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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | focdmex 13701 | The codomain of an onto function is a set if its domain is a set. (Contributed by AV, 4-May-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ V) | ||
Theorem | hasheqf1oi 13702* | The size of two sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Alexander van der Vekens, 25-Dec-2017.) (Revised by AV, 4-May-2021.) |
⊢ (𝐴 ∈ 𝑉 → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵))) | ||
Theorem | hashf1rn 13703 | The size of a finite set which is a one-to-one function is equal to the size of the function's range. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 4-May-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (♯‘𝐹) = (♯‘ran 𝐹)) | ||
Theorem | hasheqf1od 13704 | The size of two sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by AV, 4-May-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) ⇒ ⊢ (𝜑 → (♯‘𝐴) = (♯‘𝐵)) | ||
Theorem | fz1eqb 13705 | Two possibly-empty 1-based finite sets of sequential integers are equal iff their endpoints are equal. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 29-Mar-2014.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((1...𝑀) = (1...𝑁) ↔ 𝑀 = 𝑁)) | ||
Theorem | hashcard 13706 | The size function of the cardinality function. (Contributed by Mario Carneiro, 19-Sep-2013.) (Revised by Mario Carneiro, 4-Nov-2013.) |
⊢ (𝐴 ∈ Fin → (♯‘(card‘𝐴)) = (♯‘𝐴)) | ||
Theorem | hashcl 13707 | Closure of the ♯ function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.) |
⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | ||
Theorem | hashxrcl 13708 | Extended real closure of the ♯ function. (Contributed by Mario Carneiro, 22-Apr-2015.) |
⊢ (𝐴 ∈ 𝑉 → (♯‘𝐴) ∈ ℝ*) | ||
Theorem | hashclb 13709 | Reverse closure of the ♯ function. (Contributed by Mario Carneiro, 15-Jan-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ (♯‘𝐴) ∈ ℕ0)) | ||
Theorem | nfile 13710 | The size of any infinite set is always greater than or equal to the size of any set. (Contributed by AV, 13-Nov-2020.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) ≤ (♯‘𝐵)) | ||
Theorem | hashvnfin 13711 | A set of finite size is a finite set. (Contributed by Alexander van der Vekens, 8-Dec-2017.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ((♯‘𝑆) = 𝑁 → 𝑆 ∈ Fin)) | ||
Theorem | hashnfinnn0 13712 | The size of an infinite set is not a nonnegative integer. (Contributed by Alexander van der Vekens, 21-Dec-2017.) (Proof shortened by Alexander van der Vekens, 18-Jan-2018.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) ∉ ℕ0) | ||
Theorem | isfinite4 13713 | 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 13714 | Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.) |
⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) | ||
Theorem | hashneq0 13715 | 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 13716 | 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 13717 | Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.) |
⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅)) | ||
Theorem | hash0 13718 | The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.) |
⊢ (♯‘∅) = 0 | ||
Theorem | hashelne0d 13719 | 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 13720 | 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 13721 | 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 13722 | 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 13723* | 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 13724* | 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 13725* | 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 13726 | A function is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.) |
⊢ (𝐹 Fn 𝐴 → (♯‘𝐹) = (♯‘𝐴)) | ||
Theorem | fseq1hash 13727 | 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 13728 | 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 13729 | A short expression for the 𝐺 function of hashgf1o 13329. (Contributed by Mario Carneiro, 24-Jan-2015.) |
⊢ (♯ ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) | ||
Theorem | hashdom 13730 | 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 13731 | 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 13732 | Strict dominance relation for the size function. (Contributed by Mario Carneiro, 18-Aug-2014.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) < (♯‘𝐵) ↔ 𝐴 ≺ 𝐵)) | ||
Theorem | hashun 13733 | 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 13734 | 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 13735 | 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 13736 | 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 13737 | The size of the union of disjoint sets is the result of the extended real addition of their sizes, analogous to hashun 13733. (Contributed by Alexander van der Vekens, 21-Dec-2017.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +𝑒 (♯‘𝐵))) | ||
Theorem | hashge0 13738 | 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 13739 | The cardinality of a nonempty set is greater than zero. (Contributed by Thierry Arnoux, 2-Mar-2017.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → 0 < (♯‘𝐴)) | ||
Theorem | hashge1 13740 | 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 13741 | 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 13742 | 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 13743 | 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 13744 | 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 13743. (Contributed by BTernaryTau, 9-Sep-2023.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝐵 ∈ 𝐴 → (♯‘(𝐴 ∪ {𝐵})) = ((♯‘𝐴) +𝑒 1))) | ||
Theorem | hashunsnggt 13745 | 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 13746 | 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 13747 | 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 13748 | 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 13749 | 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 13750 | There is (at least) one set with two different elements: the unordered pair containing 0 and 1. In contrast to pr0hash2ex 13759, 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 13751 | 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 13752* | 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 13753* | 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 13754 | 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 13755 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ (♯‘1o) = 1 | ||
Theorem | hash2 13756 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ (♯‘2o) = 2 | ||
Theorem | hash3 13757 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ (♯‘3o) = 3 | ||
Theorem | hash4 13758 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ (♯‘4o) = 4 | ||
Theorem | pr0hash2ex 13759 | 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 13760 | 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 13761 | 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 13762 | 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 13763 | 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 13764 | 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 13765 | 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 13766 | The size of the difference of a finite set and a proper ordered pair subset 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 13767 | The size of a singleton is either 0 or 1. (Contributed by AV, 23-Feb-2021.) |
⊢ ((♯‘{𝐴}) = 0 ∨ (♯‘{𝐴}) = 1) | ||
Theorem | hashsnle1 13768 | The size of a singleton is less than or equal to 1. (Contributed by AV, 23-Feb-2021.) |
⊢ (♯‘{𝐴}) ≤ 1 | ||
Theorem | hashsnlei 13769 | 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 13770* | 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 13771* | 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 13772 | 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 13773* | 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 13774* | 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 13775* | 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 13776 | 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 13777 | 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 13778 | 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 13779 | 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 13780 | 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 13781 | 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 13782 | 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 13783 | 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 13784 | Lemma for hashxp 13785. (Contributed by Paul Chapman, 30-Nov-2012.) |
⊢ 𝐵 ∈ Fin ⇒ ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵))) | ||
Theorem | hashxp 13785 | 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 13786 | 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 13787 | 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 13788 | 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 13789 | 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 13790 | 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 13791 | 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 13792 | 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 | resunimafz0 13793 | TODO-AV: Revise using 𝐹 ∈ Word dom 𝐼? Formerly part of proof of eupth2lem3 27943: 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 13794 | 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 13795 | 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 13796 | 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 13797 | 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 13798 | 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 13799 | 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 13800* | Lemma for hashbc 13801: 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})) |
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