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Theorem List for Metamath Proof Explorer - 30101-30200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremacunirnmpt2f 30101* 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 30102* 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 30103* 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))

Theoremofoprabco 30104* 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 30105* 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 30106* Express the preimage of a function operation as a union of preimages. This version of ofpreima 30105 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)})))

Theoremfuncnvmpt 30107* 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 30108* 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 30109* 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)))

Theorempreimane 30110 Different elements have different preimages. (Contributed by Thierry Arnoux, 7-May-2023.)
(𝜑 → Fun 𝐹)    &   (𝜑𝑋𝑌)    &   (𝜑𝑋 ∈ ran 𝐹)    &   (𝜑𝑌 ∈ ran 𝐹)       (𝜑 → (𝐹 “ {𝑋}) ≠ (𝐹 “ {𝑌}))

Theoremfnpreimac 30111* Choose a set 𝑥 containing a preimage of each element of a given set 𝐵. (Contributed by Thierry Arnoux, 7-May-2023.)
((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → ∃𝑥 ∈ 𝒫 𝐴(𝑥𝐵 ∧ (𝐹𝑥) = 𝐵))

Theoremfgreu 30112* 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 30113* 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))

Theoremrnmposs 30114* 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 30115* Deduce subset relation of mapping-to function graphs from a subset relation of domains. Alternative proof of mptss 5796. (Contributed by Thierry Arnoux, 30-May-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))

Theorempartfun 30116 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 30117* Alternative definition of the converse of a relation. (Contributed by Thierry Arnoux, 31-Mar-2018.)
(ran 𝑅𝐴𝑅 = 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥})))

Theoremfnimatp 30118 The image of a triplet under a function. (Contributed by Thierry Arnoux, 19-Sep-2023.)
(𝜑𝐹 Fn 𝐷)    &   (𝜑𝐴𝐷)    &   (𝜑𝐵𝐷)    &   (𝜑𝐶𝐷)       (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹𝐴), (𝐹𝐵), (𝐹𝐶)})

Theoremfnunres2 30119 Restriction of a disjoint union to the domain of the second function. (Contributed by Thierry Arnoux, 12-Oct-2023.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)

Theoremmpomptxf 30120* 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)

Theoremsuppovss 30121* A bound for the support of an operation. (Contributed by Thierry Arnoux, 19-Jul-2023.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   𝐺 = (𝑥𝐴 ↦ (𝑦𝐵𝐶))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝑍𝐷)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝐷)       (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐺 supp (𝐵 × {𝑍})) × 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))

20.3.4.4  Explicit Functions with one or two points as a domain

Theorembrsnop 30122 Binary relation for an ordered pair singleton. (Contributed by Thierry Arnoux, 23-Sep-2023.)
((𝐴𝑉𝐵𝑊) → (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ (𝑋 = 𝐴𝑌 = 𝐵)))

Theoremcosnopne 30123 Composition of two ordered pair singletons with non-matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)    &   (𝜑𝐴𝐷)       (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐷⟩}) = ∅)

Theoremcosnop 30124 Composition of two ordered pair singletons with matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)       (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐴⟩}) = {⟨𝐶, 𝐵⟩})

Theoremcnvprop 30125 Converse of a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑉𝐷𝑊)) → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {⟨𝐵, 𝐴⟩, ⟨𝐷, 𝐶⟩})

Theorembrprop 30126 Binary relation for a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)       (𝜑 → (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}𝑌 ↔ ((𝑋 = 𝐴𝑌 = 𝐵) ∨ (𝑋 = 𝐶𝑌 = 𝐷))))

Theoremmptprop 30127* Rewrite pairs of ordered pairs as mapping to functions. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)    &   (𝜑𝐴𝐶)       (𝜑 → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))

Theoremcoprprop 30128 Composition of two pairs of ordered pairs with matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)    &   (𝜑𝐴𝐶)    &   (𝜑𝐸𝑋)    &   (𝜑𝐹𝑋)    &   (𝜑𝐸𝐹)       (𝜑 → ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∘ {⟨𝐸, 𝐴⟩, ⟨𝐹, 𝐶⟩}) = {⟨𝐸, 𝐵⟩, ⟨𝐹, 𝐷⟩})

Theoremgtiso 30129 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 30130* 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)))

20.3.4.6  Disjointness (additional proof requiring functions)

Theoremdisjdsct 30131* 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 6298) (Contributed by Thierry Arnoux, 28-Feb-2017.)
𝑥𝜑    &   𝑥𝐴    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (𝑉 ∖ {∅}))    &   (𝜑Disj 𝑥𝐴 𝐵)       (𝜑 → Fun (𝑥𝐴𝐵))

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

Theoremdf1stres 30132* 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 30133* 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 30134 The preimage of a singleton. (Contributed by Thierry Arnoux, 27-Apr-2020.)
((Rel 𝐴𝑋𝑉) → ((1st𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋})))

Theorem1stpreima 30135 The preimage by 1st is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
(𝐴𝐵 → ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶))

Theorem2ndpreima 30136 The preimage by 2nd is an 'horizontal band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
(𝐴𝐶 → ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))

Theoremcurry2ima 30137* 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 30138* 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 30139* 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))

20.3.4.9  Finite Sets

Theoremimafi2 30140 The image by a finite set is finite. See also imafi 8668. (Contributed by Thierry Arnoux, 25-Apr-2020.)
(𝐴 ∈ Fin → (𝐴𝐵) ∈ Fin)

Theoremunifi3 30141 If a union is finite, then all its elements are finite. See unifi 8664. (Contributed by Thierry Arnoux, 27-Aug-2017.)
( 𝐴 ∈ Fin → 𝐴 ⊆ Fin)

20.3.4.10  Countable Sets

Theoremsnct 30142 A singleton is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.)
(𝐴𝑉 → {𝐴} ≼ ω)

Theoremprct 30143 An unordered pair is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ≼ ω)

Theoremmpocti 30144* 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 30145* An image set of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≼ ω)

Theoremmptctf 30146 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 30147* 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 30148* 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)))

TheoremcnvoprabOLD 30149* The converse of a class abstraction of nested ordered pairs. Obsolete version of cnvoprab 7619 as of 16-Oct-2022, which has nonfreeness hypotheses instead of disjoint variable conditions. (Contributed by Thierry Arnoux, 17-Aug-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   𝑦𝜓    &   (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓𝜑))    &   (𝜓𝑎 ∈ (V × V))       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}

Theoremf1od2 30150* Sufficient condition for a binary function expressed in maps-to notation to be bijective. (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 30151* 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 30152* Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also mapfien 8722. (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 30153* 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))

Theoremfsuppcurry1 30154* Finite support of a curried function with a constant first argument. (Contributed by Thierry Arnoux, 7-Jul-2023.)
𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥))    &   (𝜑𝑍𝑈)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐹 Fn (𝐴 × 𝐵))    &   (𝜑𝐶𝐴)    &   (𝜑𝐹 finSupp 𝑍)       (𝜑𝐺 finSupp 𝑍)

Theoremfsuppcurry2 30155* Finite support of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 7-Jul-2023.)
𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶))    &   (𝜑𝑍𝑈)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐹 Fn (𝐴 × 𝐵))    &   (𝜑𝐶𝐵)    &   (𝜑𝐹 finSupp 𝑍)       (𝜑𝐺 finSupp 𝑍)

Theoremoffinsupp1 30156* Finite support for a function operation. (Contributed by Thierry Arnoux, 8-Jul-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝑌𝑈)    &   (𝜑𝑍𝑊)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑇)    &   (𝜑𝐹 finSupp 𝑌)    &   ((𝜑𝑥𝑇) → (𝑌𝑅𝑥) = 𝑍)       (𝜑 → (𝐹𝑓 𝑅𝐺) finSupp 𝑍)

Theoremffs2 30157 Rewrite a function's support based with its range rather than the universal class. See also frnsuppeq 7698. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝐶 = (𝐵 ∖ {𝑍})       ((𝐴𝑉𝑍𝑊𝐹:𝐴𝐵) → (𝐹 supp 𝑍) = (𝐹𝐶))

Theoremffsrn 30158 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 30159* 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 30160* 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 30161* Lemma for fpwrelmapffs 30163. 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 30162* 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 9721 and marypha2lem1 8750. (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 30163* 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

Theoremcreq0 30164 The real representation of complex numbers is zero iff both its terms are zero. Cf. crne0 11484. (Contributed by Thierry Arnoux, 20-Aug-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 = 0 ∧ 𝐵 = 0) ↔ (𝐴 + (i · 𝐵)) = 0))

Theorem1nei 30165 The imaginary unit i is not one. (Contributed by Thierry Arnoux, 20-Aug-2023.)
1 ≠ i

Theorem1neg1t1neg1 30166 An integer unit times itself. (Contributed by Thierry Arnoux, 23-Aug-2020.)
(𝑁 ∈ {-1, 1} → (𝑁 · 𝑁) = 1)

Theoremnnmulge 30167 Multiplying by a positive integer 𝑀 yields greater than or equal nonnegative integers. (Contributed by Thierry Arnoux, 13-Dec-2021.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (𝑀 · 𝑁))

20.3.5.2  Ordering on reals - misc additions

Theoremlt2addrd 30168* 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

Theoremxrlelttric 30169 Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵𝐵 < 𝐴))

Theoremxaddeq0 30170 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))

Theoremxrinfm 30171 The extended real numbers are unbounded below. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Revised by AV, 28-Sep-2020.)
inf(ℝ*, ℝ*, < ) = -∞

Theoremle2halvesd 30172 A sum is less than the whole if each term is less than half. (Contributed by Thierry Arnoux, 29-Nov-2017.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 ≤ (𝐶 / 2))    &   (𝜑𝐵 ≤ (𝐶 / 2))       (𝜑 → (𝐴 + 𝐵) ≤ 𝐶)

Theoremxraddge02 30173 A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 28-Dec-2016.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (0 ≤ 𝐵𝐴 ≤ (𝐴 +𝑒 𝐵)))

Theoremxrge0addge 30174 A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 19-Jul-2020.)
((𝐴 ∈ ℝ*𝐵 ∈ (0[,]+∞)) → 𝐴 ≤ (𝐴 +𝑒 𝐵))

Theoremxlt2addrd 30175* 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*)    &   (𝜑𝐵 ≠ -∞)    &   (𝜑𝐶 ≠ -∞)    &   (𝜑𝐴 < (𝐵 +𝑒 𝐶))       (𝜑 → ∃𝑏 ∈ ℝ*𝑐 ∈ ℝ* (𝐴 = (𝑏 +𝑒 𝑐) ∧ 𝑏 < 𝐵𝑐 < 𝐶))

Theoremxrsupssd 30176 Inequality deduction for supremum of an extended real subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)
(𝜑𝐵𝐶)    &   (𝜑𝐶 ⊆ ℝ*)       (𝜑 → sup(𝐵, ℝ*, < ) ≤ sup(𝐶, ℝ*, < ))

Theoremxrge0infss 30177* Any subset of nonnegative extended reals has an infimum. (Contributed by Thierry Arnoux, 16-Sep-2019.) (Revised by AV, 4-Oct-2020.)
(𝐴 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))

Theoremxrge0infssd 30178 Inequality deduction for infimum of a nonnegative extended real subset. (Contributed by Thierry Arnoux, 16-Sep-2019.) (Revised by AV, 4-Oct-2020.)
(𝜑𝐶𝐵)    &   (𝜑𝐵 ⊆ (0[,]+∞))       (𝜑 → inf(𝐵, (0[,]+∞), < ) ≤ inf(𝐶, (0[,]+∞), < ))

Theoremxrge0addcld 30179 Nonnegative extended reals are closed under addition. (Contributed by Thierry Arnoux, 16-Sep-2019.)
(𝜑𝐴 ∈ (0[,]+∞))    &   (𝜑𝐵 ∈ (0[,]+∞))       (𝜑 → (𝐴 +𝑒 𝐵) ∈ (0[,]+∞))

Theoremxrge0subcld 30180 Condition for closure of nonnegative extended reals under subtraction. (Contributed by Thierry Arnoux, 27-May-2020.)
(𝜑𝐴 ∈ (0[,]+∞))    &   (𝜑𝐵 ∈ (0[,]+∞))    &   (𝜑𝐵𝐴)       (𝜑 → (𝐴 +𝑒 -𝑒𝐵) ∈ (0[,]+∞))

Theoreminfxrge0lb 30181 A member of a set of nonnegative extended reals is greater than or equal to the set's infimum. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.)
(𝜑𝐴 ⊆ (0[,]+∞))    &   (𝜑𝐵𝐴)       (𝜑 → inf(𝐴, (0[,]+∞), < ) ≤ 𝐵)

Theoreminfxrge0glb 30182* The infimum of a set of nonnegative extended reals is the greatest lower bound. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.)
(𝜑𝐴 ⊆ (0[,]+∞))    &   (𝜑𝐵 ∈ (0[,]+∞))       (𝜑 → (inf(𝐴, (0[,]+∞), < ) < 𝐵 ↔ ∃𝑥𝐴 𝑥 < 𝐵))

Theoreminfxrge0gelb 30183* The infimum of a set of nonnegative extended reals is greater than or equal to a lower bound. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.)
(𝜑𝐴 ⊆ (0[,]+∞))    &   (𝜑𝐵 ∈ (0[,]+∞))       (𝜑 → (𝐵 ≤ inf(𝐴, (0[,]+∞), < ) ↔ ∀𝑥𝐴 𝐵𝑥))

Theoremdfrp2 30184 Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.)
+ = (0(,)+∞)

Theoremxrofsup 30185 The supremum is preserved by extended addition set operation. (Provided minus infinity is not involved as it does not behave well with addition.) (Contributed by Thierry Arnoux, 20-Mar-2017.)
(𝜑𝑋 ⊆ ℝ*)    &   (𝜑𝑌 ⊆ ℝ*)    &   (𝜑 → sup(𝑋, ℝ*, < ) ≠ -∞)    &   (𝜑 → sup(𝑌, ℝ*, < ) ≠ -∞)    &   (𝜑𝑍 = ( +𝑒 “ (𝑋 × 𝑌)))       (𝜑 → sup(𝑍, ℝ*, < ) = (sup(𝑋, ℝ*, < ) +𝑒 sup(𝑌, ℝ*, < )))

Theoremsupxrnemnf 30186 The supremum of a nonempty set of extended reals which does not contain minus infinity is not minus infinity. (Contributed by Thierry Arnoux, 21-Mar-2017.)
((𝐴 ⊆ ℝ*𝐴 ≠ ∅ ∧ ¬ -∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) ≠ -∞)

20.3.5.4  Extended nonnegative integers - misc additions

Theoremxnn0gt0 30187 Non-zero extended nonnegative integers are strictly greater than zero. (Contributed by Thierry Arnoux, 30-Jul-2023.)
((𝑁 ∈ ℕ0*𝑁 ≠ 0) → 0 < 𝑁)

Theoremxnn01gt 30188 An extended nonnegative integer is neither 0 nor 1 if and only if it is greater than 1. (Contributed by Thierry Arnoux, 21-Nov-2023.)
(𝑁 ∈ ℕ0* → (¬ 𝑁 ∈ {0, 1} ↔ 1 < 𝑁))

Theoremnn0xmulclb 30189 Finite multiplication in the extended nonnegative integers. (Contributed by Thierry Arnoux, 30-Jul-2023.)
(((𝐴 ∈ ℕ0*𝐵 ∈ ℕ0*) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → ((𝐴 ·e 𝐵) ∈ ℕ0 ↔ (𝐴 ∈ ℕ0𝐵 ∈ ℕ0)))

20.3.5.5  Real number intervals - misc additions

Theoremjoiniooico 30190 Disjoint joining an open interval with a closed-below, open-above interval to form a closed-below, open-above interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵𝐵𝐶)) → (((𝐴(,)𝐵) ∩ (𝐵[,)𝐶)) = ∅ ∧ ((𝐴(,)𝐵) ∪ (𝐵[,)𝐶)) = (𝐴(,)𝐶)))

Theoremubico 30191 A right-open interval does not contain its right endpoint. (Contributed by Thierry Arnoux, 5-Apr-2017.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → ¬ 𝐵 ∈ (𝐴[,)𝐵))

Theoremxeqlelt 30192 Equality in terms of 'less than or equal to', 'less than'. (Contributed by Thierry Arnoux, 5-Jul-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 < 𝐵)))

Theoremeliccelico 30193 Relate elementhood to a closed interval with elementhood to the same closed-below, open-above interval or to its upper bound. (Contributed by Thierry Arnoux, 3-Jul-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ (𝐴[,)𝐵) ∨ 𝐶 = 𝐵)))

Theoremelicoelioo 30194 Relate elementhood to a closed-below, open-above interval with elementhood to the same open interval or to its lower bound. (Contributed by Thierry Arnoux, 6-Jul-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 = 𝐴𝐶 ∈ (𝐴(,)𝐵))))

Theoremiocinioc2 30195 Intersection between two open-below, closed-above intervals sharing the same upper bound. (Contributed by Thierry Arnoux, 7-Aug-2017.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ 𝐴𝐵) → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐵(,]𝐶))

Theoremxrdifh 30196 Class difference of a half-open interval in the extended reals. (Contributed by Thierry Arnoux, 1-Aug-2017.)
𝐴 ∈ ℝ*       (ℝ* ∖ (𝐴[,]+∞)) = (-∞[,)𝐴)

Theoremiocinif 30197 Relate intersection of two open-below, closed-above intervals with the same upper bound with a conditional construct. (Contributed by Thierry Arnoux, 7-Aug-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = if(𝐴 < 𝐵, (𝐵(,]𝐶), (𝐴(,]𝐶)))

Theoremdifioo 30198 The difference between two open intervals sharing the same lower bound. (Contributed by Thierry Arnoux, 26-Sep-2017.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐶) ∖ (𝐴(,)𝐵)) = (𝐵[,)𝐶))

Theoremdifico 30199 The difference between two closed-below, open-above intervals sharing the same upper bound. (Contributed by Thierry Arnoux, 13-Oct-2017.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴𝐵𝐵𝐶)) → ((𝐴[,)𝐶) ∖ (𝐵[,)𝐶)) = (𝐴[,)𝐵))

20.3.5.6  Finite intervals of integers - misc additions

Theoremuzssico 30200 Upper integer sets are a subset of the corresponding closed-below, open-above intervals. (Contributed by Thierry Arnoux, 29-Dec-2021.)
(𝑀 ∈ ℤ → (ℤ𝑀) ⊆ (𝑀[,)+∞))

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