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Theorem List for Metamath Proof Explorer - 43301-43400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempssnssi 43301 A proper subclass does not include the other class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴𝐵        ¬ 𝐵𝐴
 
Theoremrabidim2 43302 Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝑥 ∈ {𝑥𝐴𝜑} → 𝜑)
 
Theoremeluni2f 43303* Membership in class union. Restricted quantifier version. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴    &   𝑥𝐵       (𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)
 
Theoremeliin2f 43304* Membership in indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐵       (𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
 
Theoremnssd 43305 Negation of subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑋𝐴)    &   (𝜑 → ¬ 𝑋𝐵)       (𝜑 → ¬ 𝐴𝐵)
 
Theoremiineq12dv 43306* Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐵) → 𝐶 = 𝐷)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
 
Theoremsupxrcld 43307 The supremum of an arbitrary set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ⊆ ℝ*)       (𝜑 → sup(𝐴, ℝ*, < ) ∈ ℝ*)
 
Theoremelrestd 43308 A sufficient condition for being an open set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐽𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝑋𝐽)    &   𝐴 = (𝑋𝐵)       (𝜑𝐴 ∈ (𝐽t 𝐵))
 
Theoremeliuniincex 43309* Counterexample to show that the additional conditions in eliuniin 43299 and eliuniin2 43320 are actually needed. Notice that the definition of 𝐴 is not even needed (it can be any class). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐵 = {∅}    &   𝐶 = ∅    &   𝐷 = ∅    &   𝑍 = V        ¬ (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷)
 
Theoremeliincex 43310* Counterexample to show that the additional conditions in eliin 4959 and eliin2 43316 are actually needed. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴 = V    &   𝐵 = ∅        ¬ (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶)
 
Theoremeliinid 43311* Membership in an indexed intersection implies membership in any intersected set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
((𝐴 𝑥𝐵 𝐶𝑥𝐵) → 𝐴𝐶)
 
Theoremabssf 43312 Class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴       ({𝑥𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
 
Theoremsupxrubd 43313 A member of a set of extended reals is less than or equal to the set's supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ⊆ ℝ*)    &   (𝜑𝐵𝐴)    &   𝑆 = sup(𝐴, ℝ*, < )       (𝜑𝐵𝑆)
 
Theoremssrabf 43314 Subclass of a restricted class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐵    &   𝑥𝐴       (𝐵 ⊆ {𝑥𝐴𝜑} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))
 
Theoremssrabdf 43315 Subclass of a restricted class abstraction (deduction form). (Contributed by Glauco Siliprandi, 5-Jan-2025.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝜑    &   (𝜑𝐵𝐴)    &   ((𝜑𝑥𝐵) → 𝜓)       (𝜑𝐵 ⊆ {𝑥𝐴𝜓})
 
Theoremeliin2 43316* Membership in indexed intersection. See eliincex 43310 for a counterexample showing that the precondition 𝐵 ≠ ∅ cannot be simply dropped. eliin 4959 uses an alternative precondition (and it doesn't have a disjoint var constraint between 𝐵 and 𝑥; see eliin2f 43304). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
 
Theoremssrab2f 43317 Subclass relation for a restricted class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴       {𝑥𝐴𝜑} ⊆ 𝐴
 
Theoremrestuni3 43318 The underlying set of a subspace induced by the subspace operator t. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 (𝐴t 𝐵) = ( 𝐴𝐵))
 
Theoremrabssf 43319 Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐵       ({𝑥𝐴𝜑} ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝜑𝑥𝐵))
 
Theoremeliuniin2 43320* Indexed union of indexed intersections. See eliincex 43310 for a counterexample showing that the precondition 𝐶 ≠ ∅ cannot be simply dropped. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐶    &   𝐴 = 𝑥𝐵 𝑦𝐶 𝐷       (𝐶 ≠ ∅ → (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷))
 
Theoremrestuni4 43321 The underlying set of a subspace induced by the t operator. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵 𝐴)       (𝜑 (𝐴t 𝐵) = 𝐵)
 
Theoremrestuni6 43322 The underlying set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 (𝐴t 𝐵) = ( 𝐴𝐵))
 
Theoremrestuni5 43323 The underlying set of a subspace induced by the t operator. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑋 = 𝐽       ((𝐽𝑉𝐴𝑋) → 𝐴 = (𝐽t 𝐴))
 
Theoremunirestss 43324 The union of an elementwise intersection is a subset of the underlying set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 (𝐴t 𝐵) ⊆ 𝐴)
 
Theoreminiin1 43325* Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐴 ≠ ∅ → ( 𝑥𝐴 𝐶𝐵) = 𝑥𝐴 (𝐶𝐵))
 
Theoreminiin2 43326* Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐴 ≠ ∅ → (𝐵 𝑥𝐴 𝐶) = 𝑥𝐴 (𝐵𝐶))
 
Theoremcbvrabv2 43327* A more general version of cbvrabv 3417. Usage of this theorem is discouraged because it depends on ax-13 2370. Use of cbvrabv2w 43328 is preferred. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (New usage is discouraged.)
(𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑦𝐵𝜓}
 
Theoremcbvrabv2w 43328* A more general version of cbvrabv 3417. Version of cbvrabv2 43327 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Revised by Gino Giotto, 16-Apr-2024.)
(𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑦𝐵𝜓}
 
Theoremiinssiin 43329 Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 𝑥𝐴 𝐵 𝑥𝐴 𝐶)
 
Theoremeliind2 43330* Membership in indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐵) → 𝐴𝐶)       (𝜑𝐴 𝑥𝐵 𝐶)
 
Theoremiinssd 43331* Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑋𝐴)    &   (𝑥 = 𝑋𝐵 = 𝐷)    &   (𝜑𝐷𝐶)       (𝜑 𝑥𝐴 𝐵𝐶)
 
Theoremrabbida2 43332 Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 
Theoremiinexd 43333* The existence of an indexed union. 𝑥 is normally a free-variable parameter in 𝐵, which should be read 𝐵(𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐴 ≠ ∅)    &   (𝜑 → ∀𝑥𝐴 𝐵𝐶)       (𝜑 𝑥𝐴 𝐵 ∈ V)
 
Theoremrabexf 43334 Separation Scheme in terms of a restricted class abstraction. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐴    &   𝐴𝑉       {𝑥𝐴𝜑} ∈ V
 
Theoremrabbida3 43335 Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 
Theoremr19.36vf 43336 Restricted quantifier version of one direction of 19.36 2223. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜓       (∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑𝜓))
 
Theoremraleqd 43337 Equality deduction for restricted universal quantifier. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐴    &   𝑥𝐵    &   (𝜑𝐴 = 𝐵)       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))
 
Theoremiinssf 43338 Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐶       (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
 
Theoremiinssdf 43339 Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐴    &   𝑥𝑋    &   𝑥𝐶    &   𝑥𝐷    &   (𝜑𝑋𝐴)    &   (𝑥 = 𝑋𝐵 = 𝐷)    &   (𝜑𝐷𝐶)       (𝜑 𝑥𝐴 𝐵𝐶)
 
Theoremresabs2i 43340 Absorption law for restriction. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐵𝐶       ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐵)
 
Theoremssdf2 43341 A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   ((𝜑𝑥𝐴) → 𝑥𝐵)       (𝜑𝐴𝐵)
 
Theoremrabssd 43342 Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   𝑥𝐵    &   ((𝜑𝑥𝐴𝜒) → 𝑥𝐵)       (𝜑 → {𝑥𝐴𝜒} ⊆ 𝐵)
 
Theoremrexnegd 43343 Minus a real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → -𝑒𝐴 = -𝐴)
 
Theoremrexlimd3 43344 * Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   𝑥𝜒    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremresabs1i 43345 Absorption law for restriction. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐵𝐶       ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵)
 
Theoremnel1nelin 43346 Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐴𝐵 → ¬ 𝐴 ∈ (𝐵𝐶))
 
Theoremnel2nelin 43347 Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐴𝐶 → ¬ 𝐴 ∈ (𝐵𝐶))
 
Theoremnel1nelini 43348 Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
¬ 𝐴𝐵        ¬ 𝐴 ∈ (𝐵𝐶)
 
Theoremnel2nelini 43349 Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
¬ 𝐴𝐶        ¬ 𝐴 ∈ (𝐵𝐶)
 
Theoremeliunid 43350* Membership in indexed union. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
((𝑥𝐴𝐶𝐵) → 𝐶 𝑥𝐴 𝐵)
 
Theoremreximddv3 43351* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑 → ∃𝑥𝐴 𝜒)
 
Theoremreximdd 43352 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑥𝜑    &   ((𝜑𝑥𝐴𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑 → ∃𝑥𝐴 𝜒)
 
Theoremunfid 43353 The union of two finite sets is finite. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)       (𝜑 → (𝐴𝐵) ∈ Fin)
 
Theoreminopnd 43354 The intersection of two open sets of a topology is an open set. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
(𝜑𝐽 ∈ Top)    &   (𝜑𝐴𝐽)    &   (𝜑𝐵𝐽)       (𝜑 → (𝐴𝐵) ∈ 𝐽)
 
Theoremss2rabdf 43355 Deduction of restricted abstraction subclass from implication. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})
 
Theoremrestopn3 43356 If 𝐴 is open, then 𝐴 is open in the restriction to itself. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴 ∈ (𝐽t 𝐴))
 
Theoremrestopnssd 43357 A topology restricted to an open set is a subset of the original topology. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
(𝜑𝐽 ∈ Top)    &   (𝜑𝐴𝐽)       (𝜑 → (𝐽t 𝐴) ⊆ 𝐽)
 
Theoremrestsubel 43358 A subset belongs in the space it generates via restriction. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
(𝜑𝐽𝑉)    &   (𝜑 𝐽𝐽)    &   (𝜑𝐴 𝐽)       (𝜑𝐴 ∈ (𝐽t 𝐴))
 
Theoremtoprestsubel 43359 A subset is open in the topology it generates via restriction. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
(𝜑𝐽 ∈ Top)    &   (𝜑𝐴 𝐽)       (𝜑𝐴 ∈ (𝐽t 𝐴))
 
Theoremrabidd 43360 An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
(𝜑𝑥𝐴)    &   (𝜑𝜒)       (𝜑𝑥 ∈ {𝑥𝐴𝜒})
 
Theoremiunssdf 43361 Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
𝑥𝜑    &   𝑥𝐶    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 𝑥𝐴 𝐵𝐶)
 
Theoremiinss2d 43362 Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐶    &   (𝜑𝐴 ≠ ∅)    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 𝑥𝐴 𝐵𝐶)
 
Theoremr19.3rzf 43363 Restricted quantification of wff not containing quantified variable. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
𝑥𝜑    &   𝑥𝐴       (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))
 
Theoremr19.28zf 43364 Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
𝑥𝜑    &   𝑥𝐴       (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))
 
Theoremiindif2f 43365 Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws". (Contributed by Glauco Siliprandi, 24-Jan-2025.)
𝑥𝐴    &   𝑥𝐵       (𝐴 ≠ ∅ → 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
 
Theoremralfal 43366 Two ways of expressing empty set. (Contributed by Glauco Siliprandi, 24-Jan-2024.)
𝑥𝐴       (𝐴 = ∅ ↔ ∀𝑥𝐴 ⊥)
 
Theoremarchd 43367* Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed by Glauco Siliprandi, 1-Feb-2025.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → ∃𝑛 ∈ ℕ 𝐴 < 𝑛)
 
Theoremeliund 43368* Membership in indexed union. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
(𝜑 → ∃𝑥𝐵 𝐴𝐶)       (𝜑𝐴 𝑥𝐵 𝐶)
 
Theoremnimnbi 43369 If an implication is false, the biconditional is false. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
¬ (𝜑𝜓)        ¬ (𝜑𝜓)
 
Theoremnimnbi2 43370 If an implication is false, the biconditional is false. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
¬ (𝜓𝜑)        ¬ (𝜑𝜓)
 
Theoremnotbicom 43371 Commutative law for the negation of a biconditional. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
¬ (𝜑𝜓)        ¬ (𝜓𝜑)
 
Theoremrexeqif 43372 Equality inference for restricted existential quantifier. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
𝑥𝐴    &   𝑥𝐵    &   𝐴 = 𝐵       (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑)
 
Theoremrspced 43373 Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
𝑥𝜒    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑𝐴𝐵)    &   (𝜑𝜒)    &   (𝑥 = 𝐴 → (𝜓𝜒))       (𝜑 → ∃𝑥𝐵 𝜓)
 
21.38.2  Functions
 
Theoremfeq1dd 43374 Equality deduction for functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐹:𝐴𝐵)       (𝜑𝐺:𝐴𝐵)
 
Theoremfnresdmss 43375 A function does not change when restricted to a set that contains its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐹 Fn 𝐴𝐴𝐵) → (𝐹𝐵) = 𝐹)
 
Theoremfmptsnxp 43376* Maps-to notation and Cartesian product for a singleton function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴𝑉𝐵𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) = ({𝐴} × {𝐵}))
 
Theoremfvmpt2bd 43377* Value of a function given by the maps-to notation. Deduction version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹 = (𝑥𝐴𝐵))       ((𝜑𝑥𝐴𝐵𝐶) → (𝐹𝑥) = 𝐵)
 
Theoremrnmptfi 43378* The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴 = (𝑥𝐵𝐶)       (𝐵 ∈ Fin → ran 𝐴 ∈ Fin)
 
Theoremfresin2 43379 Restriction of a function with respect to the intersection with its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐹:𝐴𝐵 → (𝐹 ↾ (𝐶𝐴)) = (𝐹𝐶))
 
Theoremffi 43380 A function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐹:𝐴𝐵𝐴 ∈ Fin) → 𝐹 ∈ Fin)
 
Theoremsuprnmpt 43381* An explicit bound for the range of a bounded function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ≠ ∅)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)    &   𝐹 = (𝑥𝐴𝐵)    &   𝐶 = sup(ran 𝐹, ℝ, < )       (𝜑 → (𝐶 ∈ ℝ ∧ ∀𝑥𝐴 𝐵𝐶))
 
Theoremrnffi 43382 The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐹:𝐴𝐵𝐴 ∈ Fin) → ran 𝐹 ∈ Fin)
 
Theoremmptelpm 43383* A function in maps-to notation is a partial map . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
((𝜑𝑥𝐴) → 𝐵𝐶)    &   (𝜑𝐴𝐷)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)       (𝜑 → (𝑥𝐴𝐵) ∈ (𝐶pm 𝐷))
 
Theoremrnmptpr 43384* Range of a function defined on an unordered pair. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶)    &   (𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑥 = 𝐵𝐶 = 𝐸)       (𝜑 → ran 𝐹 = {𝐷, 𝐸})
 
Theoremresmpti 43385* Restriction of the mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐵𝐴       ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶)
 
Theoremfouniiun 43386* Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐹:𝐴onto𝐵 𝐵 = 𝑥𝐴 (𝐹𝑥))
 
Theoremrnresun 43387 Distribution law for range of a restriction over a union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
ran (𝐹 ↾ (𝐴𝐵)) = (ran (𝐹𝐴) ∪ ran (𝐹𝐵))
 
Theoremdffo3f 43388* An onto mapping expressed in terms of function values. As dffo3 7052 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝐹       (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
 
Theoremelrnmptf 43389 The range of a function in maps-to notation. Same as elrnmpt 5911, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝐶    &   𝐹 = (𝑥𝐴𝐵)       (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
 
Theoremrnmptssrn 43390* Inclusion relation for two ranges expressed in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑥𝐴) → ∃𝑦𝐶 𝐵 = 𝐷)       (𝜑 → ran (𝑥𝐴𝐵) ⊆ ran (𝑦𝐶𝐷))
 
Theoremdisjf1 43391* A 1 to 1 mapping built from disjoint, nonempty sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)    &   (𝜑Disj 𝑥𝐴 𝐵)       (𝜑𝐹:𝐴1-1𝑉)
 
Theoremrnsnf 43392 The range of a function whose domain is a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:{𝐴}⟶𝐵)       (𝜑 → ran 𝐹 = {(𝐹𝐴)})
 
Theoremwessf1ornlem 43393* Given a function 𝐹 on a well-ordered domain 𝐴 there exists a subset of 𝐴 such that 𝐹 restricted to such subset is injective and onto the range of 𝐹 (without using the axiom of choice). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐴𝑉)    &   (𝜑𝑅 We 𝐴)    &   𝐺 = (𝑦 ∈ ran 𝐹 ↦ (𝑥 ∈ (𝐹 “ {𝑦})∀𝑧 ∈ (𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥))       (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto→ran 𝐹)
 
Theoremwessf1orn 43394* Given a function 𝐹 on a well-ordered domain 𝐴 there exists a subset of 𝐴 such that 𝐹 restricted to such subset is injective and onto the range of 𝐹 (without using the axiom of choice). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐴𝑉)    &   (𝜑𝑅 We 𝐴)       (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto→ran 𝐹)
 
Theoremfoelrnf 43395* Property of a surjective function. As foelrn 7056 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝐹       ((𝐹:𝐴onto𝐵𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))
 
Theoremnelrnres 43396 If 𝐴 is not in the range, it is not in the range of any restriction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐴 ∈ ran 𝐵 → ¬ 𝐴 ∈ ran (𝐵𝐶))
 
Theoremdisjrnmpt2 43397* Disjointness of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐹 = (𝑥𝐴𝐵)       (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ ran 𝐹 𝑦)
 
Theoremelrnmpt1sf 43398* Elementhood in an image set. Same as elrnmpt1s 5912, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝐶    &   𝐹 = (𝑥𝐴𝐵)    &   (𝑥 = 𝐷𝐵 = 𝐶)       ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)
 
Theoremfouniiun0 43399* Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝐵 = 𝑥𝐴 (𝐹𝑥))
 
Theoremdisjf1o 43400* A bijection built from disjoint sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑Disj 𝑥𝐴 𝐵)    &   𝐶 = {𝑥𝐴𝐵 ≠ ∅}    &   𝐷 = (ran 𝐹 ∖ {∅})       (𝜑 → (𝐹𝐶):𝐶1-1-onto𝐷)
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