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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | suppssfv 8101* | 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 → ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌) ⊆ 𝐿) & ⊢ (𝜑 → (𝐹‘𝑌) = 𝑍) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐹‘𝐴)) supp 𝑍) ⊆ 𝐿) | ||
Theorem | suppofssd 8102 | Condition for the support of a function operation to be a subset of the union of the supports of the left and right function terms. (Contributed by Steven Nguyen, 28-Aug-2023.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) & ⊢ (𝜑 → (𝑍𝑋𝑍) = 𝑍) ⇒ ⊢ (𝜑 → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))) | ||
Theorem | suppofss1d 8103* | 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.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑍𝑋𝑥) = 𝑍) ⇒ ⊢ (𝜑 → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑍)) | ||
Theorem | suppofss2d 8104* | 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.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝑋𝑍) = 𝑍) ⇒ ⊢ (𝜑 → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑍)) | ||
Theorem | suppco 8105 | The support of the composition of two functions is the inverse image by the inner function of the support of the outer function. (Contributed by AV, 30-May-2019.) Extract this statement from the proof of supp0cosupp0 8107. (Revised by SN, 15-Sep-2023.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍))) | ||
Theorem | suppcoss 8106 | The support of the composition of two functions is a subset of the support of the inner function if the outer function preserves zero. Compare suppssfv 8101, which has a sethood condition on 𝐴 instead of 𝐵. (Contributed by SN, 25-May-2024.) |
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → (𝐹‘𝑌) = 𝑍) ⇒ ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑌)) | ||
Theorem | supp0cosupp0 8107 | 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.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅)) | ||
Theorem | imacosupp 8108 | The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍))) | ||
The following theorems are about maps-to operations (see df-mpo 7355) 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 "mpox" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpox 7443, ovmpox 7501 and fmpox 7988). If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpoxop", and the maps-to operations are called "x-op maps-to operations" for short. | ||
Theorem | opeliunxp2f 8109* | Membership in a union of Cartesian products, using bound-variable hypothesis for 𝐸 instead of distinct variable conditions as in opeliunxp2 5791. (Contributed by AV, 25-Oct-2020.) |
⊢ Ⅎ𝑥𝐸 & ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸) ⇒ ⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)) | ||
Theorem | mpoxeldm 8110* | 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 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) ⇒ ⊢ (𝑁 ∈ (𝑋𝐹𝑌) → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷)) | ||
Theorem | mpoxneldm 8111* | 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 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) ⇒ ⊢ ((𝑋 ∉ 𝐶 ∨ 𝑌 ∉ ⦋𝑋 / 𝑥⦌𝐷) → (𝑋𝐹𝑌) = ∅) | ||
Theorem | mpoxopn0yelv 8112* | 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 ∈ (1st ‘𝑥) ↦ 𝐶) ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → 𝐾 ∈ 𝑉)) | ||
Theorem | mpoxopynvov0g 8113* | 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 ∈ (1st ‘𝑥) ↦ 𝐶) ⇒ ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∉ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅) | ||
Theorem | mpoxopxnop0 8114* | 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 ∈ (1st ‘𝑥) ↦ 𝐶) ⇒ ⊢ (¬ 𝑉 ∈ (V × V) → (𝑉𝐹𝐾) = ∅) | ||
Theorem | mpoxopx0ov0 8115* | 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 ∈ (1st ‘𝑥) ↦ 𝐶) ⇒ ⊢ (∅𝐹𝐾) = ∅ | ||
Theorem | mpoxopxprcov0 8116* | 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 ∈ (1st ‘𝑥) ↦ 𝐶) ⇒ ⊢ (¬ (𝑉 ∈ V ∧ 𝑊 ∈ V) → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅) | ||
Theorem | mpoxopynvov0 8117* | 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 ∈ (1st ‘𝑥) ↦ 𝐶) ⇒ ⊢ (𝐾 ∉ 𝑉 → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅) | ||
Theorem | mpoxopoveq 8118* | 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 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑}) ⇒ ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}) | ||
Theorem | mpoxopovel 8119* | 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 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑}) ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))) | ||
Theorem | mpoxopoveqd 8120* | 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 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑}) & ⊢ (𝜓 → (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) & ⊢ ((𝜓 ∧ ¬ 𝐾 ∈ 𝑉) → {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} = ∅) ⇒ ⊢ (𝜓 → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}) | ||
Theorem | brovex 8121* | 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))) | ||
Theorem | brovmpoex 8122* | 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))) | ||
Theorem | sprmpod 8123* | 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)𝑦 ∧ 𝜒)}) & ⊢ ((𝜑 ∧ 𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜒 ↔ 𝜓)) & ⊢ (𝜑 → (𝑉 ∈ V ∧ 𝐸 ∈ V)) & ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥(𝑉𝑅𝐸)𝑦 → 𝜃)) & ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V) ⇒ ⊢ (𝜑 → (𝑉𝑀𝐸) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)}) | ||
Syntax | ctpos 8124 | The transposition of a function. |
class tpos 𝐹 | ||
Definition | df-tpos 8125* | 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})) | ||
Theorem | tposss 8126 | Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺) | ||
Theorem | tposeq 8127 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺) | ||
Theorem | tposeqd 8128 | Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.) |
⊢ (𝜑 → 𝐹 = 𝐺) ⇒ ⊢ (𝜑 → tpos 𝐹 = tpos 𝐺) | ||
Theorem | tposssxp 8129 | The transposition is a subset of a Cartesian product. (Contributed by Mario Carneiro, 12-Jan-2017.) |
⊢ tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) | ||
Theorem | reltpos 8130 | The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ Rel tpos 𝐹 | ||
Theorem | brtpos2 8131 | Value of the transposition at an ordered pair ⟨𝐴, 𝐵⟩. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵))) | ||
Theorem | brtpos0 8132 | The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). This allows to eliminate sethood hypotheses on 𝐴, 𝐵 in brtpos 8134. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴)) | ||
Theorem | reldmtpos 8133 | Necessary and sufficient condition for dom tpos 𝐹 to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹) | ||
Theorem | brtpos 8134 | The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐶 ∈ 𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶)) | ||
Theorem | ottpos 8135 | The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ (𝐶 ∈ 𝑉 → (⟨𝐴, 𝐵, 𝐶⟩ ∈ tpos 𝐹 ↔ ⟨𝐵, 𝐴, 𝐶⟩ ∈ 𝐹)) | ||
Theorem | relbrtpos 8136 | The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 3-Nov-2015.) |
⊢ (Rel 𝐹 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶)) | ||
Theorem | dmtpos 8137 | The domain of tpos 𝐹 when dom 𝐹 is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹) | ||
Theorem | rntpos 8138 | The range of tpos 𝐹 when dom 𝐹 is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹) | ||
Theorem | tposexg 8139 | The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐹 ∈ 𝑉 → tpos 𝐹 ∈ V) | ||
Theorem | ovtpos 8140 | The transposition swaps the arguments in a two-argument function. When 𝐹 is a matrix, which is to say a function from (1...𝑚) × (1...𝑛) to ℝ or some ring, tpos 𝐹 is the transposition of 𝐹, which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴) | ||
Theorem | tposfun 8141 | The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (Fun 𝐹 → Fun tpos 𝐹) | ||
Theorem | dftpos2 8142* | Alternate definition of tpos when 𝐹 has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}))) | ||
Theorem | dftpos3 8143* | Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 5639. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (Rel dom 𝐹 → tpos 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥⟩𝐹𝑧}) | ||
Theorem | dftpos4 8144* | Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ ∪ ◡{𝑥})) | ||
Theorem | tpostpos 8145 | Value of the double transposition for a general class 𝐹. (Contributed by Mario Carneiro, 16-Sep-2015.) |
⊢ tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V)) | ||
Theorem | tpostpos2 8146 | Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.) |
⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → tpos tpos 𝐹 = 𝐹) | ||
Theorem | tposfn2 8147 | The domain of a transposition. (Contributed by NM, 10-Sep-2015.) |
⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ◡𝐴)) | ||
Theorem | tposfo2 8148 | Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.) |
⊢ (Rel 𝐴 → (𝐹:𝐴–onto→𝐵 → tpos 𝐹:◡𝐴–onto→𝐵)) | ||
Theorem | tposf2 8149 | The domain and codomain of a transposition. (Contributed by NM, 10-Sep-2015.) |
⊢ (Rel 𝐴 → (𝐹:𝐴⟶𝐵 → tpos 𝐹:◡𝐴⟶𝐵)) | ||
Theorem | tposf12 8150 | Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.) |
⊢ (Rel 𝐴 → (𝐹:𝐴–1-1→𝐵 → tpos 𝐹:◡𝐴–1-1→𝐵)) | ||
Theorem | tposf1o2 8151 | Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.) |
⊢ (Rel 𝐴 → (𝐹:𝐴–1-1-onto→𝐵 → tpos 𝐹:◡𝐴–1-1-onto→𝐵)) | ||
Theorem | tposfo 8152 | The domain and codomain/range of a transposition. (Contributed by NM, 10-Sep-2015.) |
⊢ (𝐹:(𝐴 × 𝐵)–onto→𝐶 → tpos 𝐹:(𝐵 × 𝐴)–onto→𝐶) | ||
Theorem | tposf 8153 | The domain and codomain of a transposition. (Contributed by NM, 10-Sep-2015.) |
⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → tpos 𝐹:(𝐵 × 𝐴)⟶𝐶) | ||
Theorem | tposfn 8154 | Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝐹 Fn (𝐴 × 𝐵) → tpos 𝐹 Fn (𝐵 × 𝐴)) | ||
Theorem | tpos0 8155 | Transposition of the empty set. (Contributed by NM, 10-Sep-2015.) |
⊢ tpos ∅ = ∅ | ||
Theorem | tposco 8156 | Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ tpos (𝐹 ∘ 𝐺) = (𝐹 ∘ tpos 𝐺) | ||
Theorem | tpossym 8157* | Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥))) | ||
Theorem | tposeqi 8158 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ 𝐹 = 𝐺 ⇒ ⊢ tpos 𝐹 = tpos 𝐺 | ||
Theorem | tposex 8159 | A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ 𝐹 ∈ V ⇒ ⊢ tpos 𝐹 ∈ V | ||
Theorem | nftpos 8160 | Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ Ⅎ𝑥𝐹 ⇒ ⊢ Ⅎ𝑥tpos 𝐹 | ||
Theorem | tposoprab 8161* | Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⇒ ⊢ tpos 𝐹 = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑} | ||
Theorem | tposmpo 8162* | Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ tpos 𝐹 = (𝑦 ∈ 𝐵, 𝑥 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | tposconst 8163 | The transposition of a constant operation using the relation representation. (Contributed by SO, 11-Jul-2018.) |
⊢ tpos ((𝐴 × 𝐵) × {𝐶}) = ((𝐵 × 𝐴) × {𝐶}) | ||
Syntax | ccur 8164 | Extend class notation to include the currying function. |
class curry 𝐴 | ||
Syntax | cunc 8165 | Extend class notation to include the uncurrying function. |
class uncurry 𝐴 | ||
Definition | df-cur 8166* | Define the currying of 𝐹, which splits a function of two arguments into a function of the first argument, producing a function over the second argument. (Contributed by Mario Carneiro, 7-Jan-2017.) |
⊢ curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) | ||
Definition | df-unc 8167* | Define the uncurrying of 𝐹, which takes a function producing functions, and transforms it into a two-argument function. (Contributed by Mario Carneiro, 7-Jan-2017.) |
⊢ uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧} | ||
Theorem | mpocurryd 8168* | The currying of an operation given in maps-to notation, splitting the operation (function of two arguments) into a function of the first argument, producing a function over the second argument. (Contributed by AV, 27-Oct-2019.) |
⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ≠ ∅) ⇒ ⊢ (𝜑 → curry 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶))) | ||
Theorem | mpocurryvald 8169* | The value of a curried operation given in maps-to notation is a function over the second argument of the original operation. (Contributed by AV, 27-Oct-2019.) |
⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ≠ ∅) & ⊢ (𝜑 → 𝑌 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) ⇒ ⊢ (𝜑 → (curry 𝐹‘𝐴) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | fvmpocurryd 8170* | The value of the value of a curried operation given in maps-to notation is the operation value of the original operation. (Contributed by AV, 27-Oct-2019.) |
⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑌) ⇒ ⊢ (𝜑 → ((curry 𝐹‘𝐴)‘𝐵) = (𝐴𝐹𝐵)) | ||
Syntax | cund 8171 | Extend class notation with undefined value function. |
class Undef | ||
Definition | df-undef 8172 | Define the undefined value function, whose value at set 𝑠 is guaranteed not to be a member of 𝑠 (see pwuninel 8174). (Contributed by NM, 15-Sep-2011.) |
⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠) | ||
Theorem | pwuninel2 8173 | Direct proof of pwuninel 8174 avoiding functions and thus several ZF axioms. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ (∪ 𝐴 ∈ 𝑉 → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴) | ||
Theorem | pwuninel 8174 | The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. See also pwuninel2 8173. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) |
⊢ ¬ 𝒫 ∪ 𝐴 ∈ 𝐴 | ||
Theorem | undefval 8175 | Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 8177 instead. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) = 𝒫 ∪ 𝑆) | ||
Theorem | undefnel2 8176 | The undefined value generated from a set is not a member of the set. (Contributed by NM, 15-Sep-2011.) |
⊢ (𝑆 ∈ 𝑉 → ¬ (Undef‘𝑆) ∈ 𝑆) | ||
Theorem | undefnel 8177 | The undefined value generated from a set is not a member of the set. (Contributed by NM, 15-Sep-2011.) |
⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) ∉ 𝑆) | ||
Theorem | undefne0 8178 | The undefined value generated from a set is not empty. (Contributed by NM, 3-Sep-2018.) |
⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) ≠ ∅) | ||
Syntax | cfrecs 8179 | Declare the syntax for the well-founded recursion generator. See df-frecs 8180. |
class frecs(𝑅, 𝐴, 𝐹) | ||
Definition | df-frecs 8180* | This is the definition for the well-founded recursion generator. Similar to df-wrecs 8211 and df-recs 8285, it is a direct definition form of normally recursive relationships. Unlike the former two definitions, it only requires a well-founded set-like relationship for its properties, not a well-ordered relationship. This proof requires either a partial order or the axiom of infinity. We develop the theorems twice, once with a partial order and once without. The second development occurs later in the database, after ax-inf 9508 has been introduced. (Contributed by Scott Fenton, 23-Dec-2021.) |
⊢ frecs(𝑅, 𝐴, 𝐹) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} | ||
Theorem | frecseq123 8181 | Equality theorem for the well-founded recursion generator. (Contributed by Scott Fenton, 23-Dec-2021.) |
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → frecs(𝑅, 𝐴, 𝐹) = frecs(𝑆, 𝐵, 𝐺)) | ||
Theorem | nffrecs 8182 | Bound-variable hypothesis builder for the well-founded recursion generator. (Contributed by Scott Fenton, 23-Dec-2021.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐹 ⇒ ⊢ Ⅎ𝑥frecs(𝑅, 𝐴, 𝐹) | ||
Theorem | csbfrecsg 8183 | Move class substitution in and out of the well-founded recursive function generator. (Contributed by Scott Fenton, 18-Nov-2024.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌frecs(𝑅, 𝐷, 𝐹) = frecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝐹)) | ||
Theorem | fpr3g 8184* | Functions defined by well-founded recursion over a partial order are identical up to relation, domain, and characteristic function. This version of frr3g 9626 does not require infinity. (Contributed by Scott Fenton, 24-Aug-2022.) |
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐹 = 𝐺) | ||
Theorem | frrlem1 8185* | Lemma for well-founded recursion. The final item we are interested in is the union of acceptable functions 𝐵. This lemma just changes bound variables for later use. (Contributed by Paul Chapman, 21-Apr-2012.) |
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ⇒ ⊢ 𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))} | ||
Theorem | frrlem2 8186* | Lemma for well-founded recursion. An acceptable function is a function. (Contributed by Paul Chapman, 21-Apr-2012.) |
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ⇒ ⊢ (𝑔 ∈ 𝐵 → Fun 𝑔) | ||
Theorem | frrlem3 8187* | Lemma for well-founded recursion. An acceptable function's domain is a subset of 𝐴. (Contributed by Paul Chapman, 21-Apr-2012.) |
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ⇒ ⊢ (𝑔 ∈ 𝐵 → dom 𝑔 ⊆ 𝐴) | ||
Theorem | frrlem4 8188* | Lemma for well-founded recursion. Properties of the restriction of an acceptable function to the domain of another acceptable function. (Contributed by Paul Chapman, 21-Apr-2012.) |
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ⇒ ⊢ ((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) → ((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) Fn (dom 𝑔 ∩ dom ℎ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom ℎ)((𝑔 ↾ (dom 𝑔 ∩ dom ℎ))‘𝑎) = (𝑎𝐺((𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ℎ), 𝑎))))) | ||
Theorem | frrlem5 8189* | Lemma for well-founded recursion. State the well-founded recursion generator in terms of the acceptable functions. (Contributed by Scott Fenton, 27-Aug-2022.) |
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} & ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ 𝐹 = ∪ 𝐵 | ||
Theorem | frrlem6 8190* | Lemma for well-founded recursion. The well-founded recursion generator is a relationship. (Contributed by Scott Fenton, 27-Aug-2022.) |
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} & ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ Rel 𝐹 | ||
Theorem | frrlem7 8191* | Lemma for well-founded recursion. The well-founded recursion generator's domain is a subclass of 𝐴. (Contributed by Scott Fenton, 27-Aug-2022.) |
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} & ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ dom 𝐹 ⊆ 𝐴 | ||
Theorem | frrlem8 8192* | Lemma for well-founded recursion. dom 𝐹 is closed under predecessor classes. (Contributed by Scott Fenton, 6-Dec-2022.) |
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} & ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ (𝑧 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹) | ||
Theorem | frrlem9 8193* | Lemma for well-founded recursion. Show that the well-founded recursive generator produces a function. Hypothesis three will be eliminated using different induction rules depending on if we use partial orders or the axiom of infinity. (Contributed by Scott Fenton, 27-Aug-2022.) |
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} & ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) & ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) ⇒ ⊢ (𝜑 → Fun 𝐹) | ||
Theorem | frrlem10 8194* | Lemma for well-founded recursion. Under the compatibility hypothesis, compute the value of 𝐹 within its domain. (Contributed by Scott Fenton, 6-Dec-2022.) |
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} & ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) & ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) ⇒ ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) | ||
Theorem | frrlem11 8195* | Lemma for well-founded recursion. For the next several theorems we will be aiming to prove that dom 𝐹 = 𝐴. To do this, we set up a function 𝐶 that supposedly contains an element of 𝐴 that is not in dom 𝐹 and we show that the element must be in dom 𝐹. Our choice of what to restrict 𝐹 to depends on if we assume partial orders or the axiom of infinity. To begin with, we establish the functionality of 𝐶. (Contributed by Scott Fenton, 7-Dec-2022.) |
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} & ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) & ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) & ⊢ 𝐶 = ((𝐹 ↾ 𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ⇒ ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) | ||
Theorem | frrlem12 8196* | Lemma for well-founded recursion. Next, we calculate the value of 𝐶. (Contributed by Scott Fenton, 7-Dec-2022.) |
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} & ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) & ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) & ⊢ 𝐶 = ((𝐹 ↾ 𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) & ⊢ (𝜑 → 𝑅 Fr 𝐴) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∀𝑤 ∈ 𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆) ⇒ ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) → (𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))) | ||
Theorem | frrlem13 8197* | Lemma for well-founded recursion. Assuming that 𝑆 is a subset of 𝐴 and that 𝑧 is 𝑅-minimal, then 𝐶 is an acceptable function. (Contributed by Scott Fenton, 7-Dec-2022.) |
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} & ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) & ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) & ⊢ 𝐶 = ((𝐹 ↾ 𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) & ⊢ (𝜑 → 𝑅 Fr 𝐴) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∀𝑤 ∈ 𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑆 ∈ V) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑆 ⊆ 𝐴) ⇒ ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 ∈ 𝐵) | ||
Theorem | frrlem14 8198* | Lemma for well-founded recursion. Finally, we tie all these threads together and show that dom 𝐹 = 𝐴 when given the right 𝑆. Specifically, we prove that there can be no 𝑅-minimal element of (𝐴 ∖ dom 𝐹). (Contributed by Scott Fenton, 7-Dec-2022.) |
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} & ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) & ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) & ⊢ 𝐶 = ((𝐹 ↾ 𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) & ⊢ (𝜑 → 𝑅 Fr 𝐴) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∀𝑤 ∈ 𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑆 ∈ V) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑆 ⊆ 𝐴) & ⊢ ((𝜑 ∧ (𝐴 ∖ dom 𝐹) ≠ ∅) → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) ⇒ ⊢ (𝜑 → dom 𝐹 = 𝐴) | ||
Theorem | fprlem1 8199* | Lemma for well-founded recursion with a partial order. Two acceptable functions are compatible. (Contributed by Scott Fenton, 11-Sep-2023.) |
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} & ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) | ||
Theorem | fprlem2 8200* | Lemma for well-founded recursion with a partial order. Establish a subset relationship. (Contributed by Scott Fenton, 11-Sep-2023.) |
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑧 ∈ 𝐴) → ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)Pred(𝑅, 𝐴, 𝑤) ⊆ Pred(𝑅, 𝐴, 𝑧)) |
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