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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fvnonrel 43301 | The function value of any class under a non-relation is empty. (Contributed by RP, 23-Oct-2020.) |
⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅ | ||
Theorem | elinlem 43302 | Two ways to say a set is a member of an intersection. (Contributed by RP, 19-Aug-2020.) |
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ( I ‘𝐴) ∈ 𝐶)) | ||
Theorem | elcnvcnvlem 43303 | Two ways to say a set is a member of the converse of the converse of a class. (Contributed by RP, 20-Aug-2020.) |
⊢ (𝐴 ∈ ◡◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵)) | ||
Original probably needs new subsection for Relation-related existence theorems. | ||
Theorem | cnvcnvintabd 43304* | Value of the relationship content of the intersection of a class. (Contributed by RP, 20-Aug-2020.) |
⊢ (𝜑 → ∃𝑥𝜓) ⇒ ⊢ (𝜑 → ◡◡∩ {𝑥 ∣ 𝜓} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜓)}) | ||
Theorem | elcnvlem 43305 | Two ways to say a set is a member of the converse of a class. (Contributed by RP, 19-Aug-2020.) |
⊢ 𝐹 = (𝑥 ∈ (V × V) ↦ 〈(2nd ‘𝑥), (1st ‘𝑥)〉) ⇒ ⊢ (𝐴 ∈ ◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ (𝐹‘𝐴) ∈ 𝐵)) | ||
Theorem | elcnvintab 43306* | Two ways of saying a set is an element of the converse of the intersection of a class. (Contributed by RP, 19-Aug-2020.) |
⊢ (𝐴 ∈ ◡∩ {𝑥 ∣ 𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑 → 𝐴 ∈ ◡𝑥))) | ||
Theorem | cnvintabd 43307* | Value of the converse of the intersection of a nonempty class. (Contributed by RP, 20-Aug-2020.) |
⊢ (𝜑 → ∃𝑥𝜓) ⇒ ⊢ (𝜑 → ◡∩ {𝑥 ∣ 𝜓} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡𝑥 ∧ 𝜓)}) | ||
Theorem | undmrnresiss 43308* | Two ways of saying the identity relation restricted to the union of the domain and range of a relation is a subset of a relation. Generalization of reflexg 43309. (Contributed by RP, 26-Sep-2020.) |
⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐵𝑥 ∧ 𝑦𝐵𝑦))) | ||
Theorem | reflexg 43309* | Two ways of saying a relation is reflexive over its domain and range. (Contributed by RP, 4-Aug-2020.) |
⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦))) | ||
Theorem | cnvssco 43310* | A condition weaker than reflexivity. (Contributed by RP, 3-Aug-2020.) |
⊢ (◡𝐴 ⊆ ◡(𝐵 ∘ 𝐶) ↔ ∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦))) | ||
Theorem | refimssco 43311 | Reflexive relations are subsets of their self-composition. (Contributed by RP, 4-Aug-2020.) |
⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 → ◡𝐴 ⊆ ◡(𝐴 ∘ 𝐴)) | ||
Theorem | cleq2lem 43312 | Equality implies bijection. (Contributed by RP, 24-Jul-2020.) |
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ 𝜑) ↔ (𝑅 ⊆ 𝐵 ∧ 𝜓))) | ||
Theorem | cbvcllem 43313* | Change of bound variable in class of supersets of a with a property. (Contributed by RP, 24-Jul-2020.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜑)} = {𝑦 ∣ (𝑋 ⊆ 𝑦 ∧ 𝜓)} | ||
Theorem | clublem 43314* | If a superset 𝑌 of 𝑋 possesses the property parameterized in 𝑥 in 𝜓, then 𝑌 is a superset of the closure of that property for the set 𝑋. (Contributed by RP, 23-Jul-2020.) |
⊢ (𝜑 → 𝑌 ∈ V) & ⊢ (𝑥 = 𝑌 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝑋 ⊆ 𝑌) & ⊢ (𝜑 → 𝜒) ⇒ ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ 𝑌) | ||
Theorem | clss2lem 43315* | The closure of a property is a superset of the closure of a less restrictive property. (Contributed by RP, 24-Jul-2020.) |
⊢ (𝜑 → (𝜒 → 𝜓)) ⇒ ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)}) | ||
Theorem | dfid7 43316* | Definition of identity relation as the trivial closure. (Contributed by RP, 26-Jul-2020.) |
⊢ I = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)}) | ||
Theorem | mptrcllem 43317* | Show two versions of a closure with reflexive properties are equal. (Contributed by RP, 19-Oct-2020.) |
⊢ (𝑥 ∈ 𝑉 → ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∈ V) & ⊢ (𝑥 ∈ 𝑉 → ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} ∈ V) & ⊢ (𝑥 ∈ 𝑉 → 𝜒) & ⊢ (𝑥 ∈ 𝑉 → 𝜃) & ⊢ (𝑥 ∈ 𝑉 → 𝜏) & ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ↔ 𝜃)) & ⊢ (𝑧 = ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} → (𝜓 ↔ 𝜏)) ⇒ ⊢ (𝑥 ∈ 𝑉 ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}) = (𝑥 ∈ 𝑉 ↦ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)}) | ||
Theorem | cotrintab 43318 | The intersection of a class is a transitive relation if membership in the class implies the member is a transitive relation. (Contributed by RP, 28-Oct-2020.) |
⊢ (𝜑 → (𝑥 ∘ 𝑥) ⊆ 𝑥) ⇒ ⊢ (∩ {𝑥 ∣ 𝜑} ∘ ∩ {𝑥 ∣ 𝜑}) ⊆ ∩ {𝑥 ∣ 𝜑} | ||
Theorem | rclexi 43319* | The reflexive closure of a set exists. (Contributed by RP, 27-Oct-2020.) |
⊢ 𝐴 ∈ 𝑉 ⇒ ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V | ||
Theorem | rtrclexlem 43320 | Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 1-Nov-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅))) ∈ V) | ||
Theorem | rtrclex 43321* | The reflexive-transitive closure of a set exists. (Contributed by RP, 1-Nov-2020.) |
⊢ (𝐴 ∈ V ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V) | ||
Theorem | trclubgNEW 43322* | If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) | ||
Theorem | trclubNEW 43323* | If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → Rel 𝑅) ⇒ ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (dom 𝑅 × ran 𝑅)) | ||
Theorem | trclexi 43324* | The transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.) |
⊢ 𝐴 ∈ 𝑉 ⇒ ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V | ||
Theorem | rtrclexi 43325* | The reflexive-transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.) |
⊢ 𝐴 ∈ 𝑉 ⇒ ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V | ||
Theorem | clrellem 43326* | When the property 𝜓 holds for a relation substituted for 𝑥, then the closure on that property is a relation if the base set is a relation. (Contributed by RP, 30-Jul-2020.) |
⊢ (𝜑 → 𝑌 ∈ V) & ⊢ (𝜑 → Rel 𝑋) & ⊢ (𝑥 = ◡◡𝑌 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝑋 ⊆ 𝑌) & ⊢ (𝜑 → 𝜒) ⇒ ⊢ (𝜑 → Rel ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}) | ||
Theorem | clcnvlem 43327* | When 𝐴, an upper bound of the closure, exists and certain substitutions hold the converse of the closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.) |
⊢ ((𝜑 ∧ 𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))) → (𝜒 → 𝜓)) & ⊢ ((𝜑 ∧ 𝑦 = ◡𝑥) → (𝜓 → 𝜒)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃)) & ⊢ (𝜑 → 𝑋 ⊆ 𝐴) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝜃) ⇒ ⊢ (𝜑 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ 𝜒)}) | ||
Theorem | cnvtrucl0 43328* | The converse of the trivial closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.) |
⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ ⊤)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ ⊤)}) | ||
Theorem | cnvrcl0 43329* | The converse of the reflexive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.) |
⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)}) | ||
Theorem | cnvtrcl0 43330* | The converse of the transitive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.) |
⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦)}) | ||
Theorem | dmtrcl 43331* | The domain of the transitive closure is equal to the domain of its base relation. (Contributed by RP, 1-Nov-2020.) |
⊢ (𝑋 ∈ 𝑉 → dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = dom 𝑋) | ||
Theorem | rntrcl 43332* | The range of the transitive closure is equal to the range of its base relation. (Contributed by RP, 1-Nov-2020.) |
⊢ (𝑋 ∈ 𝑉 → ran ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ran 𝑋) | ||
Theorem | dfrtrcl5 43333* | Definition of reflexive-transitive closure as a standard closure. (Contributed by RP, 1-Nov-2020.) |
⊢ t* = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦))}) | ||
Theorem | trcleq2lemRP 43334 | Equality implies bijection. (Contributed by RP, 5-May-2020.) (Proof modification is discouraged.) |
⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴) ↔ (𝑅 ⊆ 𝐵 ∧ (𝐵 ∘ 𝐵) ⊆ 𝐵))) | ||
This is based on the observation that the real and imaginary parts of a complex number can be calculated from the number's absolute and real part and the sign of its imaginary part. Formalization of the formula in sqrtcval 43345 was motivated by a short Michael Penn video. | ||
Theorem | sqrtcvallem1 43335 | Two ways of saying a complex number does not lie on the positive real axis. Lemma for sqrtcval 43345. (Contributed by RP, 17-May-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (((ℑ‘𝐴) = 0 → (ℜ‘𝐴) ≤ 0) ↔ ¬ 𝐴 ∈ ℝ+)) | ||
Theorem | reabsifneg 43336 | Alternate expression for the absolute value of a real number. Lemma for sqrtcval 43345. (Contributed by RP, 11-May-2024.) |
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = if(𝐴 < 0, -𝐴, 𝐴)) | ||
Theorem | reabsifnpos 43337 | Alternate expression for the absolute value of a real number. (Contributed by RP, 11-May-2024.) |
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = if(𝐴 ≤ 0, -𝐴, 𝐴)) | ||
Theorem | reabsifpos 43338 | Alternate expression for the absolute value of a real number. (Contributed by RP, 11-May-2024.) |
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = if(0 < 𝐴, 𝐴, -𝐴)) | ||
Theorem | reabsifnneg 43339 | Alternate expression for the absolute value of a real number. (Contributed by RP, 11-May-2024.) |
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = if(0 ≤ 𝐴, 𝐴, -𝐴)) | ||
Theorem | reabssgn 43340 | Alternate expression for the absolute value of a real number. (Contributed by RP, 22-May-2024.) |
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = ((sgn‘𝐴) · 𝐴)) | ||
Theorem | sqrtcvallem2 43341 | Equivalent to saying that the square of the imaginary component of the square root of a complex number is a nonnegative real number. Lemma for sqrtcval 43345. See imsqrtval 43348. (Contributed by RP, 11-May-2024.) |
⊢ (𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) − (ℜ‘𝐴)) / 2)) | ||
Theorem | sqrtcvallem3 43342 | Equivalent to saying that the absolute value of the imaginary component of the square root of a complex number is a real number. Lemma for sqrtcval 43345, sqrtcval2 43346, resqrtval 43347, and imsqrtval 43348. (Contributed by RP, 11-May-2024.) |
⊢ (𝐴 ∈ ℂ → (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2)) ∈ ℝ) | ||
Theorem | sqrtcvallem4 43343 | Equivalent to saying that the square of the real component of the square root of a complex number is a nonnegative real number. Lemma for sqrtcval 43345. See resqrtval 43347. (Contributed by RP, 11-May-2024.) |
⊢ (𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) + (ℜ‘𝐴)) / 2)) | ||
Theorem | sqrtcvallem5 43344 | Equivalent to saying that the real component of the square root of a complex number is a real number. Lemma for resqrtval 43347 and imsqrtval 43348. (Contributed by RP, 11-May-2024.) |
⊢ (𝐴 ∈ ℂ → (√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)) ∈ ℝ) | ||
Theorem | sqrtcval 43345 | Explicit formula for the complex square root in terms of the square root of nonnegative reals. The right-hand side is decomposed into real and imaginary parts in the format expected by crrei 15192 and crimi 15193. This formula can be found in section 3.7.27 of Handbook of Mathematical Functions, ed. M. Abramowitz and I. A. Stegun (1965, Dover Press). (Contributed by RP, 18-May-2024.) |
⊢ (𝐴 ∈ ℂ → (√‘𝐴) = ((√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)) + (i · (if((ℑ‘𝐴) < 0, -1, 1) · (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2)))))) | ||
Theorem | sqrtcval2 43346 | Explicit formula for the complex square root in terms of the square root of nonnegative reals. The right side is slightly more compact than sqrtcval 43345. (Contributed by RP, 18-May-2024.) |
⊢ (𝐴 ∈ ℂ → (√‘𝐴) = ((√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)) + (if((ℑ‘𝐴) < 0, -i, i) · (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2))))) | ||
Theorem | resqrtval 43347 | Real part of the complex square root. (Contributed by RP, 18-May-2024.) |
⊢ (𝐴 ∈ ℂ → (ℜ‘(√‘𝐴)) = (√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2))) | ||
Theorem | imsqrtval 43348 | Imaginary part of the complex square root. (Contributed by RP, 18-May-2024.) |
⊢ (𝐴 ∈ ℂ → (ℑ‘(√‘𝐴)) = (if((ℑ‘𝐴) < 0, -1, 1) · (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2)))) | ||
Theorem | resqrtvalex 43349 | Example for resqrtval 43347. (Contributed by RP, 21-May-2024.) |
⊢ (ℜ‘(√‘(;15 + (i · 8)))) = 4 | ||
Theorem | imsqrtvalex 43350 | Example for imsqrtval 43348. (Contributed by RP, 21-May-2024.) |
⊢ (ℑ‘(√‘(;15 + (i · 8)))) = 1 | ||
Theorem | al3im 43351 | Version of ax-4 1804 for a nested implication. (Contributed by RP, 13-Apr-2020.) |
⊢ (∀𝑥(𝜑 → (𝜓 → (𝜒 → 𝜃))) → (∀𝑥𝜑 → (∀𝑥𝜓 → (∀𝑥𝜒 → ∀𝑥𝜃)))) | ||
Theorem | intima0 43352* | Two ways of expressing the intersection of images of a class. (Contributed by RP, 13-Apr-2020.) |
⊢ ∩ 𝑎 ∈ 𝐴 (𝑎 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} | ||
Theorem | elimaint 43353* | Element of image of intersection. (Contributed by RP, 13-Apr-2020.) |
⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) | ||
Theorem | cnviun 43354* | Converse of indexed union. (Contributed by RP, 20-Jun-2020.) |
⊢ ◡∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ◡𝐵 | ||
Theorem | imaiun1 43355* | The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.) |
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶) | ||
Theorem | coiun1 43356* | Composition with an indexed union. Proof analogous to that of coiun 6259. (Contributed by RP, 20-Jun-2020.) |
⊢ (∪ 𝑥 ∈ 𝐶 𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) | ||
Theorem | elintima 43357* | Element of intersection of images. (Contributed by RP, 13-Apr-2020.) |
⊢ (𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 〈𝑏, 𝑦〉 ∈ 𝑎) | ||
Theorem | intimass 43358* | The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.) |
⊢ (∩ 𝐴 “ 𝐵) ⊆ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} | ||
Theorem | intimass2 43359* | The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.) |
⊢ (∩ 𝐴 “ 𝐵) ⊆ ∩ 𝑥 ∈ 𝐴 (𝑥 “ 𝐵) | ||
Theorem | intimag 43360* | Requirement for the image under the intersection of relations to equal the intersection of the images of those relations. (Contributed by RP, 13-Apr-2020.) |
⊢ (∀𝑦(∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 〈𝑏, 𝑦〉 ∈ 𝑎 → ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) → (∩ 𝐴 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}) | ||
Theorem | intimasn 43361* | Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.) |
⊢ (𝐵 ∈ 𝑉 → (∩ 𝐴 “ {𝐵}) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ {𝐵})}) | ||
Theorem | intimasn2 43362* | Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.) |
⊢ (𝐵 ∈ 𝑉 → (∩ 𝐴 “ {𝐵}) = ∩ 𝑥 ∈ 𝐴 (𝑥 “ {𝐵})) | ||
Theorem | ss2iundf 43363* | Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑦𝑌 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ Ⅎ𝑦𝐶 & ⊢ Ⅎ𝑥𝐷 & ⊢ Ⅎ𝑦𝐺 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐺) ⇒ ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷) | ||
Theorem | ss2iundv 43364* | Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐺) ⇒ ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷) | ||
Theorem | cbviuneq12df 43365* | Rule used to change the bound variables and classes in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by RP, 17-Jul-2020.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝑋 & ⊢ Ⅎ𝑦𝑌 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ Ⅎ𝑦𝐶 & ⊢ Ⅎ𝑥𝐷 & ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑦𝐺 & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝑋 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐹) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐺) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐷 = 𝐹) ⇒ ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷) | ||
Theorem | cbviuneq12dv 43366* | Rule used to change the bound variables and classes in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by RP, 17-Jul-2020.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝑋 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐹) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐺) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐷 = 𝐹) ⇒ ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷) | ||
Theorem | conrel1d 43367 | Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → ◡𝐴 = ∅) ⇒ ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅) | ||
Theorem | conrel2d 43368 | Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → ◡𝐴 = ∅) ⇒ ⊢ (𝜑 → (𝐵 ∘ 𝐴) = ∅) | ||
Theorem | trrelind 43369 | The intersection of transitive relations is a transitive relation. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅) & ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆) & ⊢ (𝜑 → 𝑇 = (𝑅 ∩ 𝑆)) ⇒ ⊢ (𝜑 → (𝑇 ∘ 𝑇) ⊆ 𝑇) | ||
Theorem | xpintrreld 43370 | The intersection of a transitive relation with a Cartesian product is a transitive relation. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅) & ⊢ (𝜑 → 𝑆 = (𝑅 ∩ (𝐴 × 𝐵))) ⇒ ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆) | ||
Theorem | restrreld 43371 | The restriction of a transitive relation is a transitive relation. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅) & ⊢ (𝜑 → 𝑆 = (𝑅 ↾ 𝐴)) ⇒ ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆) | ||
Theorem | trrelsuperreldg 43372 | Concrete construction of a superclass of relation 𝑅 which is a transitive relation. (Contributed by RP, 25-Dec-2019.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑆 = (dom 𝑅 × ran 𝑅)) ⇒ ⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆)) | ||
Theorem | trficl 43373* | The class of all transitive relations has the finite intersection property. (Contributed by RP, 1-Jan-2020.) (Proof shortened by RP, 3-Jan-2020.) |
⊢ 𝐴 = {𝑧 ∣ (𝑧 ∘ 𝑧) ⊆ 𝑧} ⇒ ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 | ||
Theorem | cnvtrrel 43374 | The converse of a transitive relation is a transitive relation. (Contributed by RP, 25-Dec-2019.) |
⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ (◡𝑆 ∘ ◡𝑆) ⊆ ◡𝑆) | ||
Theorem | trrelsuperrel2dg 43375 | Concrete construction of a superclass of relation 𝑅 which is a transitive relation. (Contributed by RP, 20-Jul-2020.) |
⊢ (𝜑 → 𝑆 = (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⇒ ⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆)) | ||
Syntax | crcl 43376 | Extend class notation with reflexive closure. |
class r* | ||
Definition | df-rcl 43377* | Reflexive closure of a relation. This is the smallest superset which has the reflexive property. (Contributed by RP, 5-Jun-2020.) |
⊢ r* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)}) | ||
Theorem | dfrcl2 43378 | Reflexive closure of a relation as union with restricted identity relation. (Contributed by RP, 6-Jun-2020.) |
⊢ r* = (𝑥 ∈ V ↦ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥)) | ||
Theorem | dfrcl3 43379 | Reflexive closure of a relation as union of powers of the relation. (Contributed by RP, 6-Jun-2020.) |
⊢ r* = (𝑥 ∈ V ↦ ((𝑥↑𝑟0) ∪ (𝑥↑𝑟1))) | ||
Theorem | dfrcl4 43380* | Reflexive closure of a relation as indexed union of powers of the relation. (Contributed by RP, 8-Jun-2020.) |
⊢ r* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ {0, 1} (𝑟↑𝑟𝑛)) | ||
In order for theorems on the transitive closure of a relation to be grouped together before the concept of continuity, we really need an analogue of ↑𝑟 that works on finite ordinals or finite sets instead of natural numbers. | ||
Theorem | relexp2 43381 | A set operated on by the relation exponent to the second power is equal to the composition of the set with itself. (Contributed by RP, 1-Jun-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟2) = (𝑅 ∘ 𝑅)) | ||
Theorem | relexpnul 43382 | If the domain and range of powers of a relation are disjoint then the relation raised to the sum of those exponents is empty. (Contributed by RP, 1-Jun-2020.) |
⊢ (((𝑅 ∈ 𝑉 ∧ Rel 𝑅) ∧ (𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → ((dom (𝑅↑𝑟𝑁) ∩ ran (𝑅↑𝑟𝑀)) = ∅ ↔ (𝑅↑𝑟(𝑁 + 𝑀)) = ∅)) | ||
Theorem | eliunov2 43383* | Membership in the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the element is a member of that operator value. Generalized from dfrtrclrec2 15058. (Contributed by RP, 1-Jun-2020.) |
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) ⇒ ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛))) | ||
Theorem | eltrclrec 43384* | Membership in the indexed union of relation exponentiation over the natural numbers is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.) |
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛)) ⇒ ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ ℕ 𝑋 ∈ (𝑅↑𝑟𝑛))) | ||
Theorem | elrtrclrec 43385* | Membership in the indexed union of relation exponentiation over the natural numbers (including zero) is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.) |
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) ⇒ ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ ℕ0 𝑋 ∈ (𝑅↑𝑟𝑛))) | ||
Theorem | briunov2 43386* | Two classes related by the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the two classes are related by that operator value. (Contributed by RP, 1-Jun-2020.) |
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) ⇒ ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌)) | ||
Theorem | brmptiunrelexpd 43387* | If two elements are connected by an indexed union of relational powers, then they are connected via 𝑛 instances the relation, for some 𝑛. Generalization of dfrtrclrec2 15058. (Contributed by RP, 21-Jul-2020.) |
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛)) & ⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝑁 ⊆ ℕ0) ⇒ ⊢ (𝜑 → (𝐴(𝐶‘𝑅)𝐵 ↔ ∃𝑛 ∈ 𝑁 𝐴(𝑅↑𝑟𝑛)𝐵)) | ||
Theorem | fvmptiunrelexplb0d 43388* | If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.) |
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛)) & ⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝑁 ∈ V) & ⊢ (𝜑 → 0 ∈ 𝑁) ⇒ ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅)) | ||
Theorem | fvmptiunrelexplb0da 43389* | If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.) |
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛)) & ⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝑁 ∈ V) & ⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 0 ∈ 𝑁) ⇒ ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (𝐶‘𝑅)) | ||
Theorem | fvmptiunrelexplb1d 43390* | If the indexed union ranges over the first power of the relation, then the relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.) |
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛)) & ⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝑁 ∈ V) & ⊢ (𝜑 → 1 ∈ 𝑁) ⇒ ⊢ (𝜑 → 𝑅 ⊆ (𝐶‘𝑅)) | ||
Theorem | brfvid 43391 | If two elements are connected by a value of the identity relation, then they are connected via the argument. (Contributed by RP, 21-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵)) | ||
Theorem | brfvidRP 43392 | If two elements are connected by a value of the identity relation, then they are connected via the argument. This is an example which uses brmptiunrelexpd 43387. (Contributed by RP, 21-Jul-2020.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵)) | ||
Theorem | fvilbd 43393 | A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → 𝑅 ⊆ ( I ‘𝑅)) | ||
Theorem | fvilbdRP 43394 | A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → 𝑅 ⊆ ( I ‘𝑅)) | ||
Theorem | brfvrcld 43395 | If two elements are connected by the reflexive closure of a relation, then they are connected via zero or one instances the relation. (Contributed by RP, 21-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ (𝐴(𝑅↑𝑟0)𝐵 ∨ 𝐴(𝑅↑𝑟1)𝐵))) | ||
Theorem | brfvrcld2 43396 | If two elements are connected by the reflexive closure of a relation, then they are equal or related by relation. (Contributed by RP, 21-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵) ∨ 𝐴𝑅𝐵))) | ||
Theorem | fvrcllb0d 43397 | A restriction of the identity relation is a subset of the reflexive closure of a set. (Contributed by RP, 22-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (r*‘𝑅)) | ||
Theorem | fvrcllb0da 43398 | A restriction of the identity relation is a subset of the reflexive closure of a relation. (Contributed by RP, 22-Jul-2020.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (r*‘𝑅)) | ||
Theorem | fvrcllb1d 43399 | A set is a subset of its image under the reflexive closure. (Contributed by RP, 22-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → 𝑅 ⊆ (r*‘𝑅)) | ||
Theorem | brtrclrec 43400* | Two classes related by the indexed union of relation exponentiation over the natural numbers is equivalent to the existence of at least one number such that the two classes are related by that relationship power. (Contributed by RP, 2-Jun-2020.) |
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛)) ⇒ ⊢ (𝑅 ∈ 𝑉 → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ 𝑋(𝑅↑𝑟𝑛)𝑌)) |
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