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Theorem List for Metamath Proof Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcnvf1olem 7801 Lemma for cnvf1o 7802. (Contributed by Mario Carneiro, 27-Apr-2014.)
((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → (𝐶𝐴𝐵 = {𝐶}))

Theoremcnvf1o 7802* Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)
(Rel 𝐴 → (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴)

Theoremfparlem1 7803 Lemma for fpar 7807. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
((1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)

Theoremfparlem2 7804 Lemma for fpar 7807. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
((2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦})

Theoremfparlem3 7805* Lemma for fpar 7807. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐹 Fn 𝐴 → ((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)))

Theoremfparlem4 7806* Lemma for fpar 7807. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐺 Fn 𝐵 → ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})))

Theoremfpar 7807* Merge two functions in parallel. Use as the second argument of a composition with a binary operation to build compound functions such as (𝑥 ∈ (0[,)+∞), 𝑦 ∈ ℝ ↦ ((√‘𝑥) + (sin‘𝑦))), see also ex-fpar 28250. (Contributed by NM, 17-Sep-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
𝐻 = (((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))       ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩))

Theoremfsplit 7808 A function that can be used to feed a common value to both operands of an operation. Use as the second argument of a composition with the function of fpar 7807 in order to build compound functions such as (𝑥 ∈ (0[,)+∞) ↦ ((√‘𝑥) + (sin‘𝑥))). (Contributed by NM, 17-Sep-2007.) Replace use of dfid2 5450 with df-id 5447. (Revised by BJ, 31-Dec-2023.)
(1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)

TheoremfsplitOLD 7809 Obsolete proof of fsplit 7808 as of 31-Dec-2023 . (Contributed by NM, 17-Sep-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)

Theoremfsplitfpar 7810* Merge two functions with a common argument in parallel. Combination of fsplit 7808 and fpar 7807. (Contributed by AV, 3-Jan-2024.)
𝐻 = (((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))    &   𝑆 = ((1st ↾ I ) ↾ 𝐴)       ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐻𝑆) = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))

Theoremoffsplitfpar 7811 Express the function operation map f by the functions defined in fsplit 7808 and fpar 7807. (Contributed by AV, 4-Jan-2024.)
𝐻 = (((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))    &   𝑆 = ((1st ↾ I ) ↾ 𝐴)       (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝐻𝑆)) = (𝐹f + 𝐺))

Theoremf2ndf 7812 The 2nd (second component of an ordered pair) function restricted to a function 𝐹 is a function from 𝐹 into the codomain of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
(𝐹:𝐴𝐵 → (2nd𝐹):𝐹𝐵)

Theoremfo2ndf 7813 The 2nd (second component of an ordered pair) function restricted to a function 𝐹 is a function from 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
(𝐹:𝐴𝐵 → (2nd𝐹):𝐹onto→ran 𝐹)

Theoremf1o2ndf1 7814 The 2nd (second component of an ordered pair) function restricted to a one-to-one function 𝐹 is a one-to-one function from 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
(𝐹:𝐴1-1𝐵 → (2nd𝐹):𝐹1-1-onto→ran 𝐹)

Theoremalgrflem 7815 Lemma for algrf 15915 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐵 ∈ V    &   𝐶 ∈ V       (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹𝐵)

Theoremfrxp 7816* A lexicographical ordering of two well-founded classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) (Proof shortened by Wolf Lammen, 4-Oct-2014.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st𝑥)𝑅(1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦))))}       ((𝑅 Fr 𝐴𝑆 Fr 𝐵) → 𝑇 Fr (𝐴 × 𝐵))

Theoremxporderlem 7817* Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st𝑥)𝑅(1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦))))}       (⟨𝑎, 𝑏𝑇𝑐, 𝑑⟩ ↔ (((𝑎𝐴𝑐𝐴) ∧ (𝑏𝐵𝑑𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐𝑏𝑆𝑑))))

Theorempoxp 7818* A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st𝑥)𝑅(1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦))))}       ((𝑅 Po 𝐴𝑆 Po 𝐵) → 𝑇 Po (𝐴 × 𝐵))

Theoremsoxp 7819* A lexicographical ordering of two strictly ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st𝑥)𝑅(1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦))))}       ((𝑅 Or 𝐴𝑆 Or 𝐵) → 𝑇 Or (𝐴 × 𝐵))

Theoremwexp 7820* A lexicographical ordering of two well-ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st𝑥)𝑅(1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦))))}       ((𝑅 We 𝐴𝑆 We 𝐵) → 𝑇 We (𝐴 × 𝐵))

Theoremfnwelem 7821* Lemma for fnwe 7822. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))}    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝑅 We 𝐵)    &   (𝜑𝑆 We 𝐴)    &   (𝜑 → (𝐹𝑤) ∈ V)    &   𝑄 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((1st𝑢)𝑅(1st𝑣) ∨ ((1st𝑢) = (1st𝑣) ∧ (2nd𝑢)𝑆(2nd𝑣))))}    &   𝐺 = (𝑧𝐴 ↦ ⟨(𝐹𝑧), 𝑧⟩)       (𝜑𝑇 We 𝐴)

Theoremfnwe 7822* A variant on lexicographic order, which sorts first by some function of the base set, and then by a "backup" well-ordering when the function value is equal on both elements. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))}    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝑅 We 𝐵)    &   (𝜑𝑆 We 𝐴)    &   (𝜑 → (𝐹𝑤) ∈ V)       (𝜑𝑇 We 𝐴)

Theoremfnse 7823* Condition for the well-order in fnwe 7822 to be set-like. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))}    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝑅 Se 𝐵)    &   (𝜑 → (𝐹𝑤) ∈ V)       (𝜑𝑇 Se 𝐴)

Theoremfvproj 7824* Value of a function on ordered pairs with values expressed as ordered pairs. Note that 𝐹 and 𝐺 are the projections of 𝐻 to the first and second coordinate respectively. (Contributed by Thierry Arnoux, 30-Dec-2019.)
𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)

Theoremfimaproj 7825* Image of a cartesian product for a function on ordered pairs with values expressed as ordered pairs. Note that 𝐹 and 𝐺 are the projections of 𝐻 to the first and second coordinate respectively. (Contributed by Thierry Arnoux, 30-Dec-2019.)
𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐻 “ (𝑋 × 𝑌)) = ((𝐹𝑋) × (𝐺𝑌)))

2.4.9  The support of functions

In this section, the support of functions is defined and corresponding theorems are provided. Since basic properties (see suppval 7828) are based on the Axiom of Union (usage of dmexg 7608), these definition and theorems cannot be provided earlier. Until April 2019, the support of a function was represented by the expression (𝑅 “ (V ∖ {𝑍})) (see suppimacnv 7837). The theorems which are based on this representation and which are provided in previous sections could be moved into this section to have all related theorems in one section, although they do not depend on the Axiom of Union. This was possible because they are not used before. The current theorems differ from the original ones by requiring that the classes representing the "function" (or its "domain") and the "zero element" are sets. Actually, this does not cause any problem (until now).

Syntaxcsupp 7826 Extend class definition to include the support of functions.
class supp

Definitiondf-supp 7827* Define the support of a function against a "zero" value. According to Wikipedia ("Support (mathematics)", 31-Mar-2019, https://en.wikipedia.org/wiki/Support_(mathematics)) "In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero." and "The notion of support also extends in a natural way to functions taking values in more general sets than R [the real numbers] and to other objects.". The following definition allows for such extensions, being applicable for any sets (which usually are functions) and any element (even not necessarily from the range of the function) regarded as "zero". (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.)
supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})

Theoremsuppval 7828* The value of the operation constructing the support of a function. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.)
((𝑋𝑉𝑍𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})

Theoremsupp0prc 7829 The support of a class is empty if either the class or the "zero" is a proper class. (Contributed by AV, 28-May-2019.)
(¬ (𝑋 ∈ V ∧ 𝑍 ∈ V) → (𝑋 supp 𝑍) = ∅)

Theoremsuppvalbr 7830* The value of the operation constructing the support of a function expressed by binary relations. (Contributed by AV, 7-Apr-2019.)
((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))})

Theoremsupp0 7831 The support of the empty set is the empty set. (Contributed by AV, 12-Apr-2019.)
(𝑍𝑊 → (∅ supp 𝑍) = ∅)

Theoremsuppval1 7832* The value of the operation constructing the support of a function. (Contributed by AV, 6-Apr-2019.)
((Fun 𝑋𝑋𝑉𝑍𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋𝑖) ≠ 𝑍})

Theoremsuppvalfn 7833* The value of the operation constructing the support of a function with a given domain. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 22-Apr-2019.)
((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})

Theoremelsuppfn 7834 An element of the support of a function with a given domain. (Contributed by AV, 27-May-2019.)
((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → (𝑆 ∈ (𝐹 supp 𝑍) ↔ (𝑆𝑋 ∧ (𝐹𝑆) ≠ 𝑍)))

Theoremcnvimadfsn 7835* The support of functions "defined" by inverse images expressed by binary relations. (Contributed by AV, 7-Apr-2019.)
(𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)}

Theoremsuppimacnvss 7836 The support of functions "defined" by inverse images is a subset of the support defined by df-supp 7827. (Contributed by AV, 7-Apr-2019.)
((𝑅𝑉𝑍𝑊) → (𝑅 “ (V ∖ {𝑍})) ⊆ (𝑅 supp 𝑍))

Theoremsuppimacnv 7837 Support sets of functions expressed by inverse images. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 7-Apr-2019.)
((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = (𝑅 “ (V ∖ {𝑍})))

Theoremfrnsuppeq 7838 Two ways of writing the support of a function with known codomain. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)
((𝐼𝑉𝑍𝑊) → (𝐹:𝐼𝑆 → (𝐹 supp 𝑍) = (𝐹 “ (𝑆 ∖ {𝑍}))))

Theoremsuppssdm 7839 The support of a function is a subset of the function's domain. (Contributed by AV, 30-May-2019.)
(𝐹 supp 𝑍) ⊆ dom 𝐹

Theoremsuppsnop 7840 The support of a singleton of an ordered pair. (Contributed by AV, 12-Apr-2019.)
𝐹 = {⟨𝑋, 𝑌⟩}       ((𝑋𝑉𝑌𝑊𝑍𝑈) → (𝐹 supp 𝑍) = if(𝑌 = 𝑍, ∅, {𝑋}))

Theoremsnopsuppss 7841 The support of a singleton containing an ordered pair is a subset of the singleton containing the first element of the ordered pair, i.e. it is empty or the singleton itself. (Contributed by AV, 19-Jul-2019.)
({⟨𝑋, 𝑌⟩} supp 𝑍) ⊆ {𝑋}

Theoremfvn0elsupp 7842 If the function value for a given argument is not empty, the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 2-Jul-2019.) (Revised by AV, 4-Apr-2020.)
(((𝐵𝑉𝑋𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺𝑋) ≠ ∅)) → 𝑋 ∈ (𝐺 supp ∅))

Theoremfvn0elsuppb 7843 The function value for a given argument is not empty iff the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 4-Apr-2020.)
((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → ((𝐺𝑋) ≠ ∅ ↔ 𝑋 ∈ (𝐺 supp ∅)))

Theoremrexsupp 7844* Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by AV, 27-May-2019.)
((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → (∃𝑥 ∈ (𝐹 supp 𝑍)𝜑 ↔ ∃𝑥𝑋 ((𝐹𝑥) ≠ 𝑍𝜑)))

Theoremressuppss 7845 The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.)
((𝐹𝑉𝑍𝑊) → ((𝐹𝐵) supp 𝑍) ⊆ (𝐹 supp 𝑍))

Theoremsuppun 7846 The support of a class/function is a subset of the support of the union of this class/function with another class/function. (Contributed by AV, 4-Jun-2019.)
(𝜑𝐺𝑉)       (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹𝐺) supp 𝑍))

Theoremressuppssdif 7847 The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.)
((𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) ⊆ (((𝐹𝐵) supp 𝑍) ∪ (dom 𝐹𝐵)))

Theoremmptsuppdifd 7848* The support of a function in maps-to notation with a class difference. (Contributed by AV, 28-May-2019.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝑍𝑊)       (𝜑 → (𝐹 supp 𝑍) = {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})})

Theoremmptsuppd 7849* The support of a function in maps-to notation. (Contributed by AV, 10-Apr-2019.) (Revised by AV, 28-May-2019.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝑍𝑊)    &   ((𝜑𝑥𝐴) → 𝐵𝑈)       (𝜑 → (𝐹 supp 𝑍) = {𝑥𝐴𝐵𝑍})

Theoremextmptsuppeq 7850* The support of an extended function is the same as the original. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 30-Jun-2019.)
(𝜑𝐵𝑊)    &   (𝜑𝐴𝐵)    &   ((𝜑𝑛 ∈ (𝐵𝐴)) → 𝑋 = 𝑍)       (𝜑 → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍))

Theoremsuppfnss 7851* The support of a function which has the same zero values (in its domain) as another function is a subset of the support of this other function. (Contributed by AV, 30-Apr-2019.) (Proof shortened by AV, 6-Jun-2022.)
(((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))

Theoremfunsssuppss 7852 The support of a function which is a subset of another function is a subset of the support of this other function. (Contributed by AV, 27-Jul-2019.)
((Fun 𝐺𝐹𝐺𝐺𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))

Theoremfnsuppres 7853 Two ways to express restriction of a support set. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 28-May-2019.)
((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ((𝐹 supp 𝑍) ⊆ 𝐴 ↔ (𝐹𝐵) = (𝐵 × {𝑍})))

Theoremfnsuppeq0 7854 The support of a function is empty iff it is identically zero. (Contributed by Stefan O'Rear, 22-Mar-2015.) (Revised by AV, 28-May-2019.)
((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ 𝐹 = (𝐴 × {𝑍})))

Theoremfczsupp0 7855 The support of a constant function with value zero is empty. (Contributed by AV, 30-Jun-2019.)
((𝐵 × {𝑍}) supp 𝑍) = ∅

Theoremsuppss 7856* Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.)
(𝜑𝐹:𝐴𝐵)    &   ((𝜑𝑘 ∈ (𝐴𝑊)) → (𝐹𝑘) = 𝑍)       (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)

Theoremsuppssr 7857 A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)    &   (𝜑𝐴𝑉)    &   (𝜑𝑍𝑈)       ((𝜑𝑋 ∈ (𝐴𝑊)) → (𝐹𝑋) = 𝑍)

Theoremsuppssov1 7858* Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
(𝜑 → ((𝑥𝐷𝐴) supp 𝑌) ⊆ 𝐿)    &   ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)    &   ((𝜑𝑥𝐷) → 𝐴𝑉)    &   ((𝜑𝑥𝐷) → 𝐵𝑅)    &   (𝜑𝑌𝑊)       (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)

Theoremsuppssof1 7859* Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
(𝜑 → (𝐴 supp 𝑌) ⊆ 𝐿)    &   ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)    &   (𝜑𝐴:𝐷𝑉)    &   (𝜑𝐵:𝐷𝑅)    &   (𝜑𝐷𝑊)    &   (𝜑𝑌𝑈)       (𝜑 → ((𝐴f 𝑂𝐵) supp 𝑍) ⊆ 𝐿)

Theoremsuppss2 7860* Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 22-Mar-2015.) (Revised by AV, 28-May-2019.)
((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍)    &   (𝜑𝐴𝑉)       (𝜑 → ((𝑘𝐴𝐵) supp 𝑍) ⊆ 𝑊)

Theoremsuppsssn 7861* Show that the support of a function is a subset of a singleton. (Contributed by AV, 21-Jul-2019.)
((𝜑𝑘𝐴𝑘𝑊) → 𝐵 = 𝑍)    &   (𝜑𝐴𝑉)       (𝜑 → ((𝑘𝐴𝐵) supp 𝑍) ⊆ {𝑊})

Theoremsuppssfv 7862* Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
(𝜑 → ((𝑥𝐷𝐴) supp 𝑌) ⊆ 𝐿)    &   (𝜑 → (𝐹𝑌) = 𝑍)    &   ((𝜑𝑥𝐷) → 𝐴𝑉)    &   (𝜑𝑌𝑈)       (𝜑 → ((𝑥𝐷 ↦ (𝐹𝐴)) supp 𝑍) ⊆ 𝐿)

Theoremsuppofssd 7863 Condition for the support of a function operation to be a subset of the union of the supports of the left and right function terms. (Contributed by Steven Nguyen, 28-Aug-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑 → (𝑍𝑋𝑍) = 𝑍)       (𝜑 → ((𝐹f 𝑋𝐺) supp 𝑍) ⊆ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))

Theoremsuppofss1d 7864* Condition for the support of a function operation to be a subset of the support of the left function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)
(𝜑𝐴𝑉)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   ((𝜑𝑥𝐵) → (𝑍𝑋𝑥) = 𝑍)       (𝜑 → ((𝐹f 𝑋𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑍))

Theoremsuppofss2d 7865* Condition for the support of a function operation to be a subset of the support of the right function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)
(𝜑𝐴𝑉)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   ((𝜑𝑥𝐵) → (𝑥𝑋𝑍) = 𝑍)       (𝜑 → ((𝐹f 𝑋𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑍))

Theoremsuppco 7866 The support of the composition of two functions is the inverse image by the inner function of the support of the outer function. (Contributed by AV, 30-May-2019.) Extract this statement from the proof of supp0cosupp0 7868. (Revised by SN, 15-Sep-2023.)
((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))

Theoremsuppcofnd 7867* The support of the composition of two functions. (Contributed by SN, 15-Sep-2023.)
(𝜑𝑍𝑈)    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐴𝑉)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐵𝑊)       (𝜑 → ((𝐹𝐺) supp 𝑍) = {𝑥𝐵 ∣ ((𝐺𝑥) ∈ 𝐴 ∧ (𝐹‘(𝐺𝑥)) ≠ 𝑍)})

Theoremsupp0cosupp0 7868 The support of the composition of two functions is empty if the support of the outer function is empty. (Contributed by AV, 30-May-2019.)
((𝐹𝑉𝐺𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅))

Theoremsupp0cosupp0OLD 7869 Obsolete version of supp0cosupp0 7868 as of 15-Sep-2023. The support of the composition of two functions is empty if the support of the outer function is empty. (Contributed by AV, 30-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐹𝑉𝐺𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅))

Theoremimacosupp 7870 The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.)
((𝐹𝑉𝐺𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍)))

TheoremimacosuppOLD 7871 Obsolete version of imacosupp 7870 as of 15-Sep-2023. The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐹𝑉𝐺𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍)))

2.4.10  Special maps-to operations

The following theorems are about maps-to operations (see df-mpo 7154) where the domain of the second argument depends on the domain of the first argument, especially when the first argument is a pair and the base set of the second argument is the first component of the first argument, in short "x-maps-to operations". For labels, the abbreviations "mpox" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpox 7240, ovmpox 7296 and fmpox 7760). If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpoxop", and the maps-to operations are called "x-op maps-to operations" for short.

Theoremopeliunxp2f 7872* Membership in a union of Cartesian products, using bound-variable hypothesis for 𝐸 instead of distinct variable conditions as in opeliunxp2 5696. (Contributed by AV, 25-Oct-2020.)
𝑥𝐸    &   (𝑥 = 𝐶𝐵 = 𝐸)       (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))

Theoremmpoxeldm 7873* If there is an element of the value of an operation given by a maps-to rule, then the first argument is an element of the first component of the domain and the second argument is an element of the second component of the domain depending on the first argument. (Contributed by AV, 25-Oct-2020.)
𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)       (𝑁 ∈ (𝑋𝐹𝑌) → (𝑋𝐶𝑌𝑋 / 𝑥𝐷))

Theoremmpoxneldm 7874* If the first argument of an operation given by a maps-to rule is not an element of the first component of the domain or the second argument is not an element of the second component of the domain depending on the first argument, then the value of the operation is the empty set. (Contributed by AV, 25-Oct-2020.)
𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)       ((𝑋𝐶𝑌𝑋 / 𝑥𝐷) → (𝑋𝐹𝑌) = ∅)

Theoremmpoxopn0yelv 7875* If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)       ((𝑉𝑋𝑊𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) → 𝐾𝑉))

Theoremmpoxopynvov0g 7876* If the second argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument is not element of the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)       (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = ∅)

Theoremmpoxopxnop0 7877* If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is not an ordered pair, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)       𝑉 ∈ (V × V) → (𝑉𝐹𝐾) = ∅)

Theoremmpoxopx0ov0 7878* If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is the empty set, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)       (∅𝐹𝐾) = ∅

Theoremmpoxopxprcov0 7879* If the components of the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, are not sets, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)       (¬ (𝑉 ∈ V ∧ 𝑊 ∈ V) → (⟨𝑉, 𝑊𝐹𝐾) = ∅)

Theoremmpoxopynvov0 7880* If the second argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument is not element of the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)       (𝐾𝑉 → (⟨𝑉, 𝑊𝐹𝐾) = ∅)

Theoremmpoxopoveq 7881* Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})       (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})

Theoremmpoxopovel 7882* Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.)
𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})       ((𝑉𝑋𝑊𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) ↔ (𝐾𝑉𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))

Theoremmpoxopoveqd 7883* Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, deduction version. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})    &   (𝜓 → (𝑉𝑋𝑊𝑌))    &   ((𝜓 ∧ ¬ 𝐾𝑉) → {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} = ∅)       (𝜓 → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})

Theorembrovex 7884* A binary relation of the value of an operation given by the maps-to notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.)
𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶)    &   ((𝑉 ∈ V ∧ 𝐸 ∈ V) → Rel (𝑉𝑂𝐸))       (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))

Theorembrovmpoex 7885* A binary relation of the value of an operation given by the maps-to notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.)
𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {⟨𝑧, 𝑤⟩ ∣ 𝜑})       (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))

Theoremsprmpod 7886* The extension of a binary relation which is the value of an operation given in maps-to notation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 20-Jun-2019.)
𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑣𝑅𝑒)𝑦𝜒)})    &   ((𝜑𝑣 = 𝑉𝑒 = 𝐸) → (𝜒𝜓))    &   (𝜑 → (𝑉 ∈ V ∧ 𝐸 ∈ V))    &   (𝜑 → ∀𝑥𝑦(𝑥(𝑉𝑅𝐸)𝑦𝜃))    &   (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V)       (𝜑 → (𝑉𝑀𝐸) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)})

2.4.11  Function transposition

Syntaxctpos 7887 The transposition of a function.
class tpos 𝐹

Definitiondf-tpos 7888* Define the transposition of a function, which is a function 𝐺 = tpos 𝐹 satisfying 𝐺(𝑥, 𝑦) = 𝐹(𝑦, 𝑥). (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))

Theoremtposss 7889 Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)

Theoremtposeq 7890 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)

Theoremtposeqd 7891 Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐹 = 𝐺)       (𝜑 → tpos 𝐹 = tpos 𝐺)

Theoremtposssxp 7892 The transposition is a subset of a Cartesian product. (Contributed by Mario Carneiro, 12-Jan-2017.)
tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)

Theoremreltpos 7893 The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
Rel tpos 𝐹

Theorembrtpos2 7894 Value of the transposition at a pair 𝐴, 𝐵. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐵𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵)))

Theorembrtpos0 7895 The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). This allows us to eliminate sethood hypotheses on 𝐴, 𝐵 in brtpos 7897. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐴𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴))

Theoremreldmtpos 7896 Necessary and sufficient condition for dom tpos 𝐹 to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)

Theorembrtpos 7897 The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐶𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))

Theoremottpos 7898 The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.)
(𝐶𝑉 → (⟨𝐴, 𝐵, 𝐶⟩ ∈ tpos 𝐹 ↔ ⟨𝐵, 𝐴, 𝐶⟩ ∈ 𝐹))

Theoremrelbrtpos 7899 The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 3-Nov-2015.)
(Rel 𝐹 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))

Theoremdmtpos 7900 The domain of tpos 𝐹 when dom 𝐹 is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)

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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 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45259
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