4623 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)
(A 𝑉 → (AB) V)

Theoremimai 4624 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
( I “ A) = A

Theoremrnresi 4625 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)
ran ( I ↾ A) = A

Theoremresiima 4626 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
(BA → (( I ↾ A) “ B) = B)

Theoremima0 4627 Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
(A “ ∅) = ∅

Theorem0ima 4628 Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
(∅ “ A) = ∅

Theoremcsbima12g 4629 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.)
(A 𝐶A / x(𝐹B) = (A / x𝐹A / xB))

Theoremimadisj 4630 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 4631 A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.)
(AB) ⊆ dom A

Theoremcnvimarndm 4632 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 4633* The image of a singleton. (Contributed by NM, 8-May-2005.)
(A B → (𝑅 “ {A}) = {yA𝑅y})

Theoremelreimasng 4634 Elementhood in the image of a singleton. (Contributed by Jim Kingdon, 10-Dec-2018.)
((Rel 𝑅 A 𝑉) → (B (𝑅 “ {A}) ↔ A𝑅B))

Theoremelimasn 4635 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 4636 Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)
((B 𝑉 𝐶 𝑊) → (𝐶 (A “ {B}) ↔ ⟨B, 𝐶 A))

Theoremargs 4637* 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.)
{xy(𝐹 “ {x}) = {y}} = {x∃!y x𝐹y}

Theoremeliniseg 4638 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       (B 𝑉 → (𝐶 (A “ {B}) ↔ 𝐶AB))

Theoremepini 4639 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 4640* 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.)
(B 𝑉 → (A “ {B}) = {xxAB})

Theoremdfse2 4641* Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
(𝑅 Se Ax A (A ∩ (𝑅 “ {x})) V)

Theoremexse2 4642 Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
(𝑅 𝑉𝑅 Se A)

Theoremimass1 4643 Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
(AB → (A𝐶) ⊆ (B𝐶))

Theoremimass2 4644 Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)
(AB → (𝐶A) ⊆ (𝐶B))

Theoremndmima 4645 The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)
A dom B → (B “ {A}) = ∅)

Theoremrelcnv 4646 A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)
Rel A

Theoremrelbrcnvg 4647 When 𝑅 is a relation, the sethood assumptions on brcnv 4461 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
(Rel 𝑅 → (A𝑅BB𝑅A))

Theoremrelbrcnv 4648 When 𝑅 is a relation, the sethood assumptions on brcnv 4461 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
Rel 𝑅       (A𝑅BB𝑅A)

Theoremcotr 4649* Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝑅𝑅) ⊆ 𝑅xyz((x𝑅y y𝑅z) → x𝑅z))

Theoremissref 4650* Two ways to state a relation is reflexive. Adapted from Tarski. (Contributed by FL, 15-Jan-2012.) (Revised by NM, 30-Mar-2016.)
(( I ↾ A) ⊆ 𝑅x A x𝑅x)

Theoremcnvsym 4651* Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝑅𝑅xy(x𝑅yy𝑅x))

Theoremintasym 4652* Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝑅𝑅) ⊆ I ↔ xy((x𝑅y y𝑅x) → x = y))

Theoremasymref 4653* Two ways of saying a relation is antisymmetric and reflexive. 𝑅 is the field of a relation by relfld 4789. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝑅𝑅) = ( I ↾ 𝑅) ↔ x 𝑅y((x𝑅y y𝑅x) ↔ x = y))

Theoremintirr 4654* Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝑅 ∩ I ) = ∅ ↔ x ¬ x𝑅x)

Theorembrcodir 4655* Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.)
((A 𝑉 B 𝑊) → (A(𝑅𝑅)Bz(A𝑅z B𝑅z)))

Theoremcodir 4656* Two ways of saying a relation is directed. (Contributed by Mario Carneiro, 22-Nov-2013.)
((A × B) ⊆ (𝑅𝑅) ↔ x A y B z(x𝑅z y𝑅z))

Theoremqfto 4657* A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.)
((A × B) ⊆ (𝑅𝑅) ↔ x A y B (x𝑅y y𝑅x))

Theoremxpidtr 4658 A square cross product (A × A) is a transitive relation. (Contributed by FL, 31-Jul-2009.)
((A × A) ∘ (A × A)) ⊆ (A × A)

Theoremtrin2 4659 The intersection of two transitive classes is transitive. (Contributed by FL, 31-Jul-2009.)
(((𝑅𝑅) ⊆ 𝑅 (𝑆𝑆) ⊆ 𝑆) → ((𝑅𝑆) ∘ (𝑅𝑆)) ⊆ (𝑅𝑆))

Theorempoirr2 4660 A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.)
(𝑅 Po A → (𝑅 ∩ ( I ↾ A)) = ∅)

Theoremtrinxp 4661 The relation induced by a transitive relation on a part of its field is transitive. (Taking the intersection of a relation with a square cross product is a way to restrict it to a subset of its field.) (Contributed by FL, 31-Jul-2009.)
((𝑅𝑅) ⊆ 𝑅 → ((𝑅 ∩ (A × A)) ∘ (𝑅 ∩ (A × A))) ⊆ (𝑅 ∩ (A × A)))

Theoremsoirri 4662 A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
𝑅 Or 𝑆    &   𝑅 ⊆ (𝑆 × 𝑆)        ¬ A𝑅A

Theoremsotri 4663 A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
𝑅 Or 𝑆    &   𝑅 ⊆ (𝑆 × 𝑆)       ((A𝑅B B𝑅𝐶) → A𝑅𝐶)

Theoremson2lpi 4664 A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
𝑅 Or 𝑆    &   𝑅 ⊆ (𝑆 × 𝑆)        ¬ (A𝑅B B𝑅A)

Theoremsotri2 4665 A transitivity relation. (Read ¬ B < A and B < C implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)
𝑅 Or 𝑆    &   𝑅 ⊆ (𝑆 × 𝑆)       ((A 𝑆 ¬ B𝑅A B𝑅𝐶) → A𝑅𝐶)

Theoremsotri3 4666 A transitivity relation. (Read A < B and ¬ C < B implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)
𝑅 Or 𝑆    &   𝑅 ⊆ (𝑆 × 𝑆)       ((𝐶 𝑆 A𝑅B ¬ 𝐶𝑅B) → A𝑅𝐶)

Theorempoleloe 4667 Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)
(B 𝑉 → (A(𝑅 ∪ I )B ↔ (A𝑅B A = B)))

Theorempoltletr 4668 Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)
((𝑅 Po 𝑋 (A 𝑋 B 𝑋 𝐶 𝑋)) → ((A𝑅B B(𝑅 ∪ I )𝐶) → A𝑅𝐶))

Theoremcnvopab 4669* The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
{⟨x, y⟩ ∣ φ} = {⟨y, x⟩ ∣ φ}

Theoremcnv0 4670 The converse of the empty set. (Contributed by NM, 6-Apr-1998.)
∅ = ∅

Theoremcnvi 4671 The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36. (Contributed by NM, 26-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
I = I

Theoremcnvun 4672 The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(AB) = (AB)

Theoremcnvdif 4673 Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.)
(AB) = (AB)

Theoremcnvin 4674 Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Revised by Mario Carneiro, 26-Jun-2014.)
(AB) = (AB)

Theoremrnun 4675 Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)
ran (AB) = (ran A ∪ ran B)

Theoremrnin 4676 The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)
ran (AB) ⊆ (ran A ∩ ran B)

Theoremrniun 4677 The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
ran x A B = x A ran B

Theoremrnuni 4678* The range of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 17-Mar-2004.) (Revised by Mario Carneiro, 29-May-2015.)
ran A = x A ran x

Theoremimaundi 4679 Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
(A “ (B𝐶)) = ((AB) ∪ (A𝐶))

Theoremimaundir 4680 The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)
((AB) “ 𝐶) = ((A𝐶) ∪ (B𝐶))

Theoremdminss 4681 An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising." (Contributed by NM, 11-Aug-2004.)
(dom 𝑅A) ⊆ (𝑅 “ (𝑅A))

Theoremimainss 4682 An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)
((𝑅A) ∩ B) ⊆ (𝑅 “ (A ∩ (𝑅B)))

Theoreminimass 4683 The image of an intersection (Contributed by Thierry Arnoux, 16-Dec-2017.)
((AB) “ 𝐶) ⊆ ((A𝐶) ∩ (B𝐶))

Theoreminimasn 4684 The intersection of the image of singleton (Contributed by Thierry Arnoux, 16-Dec-2017.)
(𝐶 𝑉 → ((AB) “ {𝐶}) = ((A “ {𝐶}) ∩ (B “ {𝐶})))

Theoremcnvxp 4685 The converse of a cross product. Exercise 11 of [Suppes] p. 67. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(A × B) = (B × A)

Theoremxp0 4686 The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)
(A × ∅) = ∅

Theoremxpmlem 4687* The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 11-Dec-2018.)
((x x A y y B) ↔ z z (A × B))

Theoremxpm 4688* The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 13-Dec-2018.)
((x x A y y B) ↔ z z (A × B))

Theoremxpeq0r 4689 A cross product is empty if at least one member is empty. (Contributed by Jim Kingdon, 12-Dec-2018.)
((A = ∅ B = ∅) → (A × B) = ∅)

Theoremxpdisj1 4690 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
((AB) = ∅ → ((A × 𝐶) ∩ (B × 𝐷)) = ∅)

Theoremxpdisj2 4691 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
((AB) = ∅ → ((𝐶 × A) ∩ (𝐷 × B)) = ∅)

Theoremxpsndisj 4692 Cross products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.)
(B𝐷 → ((A × {B}) ∩ (𝐶 × {𝐷})) = ∅)

Theoremdjudisj 4693* Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.)
((AB) = ∅ → ( x A ({x} × 𝐶) ∩ y B ({y} × 𝐷)) = ∅)

Theoremresdisj 4694 A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((AB) = ∅ → ((𝐶A) ↾ B) = ∅)

Theoremrnxpm 4695* The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37, with non-empty changed to inhabited. (Contributed by Jim Kingdon, 12-Dec-2018.)
(x x A → ran (A × B) = B)

Theoremdmxpss 4696 The domain of a cross product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.)
dom (A × B) ⊆ A

Theoremrnxpss 4697 The range of a cross product is a subclass of the second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
ran (A × B) ⊆ B

Theoremrnxpid 4698 The range of a square cross product. (Contributed by FL, 17-May-2010.)
ran (A × A) = A

Theoremssxpbm 4699* A cross-product subclass relationship is equivalent to the relationship for its components. (Contributed by Jim Kingdon, 12-Dec-2018.)
(x x (A × B) → ((A × B) ⊆ (𝐶 × 𝐷) ↔ (A𝐶 B𝐷)))

Theoremssxp1 4700* Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)
(x x 𝐶 → ((A × 𝐶) ⊆ (B × 𝐶) ↔ AB))

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