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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | pssnssi 43301 | A proper subclass does not include the other class. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝐴 ⊊ 𝐵 ⇒ ⊢ ¬ 𝐵 ⊆ 𝐴 | ||
Theorem | rabidim2 43302 | Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜑) | ||
Theorem | eluni2f 43303* | Membership in class union. Restricted quantifier version. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) | ||
Theorem | eliin2f 43304* | Membership in indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) | ||
Theorem | nssd 43305 | Negation of subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐴 ⊆ 𝐵) | ||
Theorem | iineq12dv 43306* | Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷) | ||
Theorem | supxrcld 43307 | The supremum of an arbitrary set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ*) ⇒ ⊢ (𝜑 → sup(𝐴, ℝ*, < ) ∈ ℝ*) | ||
Theorem | elrestd 43308 | A sufficient condition for being an open set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐽 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝑋 ∈ 𝐽) & ⊢ 𝐴 = (𝑋 ∩ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐵)) | ||
Theorem | eliuniincex 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 ⇒ ⊢ ¬ (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) | ||
Theorem | eliincex 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 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) | ||
Theorem | eliinid 43311* | Membership in an indexed intersection implies membership in any intersected set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝐶) | ||
Theorem | abssf 43312 | Class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) | ||
Theorem | supxrubd 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.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ 𝑆 = sup(𝐴, ℝ*, < ) ⇒ ⊢ (𝜑 → 𝐵 ≤ 𝑆) | ||
Theorem | ssrabf 43314 | Subclass of a restricted class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | ssrabdf 43315 | Subclass of a restricted class abstraction (deduction form). (Contributed by Glauco Siliprandi, 5-Jan-2025.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝜓) ⇒ ⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}) | ||
Theorem | eliin2 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 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) | ||
Theorem | ssrab2f 43317 | Subclass relation for a restricted class. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 | ||
Theorem | restuni3 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.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵)) | ||
Theorem | rabssf 43319 | Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵)) | ||
Theorem | eliuniin2 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.) |
⊢ Ⅎ𝑥𝐶 & ⊢ 𝐴 = ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷 ⇒ ⊢ (𝐶 ≠ ∅ → (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷)) | ||
Theorem | restuni4 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.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝐴) ⇒ ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = 𝐵) | ||
Theorem | restuni6 43322 | The underlying set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵)) | ||
Theorem | restuni5 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 ⇒ ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) | ||
Theorem | unirestss 43324 | The union of an elementwise intersection is a subset of the underlying set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) ⊆ ∪ 𝐴) | ||
Theorem | iniin1 43325* | Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝐴 ≠ ∅ → (∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵)) | ||
Theorem | iniin2 43326* | Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝐴 ≠ ∅ → (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶) = ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶)) | ||
Theorem | cbvrabv2 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 = 𝑦 → 𝐴 = 𝐵) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓} | ||
Theorem | cbvrabv2w 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 = 𝑦 → 𝐴 = 𝐵) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓} | ||
Theorem | iinssiin 43329 | Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶) | ||
Theorem | eliind2 43330* | Membership in indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶) | ||
Theorem | iinssd 43331* | Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐷) & ⊢ (𝜑 → 𝐷 ⊆ 𝐶) ⇒ ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) | ||
Theorem | rabbida2 43332 | Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) | ||
Theorem | iinexd 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.) |
⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V) | ||
Theorem | rabexf 43334 | Separation Scheme in terms of a restricted class abstraction. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑥𝐴 & ⊢ 𝐴 ∈ 𝑉 ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V | ||
Theorem | rabbida3 43335 | Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) | ||
Theorem | r19.36vf 43336 | Restricted quantifier version of one direction of 19.36 2223. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓)) | ||
Theorem | raleqd 43337 | Equality deduction for restricted universal quantifier. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓)) | ||
Theorem | iinssf 43338 | Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑥𝐶 ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) | ||
Theorem | iinssdf 43339 | Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝑋 & ⊢ Ⅎ𝑥𝐶 & ⊢ Ⅎ𝑥𝐷 & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐷) & ⊢ (𝜑 → 𝐷 ⊆ 𝐶) ⇒ ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) | ||
Theorem | resabs2i 43340 | Absorption law for restriction. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ 𝐵 ⊆ 𝐶 ⇒ ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵) | ||
Theorem | ssdf2 43341 | A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | ||
Theorem | rabssd 43342 | Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐵 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒) → 𝑥 ∈ 𝐵) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜒} ⊆ 𝐵) | ||
Theorem | rexnegd 43343 | Minus a real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → -𝑒𝐴 = -𝐴) | ||
Theorem | rexlimd3 43344 | * Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜒 & ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) | ||
Theorem | resabs1i 43345 | Absorption law for restriction. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ 𝐵 ⊆ 𝐶 ⇒ ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵) | ||
Theorem | nel1nelin 43346 | Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (¬ 𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶)) | ||
Theorem | nel2nelin 43347 | Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (¬ 𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶)) | ||
Theorem | nel1nelini 43348 | Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ ¬ 𝐴 ∈ 𝐵 ⇒ ⊢ ¬ 𝐴 ∈ (𝐵 ∩ 𝐶) | ||
Theorem | nel2nelini 43349 | Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ ¬ 𝐴 ∈ 𝐶 ⇒ ⊢ ¬ 𝐴 ∈ (𝐵 ∩ 𝐶) | ||
Theorem | eliunid 43350* | Membership in indexed union. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | reximddv3 43351* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒) | ||
Theorem | reximdd 43352 | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒) | ||
Theorem | unfid 43353 | The union of two finite sets is finite. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ Fin) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ Fin) | ||
Theorem | inopnd 43354 | The intersection of two open sets of a topology is an open set. (Contributed by Glauco Siliprandi, 21-Dec-2024.) |
⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝐴 ∈ 𝐽) & ⊢ (𝜑 → 𝐵 ∈ 𝐽) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝐽) | ||
Theorem | ss2rabdf 43355 | Deduction of restricted abstraction subclass from implication. (Contributed by Glauco Siliprandi, 21-Dec-2024.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒}) | ||
Theorem | restopn3 43356 | If 𝐴 is open, then 𝐴 is open in the restriction to itself. (Contributed by Glauco Siliprandi, 21-Dec-2024.) |
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ∈ (𝐽 ↾t 𝐴)) | ||
Theorem | restopnssd 43357 | A topology restricted to an open set is a subset of the original topology. (Contributed by Glauco Siliprandi, 21-Dec-2024.) |
⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝐴 ∈ 𝐽) ⇒ ⊢ (𝜑 → (𝐽 ↾t 𝐴) ⊆ 𝐽) | ||
Theorem | restsubel 43358 | A subset belongs in the space it generates via restriction. (Contributed by Glauco Siliprandi, 21-Dec-2024.) |
⊢ (𝜑 → 𝐽 ∈ 𝑉) & ⊢ (𝜑 → ∪ 𝐽 ∈ 𝐽) & ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽) ⇒ ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐴)) | ||
Theorem | toprestsubel 43359 | A subset is open in the topology it generates via restriction. (Contributed by Glauco Siliprandi, 21-Dec-2024.) |
⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽) ⇒ ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐴)) | ||
Theorem | rabidd 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.) |
⊢ (𝜑 → 𝑥 ∈ 𝐴) & ⊢ (𝜑 → 𝜒) ⇒ ⊢ (𝜑 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒}) | ||
Theorem | iunssdf 43361 | Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 24-Jan-2025.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐶 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) | ||
Theorem | iinss2d 43362 | Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 24-Jan-2025.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) | ||
Theorem | r19.3rzf 43363 | Restricted quantification of wff not containing quantified variable. (Contributed by Glauco Siliprandi, 24-Jan-2025.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | r19.28zf 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.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))) | ||
Theorem | iindif2f 43365 | Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws". (Contributed by Glauco Siliprandi, 24-Jan-2025.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶)) | ||
Theorem | ralfal 43366 | Two ways of expressing empty set. (Contributed by Glauco Siliprandi, 24-Jan-2024.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 ⊥) | ||
Theorem | archd 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.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → ∃𝑛 ∈ ℕ 𝐴 < 𝑛) | ||
Theorem | eliund 43368* | Membership in indexed union. (Contributed by Glauco Siliprandi, 15-Feb-2025.) |
⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶) | ||
Theorem | nimnbi 43369 | If an implication is false, the biconditional is false. (Contributed by Glauco Siliprandi, 15-Feb-2025.) |
⊢ ¬ (𝜑 → 𝜓) ⇒ ⊢ ¬ (𝜑 ↔ 𝜓) | ||
Theorem | nimnbi2 43370 | If an implication is false, the biconditional is false. (Contributed by Glauco Siliprandi, 15-Feb-2025.) |
⊢ ¬ (𝜓 → 𝜑) ⇒ ⊢ ¬ (𝜑 ↔ 𝜓) | ||
Theorem | notbicom 43371 | Commutative law for the negation of a biconditional. (Contributed by Glauco Siliprandi, 15-Feb-2025.) |
⊢ ¬ (𝜑 ↔ 𝜓) ⇒ ⊢ ¬ (𝜓 ↔ 𝜑) | ||
Theorem | rexeqif 43372 | Equality inference for restricted existential quantifier. (Contributed by Glauco Siliprandi, 15-Feb-2025.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ 𝐴 = 𝐵 ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑) | ||
Theorem | rspced 43373 | Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 15-Feb-2025.) |
⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝜒) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) | ||
Theorem | feq1dd 43374 | Equality deduction for functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) | ||
Theorem | fnresdmss 43375 | A function does not change when restricted to a set that contains its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝐹 ↾ 𝐵) = 𝐹) | ||
Theorem | fmptsnxp 43376* | Maps-to notation and Cartesian product for a singleton function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) = ({𝐴} × {𝐵})) | ||
Theorem | fvmpt2bd 43377* | Value of a function given by the maps-to notation. Deduction version. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐹‘𝑥) = 𝐵) | ||
Theorem | rnmptfi 43378* | The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (𝐵 ∈ Fin → ran 𝐴 ∈ Fin) | ||
Theorem | fresin2 43379 | Restriction of a function with respect to the intersection with its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ (𝐶 ∩ 𝐴)) = (𝐹 ↾ 𝐶)) | ||
Theorem | ffi 43380 | A function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin) | ||
Theorem | suprnmpt 43381* | An explicit bound for the range of a bounded function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ 𝐶 = sup(ran 𝐹, ℝ, < ) ⇒ ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)) | ||
Theorem | rnffi 43382 | The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ Fin) → ran 𝐹 ∈ Fin) | ||
Theorem | mptelpm 43383* | A function in maps-to notation is a partial map . (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) & ⊢ (𝜑 → 𝐴 ⊆ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐶 ↑pm 𝐷)) | ||
Theorem | rnmptpr 43384* | Range of a function defined on an unordered pair. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶) & ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷) & ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) ⇒ ⊢ (𝜑 → ran 𝐹 = {𝐷, 𝐸}) | ||
Theorem | resmpti 43385* | Restriction of the mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ 𝐵 ⊆ 𝐴 ⇒ ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶) | ||
Theorem | founiiun 43386* | Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝐹:𝐴–onto→𝐵 → ∪ 𝐵 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥)) | ||
Theorem | rnresun 43387 | Distribution law for range of a restriction over a union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵)) | ||
Theorem | dffo3f 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.) |
⊢ Ⅎ𝑥𝐹 ⇒ ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))) | ||
Theorem | elrnmptf 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.) |
⊢ Ⅎ𝑥𝐶 & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) | ||
Theorem | rnmptssrn 43390* | Inclusion relation for two ranges expressed in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐶 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ran (𝑦 ∈ 𝐶 ↦ 𝐷)) | ||
Theorem | disjf1 43391* | A 1 to 1 mapping built from disjoint, nonempty sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ Ⅎ𝑥𝜑 & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) ⇒ ⊢ (𝜑 → 𝐹:𝐴–1-1→𝑉) | ||
Theorem | rnsnf 43392 | The range of a function whose domain is a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵) ⇒ ⊢ (𝜑 → ran 𝐹 = {(𝐹‘𝐴)}) | ||
Theorem | wessf1ornlem 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.) |
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ 𝐺 = (𝑦 ∈ ran 𝐹 ↦ (℩𝑥 ∈ (◡𝐹 “ {𝑦})∀𝑧 ∈ (◡𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→ran 𝐹) | ||
Theorem | wessf1orn 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.) |
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 We 𝐴) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→ran 𝐹) | ||
Theorem | foelrnf 43395* | Property of a surjective function. As foelrn 7056 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ Ⅎ𝑥𝐹 ⇒ ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 = (𝐹‘𝑥)) | ||
Theorem | nelrnres 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 (𝐵 ↾ 𝐶)) | ||
Theorem | disjrnmpt2 43397* | Disjointness of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ ran 𝐹 𝑦) | ||
Theorem | elrnmpt1sf 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.) |
⊢ Ⅎ𝑥𝐶 & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) ⇒ ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹) | ||
Theorem | founiiun0 43399* | Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝐹:𝐴–onto→(𝐵 ∪ {∅}) → ∪ 𝐵 = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥)) | ||
Theorem | disjf1o 43400* | A bijection built from disjoint sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ Ⅎ𝑥𝜑 & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) & ⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ ∅} & ⊢ 𝐷 = (ran 𝐹 ∖ {∅}) ⇒ ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶–1-1-onto→𝐷) |
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