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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | isfinite4 14401 | A finite set is equinumerous to the range of integers from one up to the hash value of the set. In other words, counting objects with natural numbers works if and only if it is a finite collection. (Contributed by Richard Penner, 26-Feb-2020.) |
| ⊢ (𝐴 ∈ Fin ↔ (1...(♯‘𝐴)) ≈ 𝐴) | ||
| Theorem | hasheq0 14402 | Two ways of saying a set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.) |
| ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) | ||
| Theorem | hashneq0 14403 | Two ways of saying a set is not empty. (Contributed by Alexander van der Vekens, 23-Sep-2018.) |
| ⊢ (𝐴 ∈ 𝑉 → (0 < (♯‘𝐴) ↔ 𝐴 ≠ ∅)) | ||
| Theorem | hashgt0n0 14404 | If the size of a set is greater than 0, the set is not empty. (Contributed by AV, 5-Aug-2018.) (Proof shortened by AV, 18-Nov-2018.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 0 < (♯‘𝐴)) → 𝐴 ≠ ∅) | ||
| Theorem | hashnncl 14405 | Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.) |
| ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅)) | ||
| Theorem | hash0 14406 | The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.) |
| ⊢ (♯‘∅) = 0 | ||
| Theorem | hashelne0d 14407 | A set with an element has nonzero size. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
| ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → ¬ (♯‘𝐴) = 0) | ||
| Theorem | hashsng 14408 | The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.) |
| ⊢ (𝐴 ∈ 𝑉 → (♯‘{𝐴}) = 1) | ||
| Theorem | hashen1 14409 | A set has size 1 if and only if it is equinumerous to the ordinal 1. (Contributed by AV, 14-Apr-2019.) |
| ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 1 ↔ 𝐴 ≈ 1o)) | ||
| Theorem | hash1elsn 14410 | A set of size 1 with a known element is the singleton of that element. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
| ⊢ (𝜑 → (♯‘𝐴) = 1) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐴 = {𝐵}) | ||
| Theorem | hashrabrsn 14411* | The size of a restricted class abstraction restricted to a singleton is a nonnegative integer. (Contributed by Alexander van der Vekens, 22-Dec-2017.) |
| ⊢ (♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) ∈ ℕ0 | ||
| Theorem | hashrabsn01 14412* | The size of a restricted class abstraction restricted to a singleton is either 0 or 1. (Contributed by Alexander van der Vekens, 3-Sep-2018.) |
| ⊢ ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 𝑁 → (𝑁 = 0 ∨ 𝑁 = 1)) | ||
| Theorem | hashrabsn1 14413* | If the size of a restricted class abstraction restricted to a singleton is 1, the condition of the class abstraction must hold for the singleton. (Contributed by Alexander van der Vekens, 3-Sep-2018.) |
| ⊢ ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑) | ||
| Theorem | hashfn 14414 | A function is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.) |
| ⊢ (𝐹 Fn 𝐴 → (♯‘𝐹) = (♯‘𝐴)) | ||
| Theorem | fseq1hash 14415 | The value of the size function on a finite 1-based sequence. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 12-Mar-2015.) |
| ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁) | ||
| Theorem | hashgadd 14416 | 𝐺 maps ordinal addition to integer addition. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.) |
| ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ⇒ ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐺‘(𝐴 +o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵))) | ||
| Theorem | hashgval2 14417 | A short expression for the 𝐺 function of hashgf1o 14012. (Contributed by Mario Carneiro, 24-Jan-2015.) |
| ⊢ (♯ ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) | ||
| Theorem | hashdom 14418 | Dominance relation for the size function. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 22-Apr-2015.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵)) | ||
| Theorem | hashdomi 14419 | Non-strict order relation of the ♯ function on the full cardinal poset. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
| ⊢ (𝐴 ≼ 𝐵 → (♯‘𝐴) ≤ (♯‘𝐵)) | ||
| Theorem | hashsdom 14420 | Strict dominance relation for the size function. (Contributed by Mario Carneiro, 18-Aug-2014.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) < (♯‘𝐵) ↔ 𝐴 ≺ 𝐵)) | ||
| Theorem | hashun 14421 | The size of the union of disjoint finite sets is the sum of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵))) | ||
| Theorem | hashun2 14422 | The size of the union of finite sets is less than or equal to the sum of their sizes. (Contributed by Mario Carneiro, 23-Sep-2013.) (Proof shortened by Mario Carneiro, 27-Jul-2014.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) ≤ ((♯‘𝐴) + (♯‘𝐵))) | ||
| Theorem | hashun3 14423 | The size of the union of finite sets is the sum of their sizes minus the size of the intersection. (Contributed by Mario Carneiro, 6-Aug-2017.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) = (((♯‘𝐴) + (♯‘𝐵)) − (♯‘(𝐴 ∩ 𝐵)))) | ||
| Theorem | hashinfxadd 14424 | The extended real addition of the size of an infinite set with the size of an arbitrary set yields plus infinity. (Contributed by Alexander van der Vekens, 20-Dec-2017.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (♯‘𝐴) ∉ ℕ0) → ((♯‘𝐴) +𝑒 (♯‘𝐵)) = +∞) | ||
| Theorem | hashunx 14425 | The size of the union of disjoint sets is the result of the extended real addition of their sizes, analogous to hashun 14421. (Contributed by Alexander van der Vekens, 21-Dec-2017.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +𝑒 (♯‘𝐵))) | ||
| Theorem | hashge0 14426 | The cardinality of a set is greater than or equal to zero. (Contributed by Thierry Arnoux, 2-Mar-2017.) |
| ⊢ (𝐴 ∈ 𝑉 → 0 ≤ (♯‘𝐴)) | ||
| Theorem | hashgt0 14427 | The cardinality of a nonempty set is greater than zero. (Contributed by Thierry Arnoux, 2-Mar-2017.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → 0 < (♯‘𝐴)) | ||
| Theorem | hashge1 14428 | The cardinality of a nonempty set is greater than or equal to one. (Contributed by Thierry Arnoux, 20-Jun-2017.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → 1 ≤ (♯‘𝐴)) | ||
| Theorem | 1elfz0hash 14429 | 1 is an element of the finite set of sequential nonnegative integers bounded by the size of a nonempty finite set. (Contributed by AV, 9-May-2020.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → 1 ∈ (0...(♯‘𝐴))) | ||
| Theorem | hashnn0n0nn 14430 | If a nonnegative integer is the size of a set which contains at least one element, this integer is a positive integer. (Contributed by Alexander van der Vekens, 9-Jan-2018.) |
| ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑌 ∈ ℕ0) ∧ ((♯‘𝑉) = 𝑌 ∧ 𝑁 ∈ 𝑉)) → 𝑌 ∈ ℕ) | ||
| Theorem | hashunsng 14431 | The size of the union of a finite set with a disjoint singleton is one more than the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.) |
| ⊢ (𝐵 ∈ 𝑉 → ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ 𝐴) → (♯‘(𝐴 ∪ {𝐵})) = ((♯‘𝐴) + 1))) | ||
| Theorem | hashunsngx 14432 | The size of the union of a set with a disjoint singleton is the extended real addition of the size of the set and 1, analogous to hashunsng 14431. (Contributed by BTernaryTau, 9-Sep-2023.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝐵 ∈ 𝐴 → (♯‘(𝐴 ∪ {𝐵})) = ((♯‘𝐴) +𝑒 1))) | ||
| Theorem | hashunsnggt 14433 | The size of a set is greater than a nonnegative integer N if and only if the size of the union of that set with a disjoint singleton is greater than N + 1. (Contributed by BTernaryTau, 10-Sep-2023.) |
| ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑁 ∈ ℕ0) ∧ ¬ 𝐵 ∈ 𝐴) → (𝑁 < (♯‘𝐴) ↔ (𝑁 + 1) < (♯‘(𝐴 ∪ {𝐵})))) | ||
| Theorem | hashprg 14434 | The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) (Revised by AV, 18-Sep-2021.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≠ 𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2)) | ||
| Theorem | elprchashprn2 14435 | If one element of an unordered pair is not a set, the size of the unordered pair is not 2. (Contributed by Alexander van der Vekens, 7-Oct-2017.) |
| ⊢ (¬ 𝑀 ∈ V → ¬ (♯‘{𝑀, 𝑁}) = 2) | ||
| Theorem | hashprb 14436 | The size of an unordered pair is 2 if and only if its elements are different sets. (Contributed by Alexander van der Vekens, 17-Jan-2018.) |
| ⊢ ((𝑀 ∈ V ∧ 𝑁 ∈ V ∧ 𝑀 ≠ 𝑁) ↔ (♯‘{𝑀, 𝑁}) = 2) | ||
| Theorem | hashprdifel 14437 | The elements of an unordered pair of size 2 are different sets. (Contributed by AV, 27-Jan-2020.) |
| ⊢ 𝑆 = {𝐴, 𝐵} ⇒ ⊢ ((♯‘𝑆) = 2 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵)) | ||
| Theorem | prhash2ex 14438 | There is (at least) one set with two different elements: the unordered pair containing 0 and 1. In contrast to pr0hash2ex 14447, numbers are used instead of sets because their representation is shorter (and more comprehensive). (Contributed by AV, 29-Jan-2020.) |
| ⊢ (♯‘{0, 1}) = 2 | ||
| Theorem | hashle00 14439 | If the size of a set is less than or equal to zero, the set must be empty. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Proof shortened by AV, 24-Oct-2021.) |
| ⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) ≤ 0 ↔ 𝑉 = ∅)) | ||
| Theorem | hashgt0elex 14440* | If the size of a set is greater than zero, then the set must contain at least one element. (Contributed by Alexander van der Vekens, 6-Jan-2018.) |
| ⊢ ((𝑉 ∈ 𝑊 ∧ 0 < (♯‘𝑉)) → ∃𝑥 𝑥 ∈ 𝑉) | ||
| Theorem | hashgt0elexb 14441* | The size of a set is greater than zero if and only if the set contains at least one element. (Contributed by Alexander van der Vekens, 18-Jan-2018.) |
| ⊢ (𝑉 ∈ 𝑊 → (0 < (♯‘𝑉) ↔ ∃𝑥 𝑥 ∈ 𝑉)) | ||
| Theorem | hashp1i 14442 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
| ⊢ 𝐴 ∈ ω & ⊢ 𝐵 = suc 𝐴 & ⊢ (♯‘𝐴) = 𝑀 & ⊢ (𝑀 + 1) = 𝑁 ⇒ ⊢ (♯‘𝐵) = 𝑁 | ||
| Theorem | hash1 14443 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
| ⊢ (♯‘1o) = 1 | ||
| Theorem | hash2 14444 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
| ⊢ (♯‘2o) = 2 | ||
| Theorem | hash3 14445 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
| ⊢ (♯‘3o) = 3 | ||
| Theorem | hash4 14446 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
| ⊢ (♯‘4o) = 4 | ||
| Theorem | pr0hash2ex 14447 | There is (at least) one set with two different elements: the unordered pair containing the empty set and the singleton containing the empty set. (Contributed by AV, 29-Jan-2020.) |
| ⊢ (♯‘{∅, {∅}}) = 2 | ||
| Theorem | hashss 14448 | The size of a subset is less than or equal to the size of its superset. (Contributed by Alexander van der Vekens, 14-Jul-2018.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → (♯‘𝐵) ≤ (♯‘𝐴)) | ||
| Theorem | prsshashgt1 14449 | The size of a superset of a proper unordered pair is greater than 1. (Contributed by AV, 6-Feb-2021.) |
| ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐶 ∈ 𝑈) → ({𝐴, 𝐵} ⊆ 𝐶 → 2 ≤ (♯‘𝐶))) | ||
| Theorem | hashin 14450 | The size of the intersection of a set and a class is less than or equal to the size of the set. (Contributed by AV, 4-Jan-2021.) |
| ⊢ (𝐴 ∈ 𝑉 → (♯‘(𝐴 ∩ 𝐵)) ≤ (♯‘𝐴)) | ||
| Theorem | hashssdif 14451 | The size of the difference of a finite set and a subset is the set's size minus the subset's. (Contributed by Steve Rodriguez, 24-Oct-2015.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (♯‘(𝐴 ∖ 𝐵)) = ((♯‘𝐴) − (♯‘𝐵))) | ||
| Theorem | hashdif 14452 | The size of the difference of a finite set and another set is the first set's size minus that of the intersection of both. (Contributed by Steve Rodriguez, 24-Oct-2015.) |
| ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ∖ 𝐵)) = ((♯‘𝐴) − (♯‘(𝐴 ∩ 𝐵)))) | ||
| Theorem | hashdifsn 14453 | The size of the difference of a finite set and a singleton subset is the set's size minus 1. (Contributed by Alexander van der Vekens, 6-Jan-2018.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) → (♯‘(𝐴 ∖ {𝐵})) = ((♯‘𝐴) − 1)) | ||
| Theorem | hashdifpr 14454 | The size of the difference of a finite set and a proper pair of its elements is the set's size minus 2. (Contributed by AV, 16-Dec-2020.) |
| ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (♯‘(𝐴 ∖ {𝐵, 𝐶})) = ((♯‘𝐴) − 2)) | ||
| Theorem | hashsn01 14455 | The size of a singleton is either 0 or 1. (Contributed by AV, 23-Feb-2021.) |
| ⊢ ((♯‘{𝐴}) = 0 ∨ (♯‘{𝐴}) = 1) | ||
| Theorem | hashsnle1 14456 | The size of a singleton is less than or equal to 1. (Contributed by AV, 23-Feb-2021.) |
| ⊢ (♯‘{𝐴}) ≤ 1 | ||
| Theorem | hashsnlei 14457 | Get an upper bound on a concretely specified finite set. Base case: singleton set. (Contributed by Mario Carneiro, 11-Feb-2015.) (Proof shortened by AV, 23-Feb-2021.) |
| ⊢ ({𝐴} ∈ Fin ∧ (♯‘{𝐴}) ≤ 1) | ||
| Theorem | hash1snb 14458* | The size of a set is 1 if and only if it is a singleton (containing a set). (Contributed by Alexander van der Vekens, 7-Dec-2017.) |
| ⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 1 ↔ ∃𝑎 𝑉 = {𝑎})) | ||
| Theorem | euhash1 14459* | The size of a set is 1 in terms of existential uniqueness. (Contributed by Alexander van der Vekens, 8-Feb-2018.) |
| ⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 1 ↔ ∃!𝑎 𝑎 ∈ 𝑉)) | ||
| Theorem | hash1n0 14460 | If the size of a set is 1 the set is not empty. (Contributed by AV, 23-Dec-2020.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) = 1) → 𝐴 ≠ ∅) | ||
| Theorem | hashgt12el 14461* | In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.) |
| ⊢ ((𝑉 ∈ 𝑊 ∧ 1 < (♯‘𝑉)) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑎 ≠ 𝑏) | ||
| Theorem | hashgt12el2 14462* | In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.) |
| ⊢ ((𝑉 ∈ 𝑊 ∧ 1 < (♯‘𝑉) ∧ 𝐴 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 𝐴 ≠ 𝑏) | ||
| Theorem | hashgt23el 14463* | A set with more than two elements has at least three different elements. (Contributed by BTernaryTau, 21-Sep-2023.) |
| ⊢ ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) | ||
| Theorem | hashunlei 14464 | Get an upper bound on a concretely specified finite set. Induction step: union of two finite bounded sets. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ 𝐶 = (𝐴 ∪ 𝐵) & ⊢ (𝐴 ∈ Fin ∧ (♯‘𝐴) ≤ 𝐾) & ⊢ (𝐵 ∈ Fin ∧ (♯‘𝐵) ≤ 𝑀) & ⊢ 𝐾 ∈ ℕ0 & ⊢ 𝑀 ∈ ℕ0 & ⊢ (𝐾 + 𝑀) = 𝑁 ⇒ ⊢ (𝐶 ∈ Fin ∧ (♯‘𝐶) ≤ 𝑁) | ||
| Theorem | hashsslei 14465 | Get an upper bound on a concretely specified finite set. Transfer boundedness to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ 𝐵 ⊆ 𝐴 & ⊢ (𝐴 ∈ Fin ∧ (♯‘𝐴) ≤ 𝑁) & ⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ (𝐵 ∈ Fin ∧ (♯‘𝐵) ≤ 𝑁) | ||
| Theorem | hashfz 14466 | Value of the numeric cardinality of a nonempty integer range. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Proof shortened by Mario Carneiro, 15-Apr-2015.) |
| ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘(𝐴...𝐵)) = ((𝐵 − 𝐴) + 1)) | ||
| Theorem | fzsdom2 14467 | Condition for finite ranges to have a strict dominance relation. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2015.) |
| ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐴...𝐵) ≺ (𝐴...𝐶)) | ||
| Theorem | hashfzo 14468 | Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
| ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘(𝐴..^𝐵)) = (𝐵 − 𝐴)) | ||
| Theorem | hashfzo0 14469 | Cardinality of a half-open set of integers based at zero. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
| ⊢ (𝐵 ∈ ℕ0 → (♯‘(0..^𝐵)) = 𝐵) | ||
| Theorem | hashfzp1 14470 | Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.) |
| ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴)) | ||
| Theorem | hashfz0 14471 | Value of the numeric cardinality of a nonempty range of nonnegative integers. (Contributed by Alexander van der Vekens, 21-Jul-2018.) |
| ⊢ (𝐵 ∈ ℕ0 → (♯‘(0...𝐵)) = (𝐵 + 1)) | ||
| Theorem | hashxplem 14472 | Lemma for hashxp 14473. (Contributed by Paul Chapman, 30-Nov-2012.) |
| ⊢ 𝐵 ∈ Fin ⇒ ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵))) | ||
| Theorem | hashxp 14473 | The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵))) | ||
| Theorem | hashmap 14474 | The size of the set exponential of two finite sets is the exponential of their sizes. (This is the original motivation behind the notation for set exponentiation.) (Contributed by Mario Carneiro, 5-Aug-2014.) (Proof shortened by AV, 18-Jul-2022.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ↑m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵))) | ||
| Theorem | hashpw 14475 | The size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.) (Proof shortened by Mario Carneiro, 5-Aug-2014.) |
| ⊢ (𝐴 ∈ Fin → (♯‘𝒫 𝐴) = (2↑(♯‘𝐴))) | ||
| Theorem | hashfun 14476 | A finite set is a function iff it is equinumerous to its domain. (Contributed by Mario Carneiro, 26-Sep-2013.) (Revised by Mario Carneiro, 12-Mar-2015.) |
| ⊢ (𝐹 ∈ Fin → (Fun 𝐹 ↔ (♯‘𝐹) = (♯‘dom 𝐹))) | ||
| Theorem | hashres 14477 | The number of elements of a finite function restricted to a subset of its domain is equal to the number of elements of that subset. (Contributed by AV, 15-Dec-2021.) |
| ⊢ ((Fun 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴) → (♯‘(𝐴 ↾ 𝐵)) = (♯‘𝐵)) | ||
| Theorem | hashreshashfun 14478 | The number of elements of a finite function expressed by a restriction. (Contributed by AV, 15-Dec-2021.) |
| ⊢ ((Fun 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴) → (♯‘𝐴) = ((♯‘(𝐴 ↾ 𝐵)) + (♯‘(dom 𝐴 ∖ 𝐵)))) | ||
| Theorem | hashimarn 14479 | The size of the image of a one-to-one function 𝐸 under the range of a function 𝐹 which is a one-to-one function into the domain of 𝐸 equals the size of the function 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.) (Proof shortened by AV, 4-May-2021.) |
| ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → (♯‘(𝐸 “ ran 𝐹)) = (♯‘𝐹))) | ||
| Theorem | hashimarni 14480 | If the size of the image of a one-to-one function 𝐸 under the range of a function 𝐹 which is a one-to-one function into the domain of 𝐸 is a nonnegative integer, the size of the function 𝐹 is the same nonnegative integer. (Contributed by Alexander van der Vekens, 4-Feb-2018.) |
| ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ 𝑃 = (𝐸 “ ran 𝐹) ∧ (♯‘𝑃) = 𝑁) → (♯‘𝐹) = 𝑁)) | ||
| Theorem | hashfundm 14481 | The size of a set function is equal to the size of its domain. (Contributed by BTernaryTau, 30-Sep-2023.) |
| ⊢ ((𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (♯‘𝐹) = (♯‘dom 𝐹)) | ||
| Theorem | hashf1dmrn 14482 | The size of the domain of a one-to-one set function is equal to the size of its range. (Contributed by BTernaryTau, 1-Oct-2023.) |
| ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (♯‘𝐴) = (♯‘ran 𝐹)) | ||
| Theorem | hashf1dmcdm 14483 | The size of the domain of a one-to-one set function is less than or equal to the size of its codomain, if it exists. (Contributed by BTernaryTau, 1-Oct-2023.) |
| ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → (♯‘𝐴) ≤ (♯‘𝐵)) | ||
| Theorem | resunimafz0 14484 | TODO-AV: Revise using 𝐹 ∈ Word dom 𝐼? Formerly part of proof of eupth2lem3 30255: The union of a restriction by an image over an open range of nonnegative integers and a singleton of an ordered pair is a restriction by an image over an interval of nonnegative integers. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.) |
| ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) ⇒ ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉})) | ||
| Theorem | fnfz0hash 14485 | The size of a function on a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 25-Jun-2018.) |
| ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 Fn (0...𝑁)) → (♯‘𝐹) = (𝑁 + 1)) | ||
| Theorem | ffz0hash 14486 | The size of a function on a finite set of sequential nonnegative integers equals the upper bound of the sequence increased by 1. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Proof shortened by AV, 11-Apr-2021.) |
| ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(0...𝑁)⟶𝐵) → (♯‘𝐹) = (𝑁 + 1)) | ||
| Theorem | fnfz0hashnn0 14487 | The size of a function on a finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by AV, 10-Apr-2021.) |
| ⊢ (𝐹 Fn (0...𝑁) → (♯‘𝐹) ∈ ℕ0) | ||
| Theorem | ffzo0hash 14488 | The size of a function on a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 25-Mar-2018.) |
| ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 Fn (0..^𝑁)) → (♯‘𝐹) = 𝑁) | ||
| Theorem | fnfzo0hash 14489 | The size of a function on a half-open range of nonnegative integers equals the upper bound of this range. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Proof shortened by AV, 11-Apr-2021.) |
| ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(0..^𝑁)⟶𝐵) → (♯‘𝐹) = 𝑁) | ||
| Theorem | fnfzo0hashnn0 14490 | The value of the size function on a half-open range of nonnegative integers is a nonnegative integer. (Contributed by AV, 10-Apr-2021.) |
| ⊢ (𝐹 Fn (0..^𝑁) → (♯‘𝐹) ∈ ℕ0) | ||
| Theorem | hashbclem 14491* | Lemma for hashbc 14492: inductive step. (Contributed by Mario Carneiro, 13-Jul-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑗 ∈ ℤ ((♯‘𝐴)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗})) & ⊢ (𝜑 → 𝐾 ∈ ℤ) ⇒ ⊢ (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾})) | ||
| Theorem | hashbc 14492* | The binomial coefficient counts the number of subsets of a finite set of a given size. This is Metamath 100 proof #58 (formula for the number of combinations). (Contributed by Mario Carneiro, 13-Jul-2014.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐾 ∈ ℤ) → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾})) | ||
| Theorem | hashfacen 14493* | The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.) (Proof shortened by AV, 7-Aug-2024.) |
| ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ≈ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷}) | ||
| Theorem | hashf1lem1 14494* | Lemma for hashf1 14496. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 14-Aug-2024.) |
| ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴) & ⊢ (𝜑 → ((♯‘𝐴) + 1) ≤ (♯‘𝐵)) & ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵) ⇒ ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ≈ (𝐵 ∖ ran 𝐹)) | ||
| Theorem | hashf1lem2 14495* | Lemma for hashf1 14496. (Contributed by Mario Carneiro, 17-Apr-2015.) |
| ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴) & ⊢ (𝜑 → ((♯‘𝐴) + 1) ≤ (♯‘𝐵)) ⇒ ⊢ (𝜑 → (♯‘{𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵}) = (((♯‘𝐵) − (♯‘𝐴)) · (♯‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵}))) | ||
| Theorem | hashf1 14496* | The permutation number ∣ 𝐴 ∣ ! · ( ∣ 𝐵 ∣ C ∣ 𝐴 ∣ ) = ∣ 𝐵 ∣ ! / ( ∣ 𝐵 ∣ − ∣ 𝐴 ∣ )! counts the number of injections from 𝐴 to 𝐵. (Contributed by Mario Carneiro, 21-Jan-2015.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘{𝑓 ∣ 𝑓:𝐴–1-1→𝐵}) = ((!‘(♯‘𝐴)) · ((♯‘𝐵)C(♯‘𝐴)))) | ||
| Theorem | hashfac 14497* | A factorial counts the number of bijections on a finite set. (Contributed by Mario Carneiro, 21-Jan-2015.) (Proof shortened by Mario Carneiro, 17-Apr-2015.) |
| ⊢ (𝐴 ∈ Fin → (♯‘{𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴}) = (!‘(♯‘𝐴))) | ||
| Theorem | leiso 14498 | Two ways to write a strictly increasing function on the reals. (Contributed by Mario Carneiro, 9-Sep-2015.) |
| ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵))) | ||
| Theorem | leisorel 14499 | Version of isorel 7346 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.) |
| ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ≤ 𝐷 ↔ (𝐹‘𝐶) ≤ (𝐹‘𝐷))) | ||
| Theorem | fz1isolem 14500* | Lemma for fz1iso 14501. (Contributed by Mario Carneiro, 2-Apr-2014.) |
| ⊢ 𝐺 = (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) & ⊢ 𝐵 = (ℕ ∩ (◡ < “ {((♯‘𝐴) + 1)})) & ⊢ 𝐶 = (ω ∩ (◡𝐺‘((♯‘𝐴) + 1))) & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) ⇒ ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , 𝑅 ((1...(♯‘𝐴)), 𝐴)) | ||
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