Home Metamath Proof ExplorerTheorem List (p. 287 of 425) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-26951) Hilbert Space Explorer (26952-28476) Users' Mathboxes (28477-42430)

Theorem List for Metamath Proof Explorer - 28601-28700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmptexgf 28601 If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) (Revised by Thierry Arnoux, 17-May-2020.)
𝑥𝐴       (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)

Theoremac6sf2 28602* Alternate version of ac6 9065 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.) (Revised by Thierry Arnoux, 17-May-2020.)
𝑦𝐵    &   𝑦𝜓    &   𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))

Theoremidssxp 28603 A diagonal set as a subset of a Cartesian product. (Contributed by Thierry Arnoux, 29-Dec-2019.)
( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)

Theoremfnresin 28604 Restriction of a function with a subclass of its domain. (Contributed by Thierry Arnoux, 10-Oct-2017.)
(𝐹 Fn 𝐴 → (𝐹𝐵) Fn (𝐴𝐵))

Theoremf1o3d 28605* Describe an implicit one-to-one onto function. (Contributed by Thierry Arnoux, 23-Apr-2017.)
(𝜑𝐹 = (𝑥𝐴𝐶))    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑦𝐵) → 𝐷𝐴)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))       (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))

Theoremrinvf1o 28606 Sufficient conditions for the restriction of an involution to be a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)
Fun 𝐹    &   𝐹 = 𝐹    &   (𝐹𝐴) ⊆ 𝐵    &   (𝐹𝐵) ⊆ 𝐴    &   𝐴 ⊆ dom 𝐹    &   𝐵 ⊆ dom 𝐹       (𝐹𝐴):𝐴1-1-onto𝐵

Theoremfresf1o 28607 Conditions for a restriction to be a one-to-one onto function. (Contributed by Thierry Arnoux, 7-Dec-2016.)
((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1-onto𝐶)

Theoremf1mptrn 28608* Express injection for a mapping operation. (Contributed by Thierry Arnoux, 3-May-2020.)
((𝜑𝑥𝐴) → 𝐵𝐶)    &   ((𝜑𝑦𝐶) → ∃!𝑥𝐴 𝑦 = 𝐵)       (𝜑 → Fun (𝑥𝐴𝐵))

Theoremdfimafnf 28609* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Thierry Arnoux, 24-Apr-2017.)
𝑥𝐴    &   𝑥𝐹       ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})

Theoremfunimass4f 28610 Membership relation for the values of a function whose image is a subclass. (Contributed by Thierry Arnoux, 24-Apr-2017.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝐹       ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))

Theoremsuppss2f 28611* Show that the support of a function is contained in a set. (Contributed by Thierry Arnoux, 22-Jun-2017.) (Revised by AV, 1-Sep-2020.)
𝑘𝜑    &   𝑘𝐴    &   𝑘𝑊    &   ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍)    &   (𝜑𝐴𝑉)       (𝜑 → ((𝑘𝐴𝐵) supp 𝑍) ⊆ 𝑊)

Theoremfovcld 28612 Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Revised by Thierry Arnoux, 17-Feb-2017.)
(𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)       ((𝜑𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)

Theoremofrn 28613 The range of the function operation. (Contributed by Thierry Arnoux, 8-Jan-2017.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑+ :(𝐵 × 𝐵)⟶𝐶)    &   (𝜑𝐴𝑉)       (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ 𝐶)

Theoremofrn2 28614 The range of the function operation. (Contributed by Thierry Arnoux, 21-Mar-2017.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑+ :(𝐵 × 𝐵)⟶𝐶)    &   (𝜑𝐴𝑉)       (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))

Theoremoff2 28615* The function operation produces a function - alternative form with all antecedents as deduction. (Contributed by Thierry Arnoux, 17-Feb-2017.)
((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐵𝑇)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑 → (𝐴𝐵) = 𝐶)       (𝜑 → (𝐹𝑓 𝑅𝐺):𝐶𝑈)

Theoremofresid 28616 Applying an operation restricted to the range of the functions does not change the function operation. (Contributed by Thierry Arnoux, 14-Feb-2018.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑𝐴𝑉)       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝐹𝑓 (𝑅 ↾ (𝐵 × 𝐵))𝐺))

Theoremfimarab 28617* Expressing the image of a set as a restricted abstract builder. (Contributed by Thierry Arnoux, 27-Jan-2020.)
((𝐹:𝐴𝐵𝑋𝐴) → (𝐹𝑋) = {𝑦𝐵 ∣ ∃𝑥𝑋 (𝐹𝑥) = 𝑦})

Theoremunipreima 28618* Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.)
(Fun 𝐹 → (𝐹 𝐴) = 𝑥𝐴 (𝐹𝑥))

Theoremsspreima 28619 The preimage of a subset is a subset of the preimage. (Contributed by Brendan Leahy, 23-Sep-2017.)
((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))

Theoremopfv 28620 Value of a function producing ordered pairs. (Contributed by Thierry Arnoux, 3-Jan-2017.)
(((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) = ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩)

Theoremxppreima 28621 The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 6-Jun-2017.)
((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (𝐹 “ (𝑌 × 𝑍)) = (((1st𝐹) “ 𝑌) ∩ ((2nd𝐹) “ 𝑍)))

Theoremxppreima2 28622* The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 7-Jun-2017.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐶)    &   𝐻 = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)       (𝜑 → (𝐻 “ (𝑌 × 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))

Theoremelunirn2 28623 Condition for the membership in the union of the range of a function. (Contributed by Thierry Arnoux, 13-Nov-2016.)
((Fun 𝐹𝐵 ∈ (𝐹𝐴)) → 𝐵 ran 𝐹)

Theoremabfmpunirn 28624* Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 28-Sep-2016.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})    &   {𝑦𝜑} ∈ V    &   (𝑦 = 𝐵 → (𝜑𝜓))       (𝐵 ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 𝜓))

Theoremrabfmpunirn 28625* Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 30-Sep-2016.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝑊𝜑})    &   𝑊 ∈ V    &   (𝑦 = 𝐵 → (𝜑𝜓))       (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 (𝐵𝑊𝜓))

Theoremabfmpeld 28626* Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝜓})    &   (𝜑 → {𝑦𝜓} ∈ V)    &   (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒)))       (𝜑 → ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜒)))

Theoremabfmpel 28627* Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})    &   {𝑦𝜑} ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜓))

TheoremfmptdF 28628 Domain and co-domain of the mapping operation; deduction form. This version of fmptd 6181 uses bound-variable hypothesis instead of distinct variable conditions. (Contributed by Thierry Arnoux, 28-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐶    &   ((𝜑𝑥𝐴) → 𝐵𝐶)    &   𝐹 = (𝑥𝐴𝐵)       (𝜑𝐹:𝐴𝐶)

Theoremmpteq12df 28629 An equality theorem for the maps to notation. (Contributed by Thierry Arnoux, 30-May-2020.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐶    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Theoremresmptf 28630 Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))

Theoremfmptcof2 28631* Composition of two functions expressed as ordered-pair class abstractions. (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.) (Revised by Thierry Arnoux, 10-May-2017.)
𝑥𝑆    &   𝑦𝑇    &   𝑥𝐴    &   𝑥𝐵    &   𝑥𝜑    &   (𝜑 → ∀𝑥𝐴 𝑅𝐵)    &   (𝜑𝐹 = (𝑥𝐴𝑅))    &   (𝜑𝐺 = (𝑦𝐵𝑆))    &   (𝑦 = 𝑅𝑆 = 𝑇)       (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))

Theoremfcomptf 28632* Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 6195. (Contributed by Thierry Arnoux, 30-Jun-2017.)
𝑥𝐵       ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (𝐴‘(𝐵𝑥))))

Theoremacunirnmpt 28633* Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 6-Nov-2019.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)    &   𝐶 = ran (𝑗𝐴𝐵)       (𝜑 → ∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶𝑗𝐴 (𝑓𝑦) ∈ 𝐵))

Theoremacunirnmpt2 28634* Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 7-Nov-2019.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)    &   𝐶 = ran (𝑗𝐴𝐵)    &   (𝑗 = (𝑓𝑥) → 𝐵 = 𝐷)       (𝜑 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷))

Theoremacunirnmpt2f 28635* Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 7-Nov-2019.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)    &   𝑗𝐴    &   𝑗𝐶    &   𝑗𝐷    &   𝐶 = 𝑗𝐴 𝐵    &   (𝑗 = (𝑓𝑥) → 𝐵 = 𝐷)    &   ((𝜑𝑗𝐴) → 𝐵𝑊)       (𝜑 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷))

Theoremaciunf1lem 28636* Choice in an index union. (Contributed by Thierry Arnoux, 8-Nov-2019.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)    &   𝑗𝐴    &   ((𝜑𝑗𝐴) → 𝐵𝑊)       (𝜑 → ∃𝑓(𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 𝑗𝐴 𝐵(2nd ‘(𝑓𝑥)) = 𝑥))

Theoremaciunf1 28637* Choice in an index union. (Contributed by Thierry Arnoux, 4-May-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵𝑊)       (𝜑 → ∃𝑓(𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑘 𝑗𝐴 𝐵(2nd ‘(𝑓𝑘)) = 𝑘))

Theoremcofmpt 28638* Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 29-Jun-2017.)
(𝜑𝐹:𝐶𝐷)    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → (𝐹 ∘ (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ (𝐹𝐵)))

Theoremofoprabco 28639* Function operation as a composition with an operation. (Contributed by Thierry Arnoux, 4-Jun-2017.)
𝑎𝑀    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝑀 = (𝑎𝐴 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩))    &   (𝜑𝑁 = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)))       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑁𝑀))

Theoremofpreima 28640* Express the preimage of a function operation as a union of preimages. (Contributed by Thierry Arnoux, 8-Mar-2018.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝑅 Fn (𝐵 × 𝐶))       (𝜑 → ((𝐹𝑓 𝑅𝐺) “ 𝐷) = 𝑝 ∈ (𝑅𝐷)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))

Theoremofpreima2 28641* Express the preimage of a function operation as a union of preimages. This version of ofpreima 28640 iterates the union over a smaller set. (Contributed by Thierry Arnoux, 8-Mar-2018.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝑅 Fn (𝐵 × 𝐶))       (𝜑 → ((𝐹𝑓 𝑅𝐺) “ 𝐷) = 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))

TheoremfuncnvmptOLD 28642* Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐹    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (Fun 𝐹 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦 = 𝐵)))

Theoremfuncnvmpt 28643* Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐹    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (Fun 𝐹 ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))

Theoremfuncnv5mpt 28644* Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 1-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐹    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝑥 = 𝑧𝐵 = 𝐶)       (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝐴𝑧𝐴 (𝑥 = 𝑧𝐵𝐶)))

Theoremfuncnv4mpt 28645* Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 2-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐹    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (Fun 𝐹 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)))

Theoremfgreu 28646* Exactly one point of a function's graph has a given first element. (Contributed by Thierry Arnoux, 1-Apr-2018.)
((Fun 𝐹𝑋 ∈ dom 𝐹) → ∃!𝑝𝐹 𝑋 = (1st𝑝))

Theoremfcnvgreu 28647* If the converse of a relation 𝐴 is a function, exactly one point of its graph has a given second element (that is, function value). (Contributed by Thierry Arnoux, 1-Apr-2018.)
(((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ ran 𝐴) → ∃!𝑝𝐴 𝑌 = (2nd𝑝))

Theoremrnmpt2ss 28648* The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 23-May-2017.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → ran 𝐹𝐷)

TheoremmptssALT 28649* Deduce subset relation of mapping-to function graphs from a subset relation of domains. Alternative proof of mptss 5264. (Contributed by Thierry Arnoux, 30-May-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))

Theorempartfun 28650 Rewrite a function defined by parts, using a mapping and an if construct, into a union of functions on disjoint domains. (Contributed by Thierry Arnoux, 30-Mar-2017.)
(𝑥𝐴 ↦ if(𝑥𝐵, 𝐶, 𝐷)) = ((𝑥 ∈ (𝐴𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴𝐵) ↦ 𝐷))

Theoremdfcnv2 28651* Alternative definition of the converse of a relation. (Contributed by Thierry Arnoux, 31-Mar-2018.)
(ran 𝑅𝐴𝑅 = 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥})))

Theoremmpt2mptxf 28652* Express a two-argument function as a one-argument function, or vice-versa. In this version 𝐵(𝑥) is not assumed to be constant w.r.t 𝑥. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Thierry Arnoux, 31-Mar-2018.)
𝑥𝐶    &   𝑦𝐶    &   (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)       (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)

Theoremgtiso 28653 Two ways to write a strictly decreasing function on the reals. (Contributed by Thierry Arnoux, 6-Apr-2017.)
((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))

Theoremisoun 28654* Infer an isomorphism from a union of two isomorphisms. (Contributed by Thierry Arnoux, 30-Mar-2017.)
(𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷))    &   ((𝜑𝑥𝐴𝑦𝐶) → 𝑥𝑅𝑦)    &   ((𝜑𝑧𝐵𝑤𝐷) → 𝑧𝑆𝑤)    &   ((𝜑𝑥𝐶𝑦𝐴) → ¬ 𝑥𝑅𝑦)    &   ((𝜑𝑧𝐷𝑤𝐵) → ¬ 𝑧𝑆𝑤)    &   (𝜑 → (𝐴𝐶) = ∅)    &   (𝜑 → (𝐵𝐷) = ∅)       (𝜑 → (𝐻𝐺) Isom 𝑅, 𝑆 ((𝐴𝐶), (𝐵𝐷)))

20.3.4.5  Disjointness (additional proof requiring functions)

Theoremdisjdsct 28655* A disjoint collection is distinct, i.e. each set in this collection is different of all others, provided that it does not contain the empty set This can be expressed as "the converse of the mapping function is a function", or "the mapping function is single-rooted". (Cf. funcnv 5762) (Contributed by Thierry Arnoux, 28-Feb-2017.)
𝑥𝜑    &   𝑥𝐴    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (𝑉 ∖ {∅}))    &   (𝜑Disj 𝑥𝐴 𝐵)       (𝜑 → Fun (𝑥𝐴𝐵))

20.3.4.6  First and second members of an ordered pair - misc additions

Theoremdf1stres 28656* Definition for a restriction of the 1st (first member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
(1st ↾ (𝐴 × 𝐵)) = (𝑥𝐴, 𝑦𝐵𝑥)

Theoremdf2ndres 28657* Definition for a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
(2nd ↾ (𝐴 × 𝐵)) = (𝑥𝐴, 𝑦𝐵𝑦)

Theorem1stpreimas 28658 The preimage of a singleton. (Contributed by Thierry Arnoux, 27-Apr-2020.)
((Rel 𝐴𝑋𝑉) → ((1st𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋})))

Theorem1stpreima 28659 The preimage by 1st is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
(𝐴𝐵 → ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶))

Theorem2ndpreima 28660 The preimage by 2nd is an 'horizontal band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
(𝐴𝐶 → ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))

Theoremcurry2ima 28661* The image of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 25-Sep-2017.)
𝐺 = (𝐹(1st ↾ (V × {𝐶})))       ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (𝐺𝐷) = {𝑦 ∣ ∃𝑥𝐷 𝑦 = (𝑥𝐹𝐶)})

Theoremsupssd 28662* Inequality deduction for supremum of a subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝐵𝐶)    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))       (𝜑 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅))

Theoreminfssd 28663* Inequality deduction for infimum of a subset. (Contributed by AV, 4-Oct-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝐶𝐵)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐶 𝑧𝑅𝑦)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ¬ inf(𝐶, 𝐴, 𝑅)𝑅inf(𝐵, 𝐴, 𝑅))

20.3.4.8  Finite Sets

Theoremimafi2 28664 The image by a finite set is finite. See also imafi 8022. (Contributed by Thierry Arnoux, 25-Apr-2020.)
(𝐴 ∈ Fin → (𝐴𝐵) ∈ Fin)

Theoremunifi3 28665 If a union is finite, then all its elements are finite. See unifi 8018. (Contributed by Thierry Arnoux, 27-Aug-2017.)
( 𝐴 ∈ Fin → 𝐴 ⊆ Fin)

20.3.4.9  Countable Sets

Theoremsnct 28666 A singleton is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.)
(𝐴𝑉 → {𝐴} ≼ ω)

Theoremprct 28667 An unordered pair is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ≼ ω)

Theoremfnct 28668 If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 ≼ ω)

Theoremdmct 28669 The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → dom 𝐴 ≼ ω)

Theoremcnvct 28670 If a set is countable, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → 𝐴 ≼ ω)

Theoremrnct 28671 The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → ran 𝐴 ≼ ω)

Theoremmptct 28672* A countable mapping set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → (𝑥𝐴𝐵) ≼ ω)

Theoremmpt2cti 28673* An operation is countable if both its domains are countable. (Contributed by Thierry Arnoux, 17-Sep-2017.)
𝑥𝐴𝑦𝐵 𝐶𝑉       ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝑥𝐴, 𝑦𝐵𝐶) ≼ ω)

Theoremabrexct 28674* An image set of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≼ ω)

Theoremmptctf 28675 A countable mapping set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝐴       (𝐴 ≼ ω → (𝑥𝐴𝐵) ≼ ω)

Theoremabrexctf 28676* An image set of a countable set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝐴       (𝐴 ≼ ω → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≼ ω)

Theorempadct 28677* Index a countable set with integers and pad with 𝑍. (Contributed by Thierry Arnoux, 1-Jun-2020.)
((𝐴 ≼ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))

Theoremcnvoprab 28678* The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.)
𝑥𝜓    &   𝑦𝜓    &   (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓𝜑))    &   (𝜓𝑎 ∈ (V × V))       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}

Theoremf1od2 28679* Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝑊)    &   ((𝜑𝑧𝐷) → (𝐼𝑋𝐽𝑌))    &   (𝜑 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑧𝐷 ∧ (𝑥 = 𝐼𝑦 = 𝐽))))       (𝜑𝐹:(𝐴 × 𝐵)–1-1-onto𝐷)

Theoremfcobij 28680* Composing functions with a bijection yields a bijection between sets of functions. (Contributed by Thierry Arnoux, 25-Aug-2017.)
(𝜑𝐺:𝑆1-1-onto𝑇)    &   (𝜑𝑅𝑈)    &   (𝜑𝑆𝑉)    &   (𝜑𝑇𝑊)       (𝜑 → (𝑓 ∈ (𝑆𝑚 𝑅) ↦ (𝐺𝑓)):(𝑆𝑚 𝑅)–1-1-onto→(𝑇𝑚 𝑅))

Theoremfcobijfs 28681* Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also mapfien 8076. (Contributed by Thierry Arnoux, 25-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
(𝜑𝐺:𝑆1-1-onto𝑇)    &   (𝜑𝑅𝑈)    &   (𝜑𝑆𝑉)    &   (𝜑𝑇𝑊)    &   (𝜑𝑂𝑆)    &   𝑄 = (𝐺𝑂)    &   𝑋 = {𝑔 ∈ (𝑆𝑚 𝑅) ∣ 𝑔 finSupp 𝑂}    &   𝑌 = { ∈ (𝑇𝑚 𝑅) ∣ finSupp 𝑄}       (𝜑 → (𝑓𝑋 ↦ (𝐺𝑓)):𝑋1-1-onto𝑌)

Theoremsuppss3 28682* Deduce a function's support's inclusion in another function's support. (Contributed by Thierry Arnoux, 7-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝐺 = (𝑥𝐴𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝑍𝑊)    &   (𝜑𝐹 Fn 𝐴)    &   ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑍) → 𝐵 = 𝑍)       (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))

Theoremffs2 28683 Rewrite a function's support based with its range rather than the universal class. See also frnsuppeq 7074. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝐶 = (𝐵 ∖ {𝑍})       ((𝐴𝑉𝑍𝑊𝐹:𝐴𝐵) → (𝐹 supp 𝑍) = (𝐹𝐶))

Theoremffsrn 28684 The range of a finitely supported function is finite. (Contributed by Thierry Arnoux, 27-Aug-2017.)
(𝜑𝑍𝑊)    &   (𝜑𝐹𝑉)    &   (𝜑 → Fun 𝐹)    &   (𝜑 → (𝐹 supp 𝑍) ∈ Fin)       (𝜑 → ran 𝐹 ∈ Fin)

Theoremresf1o 28685* Restriction of functions to a superset of their support creates a bijection. (Contributed by Thierry Arnoux, 12-Sep-2017.)
𝑋 = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶}    &   𝐹 = (𝑓𝑋 ↦ (𝑓𝐶))       (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → 𝐹:𝑋1-1-onto→(𝐵𝑚 𝐶))

Theoremmaprnin 28686* Restricting the range of the mapping operator. (Contributed by Thierry Arnoux, 30-Aug-2017.)
𝐴 ∈ V    &   𝐵 ∈ V       ((𝐵𝐶) ↑𝑚 𝐴) = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ ran 𝑓𝐶}

Theoremfpwrelmapffslem 28687* Lemma for fpwrelmapffs 28689. For this theorem, the sets 𝐴 and 𝐵 could be infinite, but the relation 𝑅 itself is finite. (Contributed by Thierry Arnoux, 1-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝜑𝐹:𝐴⟶𝒫 𝐵)    &   (𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))})       (𝜑 → (𝑅 ∈ Fin ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin)))

Theoremfpwrelmap 28688* Define a canonical mapping between functions from 𝐴 into subsets of 𝐵 and the relations with domain 𝐴 and range within 𝐵. Note that the same relation is used in axdc2lem 9033 and marypha2lem1 8104. (Contributed by Thierry Arnoux, 28-Aug-2017.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝑀 = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})       𝑀:(𝒫 𝐵𝑚 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)

Theoremfpwrelmapffs 28689* Define a canonical mapping between finite relations (finite subsets of a cartesian product) and functions with finite support into finite subsets. (Contributed by Thierry Arnoux, 28-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝑀 = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})    &   𝑆 = {𝑓 ∈ ((𝒫 𝐵 ∩ Fin) ↑𝑚 𝐴) ∣ (𝑓 supp ∅) ∈ Fin}       (𝑀𝑆):𝑆1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin)

20.3.5  Real and Complex Numbers

20.3.5.1  Complex operations - misc. additions

Theoremaddeq0 28690 Two complex which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 2-May-2017.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) = 0 ↔ 𝐴 = -𝐵))

Theoremsubeqxfrd 28691 Transfer two terms of a subtraction in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑 → (𝐴𝐵) = (𝐶𝐷))       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremznsqcld 28692 Squaring of nonzero relative numbers. (Contributed by Thierry Arnoux, 2-Feb-2020.)
(𝜑𝑁 ∈ ℤ)    &   (𝜑𝑁 ≠ 0)       (𝜑 → (𝑁↑2) ∈ ℕ)

Theoremnn0sqeq1 28693 Integer square one. (Contributed by Thierry Arnoux, 2-Feb-2020.)
((𝑁 ∈ ℕ0 ∧ (𝑁↑2) = 1) → 𝑁 = 1)

Theorem1neg1t1neg1 28694 An integer unit times itself. (Contributed by Thierry Arnoux, 23-Aug-2020.)
(𝑁 ∈ {-1, 1} → (𝑁 · 𝑁) = 1)

20.3.5.2  Ordering on reals - misc additions

Theoremlt2addrd 28695* If the right-hand side of a 'less than' relationship is an addition, then we can express the left-hand side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < (𝐵 + 𝐶))       (𝜑 → ∃𝑏 ∈ ℝ ∃𝑐 ∈ ℝ (𝐴 = (𝑏 + 𝑐) ∧ 𝑏 < 𝐵𝑐 < 𝐶))

20.3.5.3  Extended reals - misc additions

Theoremxgepnf 28696 An extended real which is greater than plus infinity is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ ℝ* → (+∞ ≤ 𝐴𝐴 = +∞))

Theoremxlemnf 28697 An extended real which is less than minus infinity is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)
(𝐴 ∈ ℝ* → (𝐴 ≤ -∞ ↔ 𝐴 = -∞))

Theoremxrlelttric 28698 Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵𝐵 < 𝐴))

Theoremxaddeq0 28699 Two extended reals which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 13-Jun-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 +𝑒 𝐵) = 0 ↔ 𝐴 = -𝑒𝐵))

Theoreminfxrmnf 28700 The infinimum of a set of extended reals containing minus infinity is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Revised by AV, 28-Sep-2020.)
((𝐴 ⊆ ℝ* ∧ -∞ ∈ 𝐴) → inf(𝐴, ℝ*, < ) = -∞)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42430
 Copyright terms: Public domain < Previous  Next >