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Theorem List for Metamath Proof Explorer - 7301-7400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembropfvvvvlem 7301* Lemma for bropfvvvv 7302. (Contributed by AV, 31-Dec-2020.) (Revised by AV, 16-Jan-2021.)
𝑂 = (𝑎𝑈 ↦ (𝑏𝑉, 𝑐𝑊 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜑}))    &   ((𝐴𝑈𝐵𝑆𝐶𝑇) → (𝐵(𝑂𝐴)𝐶) = {⟨𝑑, 𝑒⟩ ∣ 𝜃})       ((⟨𝐵, 𝐶⟩ ∈ (𝑆 × 𝑇) ∧ 𝐷(𝐵(𝑂𝐴)𝐶)𝐸) → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V)))

Theorembropfvvvv 7302* If a binary relation holds for the result of an operation which is a function value, the involved classes are sets. (Contributed by AV, 31-Dec-2020.) (Revised by AV, 16-Jan-2021.)
𝑂 = (𝑎𝑈 ↦ (𝑏𝑉, 𝑐𝑊 ↦ {⟨𝑑, 𝑒⟩ ∣ 𝜑}))    &   ((𝐴𝑈𝐵𝑆𝐶𝑇) → (𝐵(𝑂𝐴)𝐶) = {⟨𝑑, 𝑒⟩ ∣ 𝜃})    &   (𝑎 = 𝐴𝑉 = 𝑆)    &   (𝑎 = 𝐴𝑊 = 𝑇)    &   (𝑎 = 𝐴 → (𝜑𝜓))       ((𝑆𝑋𝑇𝑌) → (𝐷(𝐵(𝑂𝐴)𝐶)𝐸 → (𝐴𝑈 ∧ (𝐵𝑆𝐶𝑇) ∧ (𝐷 ∈ V ∧ 𝐸 ∈ V))))

Theoremovmptss 7303* If all the values of the mapping are subsets of a class 𝑋, then so is any evaluation of the mapping. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (∀𝑥𝐴𝑦𝐵 𝐶𝑋 → (𝐸𝐹𝐺) ⊆ 𝑋)

Theoremrelmpt2opab 7304* Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ {⟨𝑧, 𝑤⟩ ∣ 𝜑})       Rel (𝐶𝐹𝐷)

Theoremfmpt2co 7305* Composition of two functions. Variation of fmptco 6436 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝑅𝐶)    &   (𝜑𝐹 = (𝑥𝐴, 𝑦𝐵𝑅))    &   (𝜑𝐺 = (𝑧𝐶𝑆))    &   (𝑧 = 𝑅𝑆 = 𝑇)       (𝜑 → (𝐺𝐹) = (𝑥𝐴, 𝑦𝐵𝑇))

Theoremoprabco 7306* Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
((𝑥𝐴𝑦𝐵) → 𝐶𝐷)    &   𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   𝐺 = (𝑥𝐴, 𝑦𝐵 ↦ (𝐻𝐶))       (𝐻 Fn 𝐷𝐺 = (𝐻𝐹))

Theoremoprab2co 7307* Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)
((𝑥𝐴𝑦𝐵) → 𝐶𝑅)    &   ((𝑥𝐴𝑦𝐵) → 𝐷𝑆)    &   𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨𝐶, 𝐷⟩)    &   𝐺 = (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑀𝐷))       (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀𝐹))

Theoremdf1st2 7308* An alternate possible definition of the 1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))

Theoremdf2nd2 7309* An alternate possible definition of the 2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))

Theorem1stconst 7310 The mapping of a restriction of the 1st function to a constant function. (Contributed by NM, 14-Dec-2008.)
(𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴)

Theorem2ndconst 7311 The mapping of a restriction of the 2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)
(𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵)

Theoremdfmpt2 7312* Alternate definition for the "maps to" notation df-mpt2 6695 (although it requires that 𝐶 be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐶 ∈ V       (𝑥𝐴, 𝑦𝐵𝐶) = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}

Theoremmpt2sn 7313* An operation (in maps-to notation) on two singletons. (Contributed by AV, 4-Aug-2019.)
𝐹 = (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶)    &   (𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑦 = 𝐵𝐷 = 𝐸)       ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐹 = {⟨⟨𝐴, 𝐵⟩, 𝐸⟩})

Theoremcurry1 7314* Composition with (2nd ↾ ({𝐶} × V)) turns any binary operation 𝐹 with a constant first operand into a function 𝐺 of the second operand only. This transformation is called "currying." (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Dec-2014.)
𝐺 = (𝐹(2nd ↾ ({𝐶} × V)))       ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))

Theoremcurry1val 7315 The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐺 = (𝐹(2nd ↾ ({𝐶} × V)))       ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → (𝐺𝐷) = (𝐶𝐹𝐷))

Theoremcurry1f 7316 Functionality of a curried function with a constant first argument. (Contributed by NM, 29-Mar-2008.)
𝐺 = (𝐹(2nd ↾ ({𝐶} × V)))       ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐴) → 𝐺:𝐵𝐷)

Theoremcurry2 7317* Composition with (1st ↾ (V × {𝐶})) turns any binary operation 𝐹 with a constant second operand into a function 𝐺 of the first operand only. This transformation is called "currying." (If this becomes frequently used, we can introduce a new notation for the hypothesis.) (Contributed by NM, 16-Dec-2008.)
𝐺 = (𝐹(1st ↾ (V × {𝐶})))       ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))

Theoremcurry2f 7318 Functionality of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)
𝐺 = (𝐹(1st ↾ (V × {𝐶})))       ((𝐹:(𝐴 × 𝐵)⟶𝐷𝐶𝐵) → 𝐺:𝐴𝐷)

Theoremcurry2val 7319 The value of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)
𝐺 = (𝐹(1st ↾ (V × {𝐶})))       ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝐺𝐷) = (𝐷𝐹𝐶))

Theoremcnvf1olem 7320 Lemma for cnvf1o 7321. (Contributed by Mario Carneiro, 27-Apr-2014.)
((Rel 𝐴 ∧ (𝐵𝐴𝐶 = {𝐵})) → (𝐶𝐴𝐵 = {𝐶}))

Theoremcnvf1o 7321* 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 7322 Lemma for fpar 7326. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
((1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)

Theoremfparlem2 7323 Lemma for fpar 7326. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
((2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦})

Theoremfparlem3 7324* Lemma for fpar 7326. (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 7325* Lemma for fpar 7326. (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 7326* Merge two functions in parallel. Use as the second argument of a composition with a (2-place) operation to build compound operations such as 𝑧 = ((√‘𝑥) + (abs‘𝑦)). (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 7327 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 7326 in order to build compound functions such as 𝑦 = ((√‘𝑥) + (abs‘𝑥)). (Contributed by NM, 17-Sep-2007.)
(1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)

Theoremf2ndf 7328 The 2nd (second member of an ordered pair) function restricted to a function 𝐹 is a function of 𝐹 into the codomain of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
(𝐹:𝐴𝐵 → (2nd𝐹):𝐹𝐵)

Theoremfo2ndf 7329 The 2nd (second member of an ordered pair) function restricted to a function 𝐹 is a function of 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
(𝐹:𝐴𝐵 → (2nd𝐹):𝐹onto→ran 𝐹)

Theoremf1o2ndf1 7330 The 2nd (second member of an ordered pair) function restricted to a one-to-one function 𝐹 is a one-to-one function of 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 7331 Lemma for algrf 15333 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 7332* 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 7333* 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 7334* 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 7335* 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 7336* 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 7337* Lemma for fnwe 7338. (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 7338* 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 7339* Condition for the well-order in fnwe 7338 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)

2.4.8  The support of functions

In this section, the support of functions is defined and corresponding theorems are provided. Since basic properties (see suppval 7342) are based on the Axiom of Union (usage of dmexg 7139), 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 7351). 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 7340 Extend class definition to include the support of functions.
class supp

Definitiondf-supp 7341* 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 7342* 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 7343 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 7344* 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 7345 The support of the empty set is the empty set. (Contributed by AV, 12-Apr-2019.)
(𝑍𝑊 → (∅ supp 𝑍) = ∅)

Theoremsuppval1 7346* 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 7347* 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 7348 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 7349* 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 7350 The support of functions "defined" by inverse images is a subset of the support defined by df-supp 7341. (Contributed by AV, 7-Apr-2019.)
((𝑅𝑉𝑍𝑊) → (𝑅 “ (V ∖ {𝑍})) ⊆ (𝑅 supp 𝑍))

Theoremsuppimacnv 7351 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 7352 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 7353 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 7354 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 7355 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 7356 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 7357 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 7358* 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 7359 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 7360 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 7361 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 7362* 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 7363* 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 7364* 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 7365* 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.)
(((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))

Theoremfunsssuppss 7366 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 7367 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 7368 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 7369 The support of a constant function with value zero is empty. (Contributed by AV, 30-Jun-2019.)
((𝐵 × {𝑍}) supp 𝑍) = ∅

Theoremsuppss 7370* 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 7371 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 7372* 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 7373* 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 7374* 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 7375* 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 7376* 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)

Theoremsuppofss1d 7377* 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 7378* 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))

Theoremsupp0cosupp0 7379 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) = ∅))

Theoremimacosupp 7380 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)))

2.4.9  Special "Maps to" operations

The following theorems are about maps-to operations (see df-mpt2 6695) 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 "mpt2x" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpt2x 6775, ovmpt2x 6831 and fmpt2x 7281). However, there is a proposal by Norman Megill to use the abbreviation "mpo" or "mpto" instead of "mpt2" (see beginning of set.mm). If this proposal will be realized, the labels in the following should also be adapted. If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpt2xop", and the maps-to operations are called "x-op maps-to operations" for short.

Theoremopeliunxp2f 7381* Membership in a union of Cartesian products, using bound-variable hypothesis for 𝐸 instead of distinct variable conditions as in opeliunxp2 5293. (Contributed by AV, 25-Oct-2020.)
𝑥𝐸    &   (𝑥 = 𝐶𝐵 = 𝐸)       (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))

Theoremmpt2xeldm 7382* 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))

Theoremmpt2xneldm 7383* 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) = ∅)

Theoremmpt2xopn0yelv 7384* 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))

Theoremmpt2xopynvov0g 7385* 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) = ∅)

Theoremmpt2xopxnop0 7386* 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) = ∅)

Theoremmpt2xopx0ov0 7387* 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) = ∅

Theoremmpt2xopxprcov0 7388* 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) = ∅)

Theoremmpt2xopynvov0 7389* 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) = ∅)

Theoremmpt2xopoveq 7390* 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})

Theoremmpt2xopovel 7391* 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)))

Theoremmpt2xopoveqd 7392* 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 7393* 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)))

Theorembrovmpt2ex 7394* 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)))

Theoremsprmpt2d 7395* 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)})

2.4.10  Function transposition

Syntaxctpos 7396 The transposition of a function.
class tpos 𝐹

Definitiondf-tpos 7397* 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 7398 Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)

Theoremtposeq 7399 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)

Theoremtposeqd 7400 Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐹 = 𝐺)       (𝜑 → tpos 𝐹 = tpos 𝐺)

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