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Theorem List for Metamath Proof Explorer - 13701-13800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfocdmex 13701 The codomain of an onto function is a set if its domain is a set. (Contributed by AV, 4-May-2021.)
((𝐴𝑉𝐹:𝐴onto𝐵) → 𝐵 ∈ V)
 
Theoremhasheqf1oi 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.)
(𝐴𝑉 → (∃𝑓 𝑓:𝐴1-1-onto𝐵 → (♯‘𝐴) = (♯‘𝐵)))
 
Theoremhashf1rn 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.)
((𝐴𝑉𝐹:𝐴1-1𝐵) → (♯‘𝐹) = (♯‘ran 𝐹))
 
Theoremhasheqf1od 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.)
(𝜑𝐴𝑈)    &   (𝜑𝐹:𝐴1-1-onto𝐵)       (𝜑 → (♯‘𝐴) = (♯‘𝐵))
 
Theoremfz1eqb 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 ∈ ℕ0𝑁 ∈ ℕ0) → ((1...𝑀) = (1...𝑁) ↔ 𝑀 = 𝑁))
 
Theoremhashcard 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))
 
Theoremhashcl 13707 Closure of the function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.)
(𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
 
Theoremhashxrcl 13708 Extended real closure of the function. (Contributed by Mario Carneiro, 22-Apr-2015.)
(𝐴𝑉 → (♯‘𝐴) ∈ ℝ*)
 
Theoremhashclb 13709 Reverse closure of the function. (Contributed by Mario Carneiro, 15-Jan-2015.)
(𝐴𝑉 → (𝐴 ∈ Fin ↔ (♯‘𝐴) ∈ ℕ0))
 
Theoremnfile 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.)
((𝐴𝑉𝐵𝑊 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) ≤ (♯‘𝐵))
 
Theoremhashvnfin 13711 A set of finite size is a finite set. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
((𝑆𝑉𝑁 ∈ ℕ0) → ((♯‘𝑆) = 𝑁𝑆 ∈ Fin))
 
Theoremhashnfinnn0 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.)
((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) ∉ ℕ0)
 
Theoremisfinite4 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)
 
Theoremhasheq0 13714 Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.)
(𝐴𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
 
Theoremhashneq0 13715 Two ways of saying a set is not empty. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
(𝐴𝑉 → (0 < (♯‘𝐴) ↔ 𝐴 ≠ ∅))
 
Theoremhashgt0n0 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.)
((𝐴𝑉 ∧ 0 < (♯‘𝐴)) → 𝐴 ≠ ∅)
 
Theoremhashnncl 13717 Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)
(𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅))
 
Theoremhash0 13718 The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)
(♯‘∅) = 0
 
Theoremhashelne0d 13719 A set with an element has nonzero size. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑𝐵𝐴)    &   (𝜑𝐴𝑉)       (𝜑 → ¬ (♯‘𝐴) = 0)
 
Theoremhashsng 13720 The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
(𝐴𝑉 → (♯‘{𝐴}) = 1)
 
Theoremhashen1 13721 A set has size 1 if and only if it is equinumerous to the ordinal 1. (Contributed by AV, 14-Apr-2019.)
(𝐴𝑉 → ((♯‘𝐴) = 1 ↔ 𝐴 ≈ 1o))
 
Theoremhash1elsn 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)    &   (𝜑𝐵𝐴)    &   (𝜑𝐴𝑉)       (𝜑𝐴 = {𝐵})
 
Theoremhashrabrsn 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
 
Theoremhashrabsn01 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))
 
Theoremhashrabsn1 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)
 
Theoremhashfn 13726 A function is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.)
(𝐹 Fn 𝐴 → (♯‘𝐹) = (♯‘𝐴))
 
Theoremfseq1hash 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 ∈ ℕ0𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁)
 
Theoremhashgadd 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)) = ((𝐺𝐴) + (𝐺𝐵)))
 
Theoremhashgval2 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)
 
Theoremhashdom 13730 Dominance relation for the size function. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 22-Apr-2015.)
((𝐴 ∈ Fin ∧ 𝐵𝑉) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴𝐵))
 
Theoremhashdomi 13731 Non-strict order relation of the function on the full cardinal poset. (Contributed by Stefan O'Rear, 12-Sep-2015.)
(𝐴𝐵 → (♯‘𝐴) ≤ (♯‘𝐵))
 
Theoremhashsdom 13732 Strict dominance relation for the size function. (Contributed by Mario Carneiro, 18-Aug-2014.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) < (♯‘𝐵) ↔ 𝐴𝐵))
 
Theoremhashun 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 ∧ (𝐴𝐵) = ∅) → (♯‘(𝐴𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
 
Theoremhashun2 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) → (♯‘(𝐴𝐵)) ≤ ((♯‘𝐴) + (♯‘𝐵)))
 
Theoremhashun3 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) → (♯‘(𝐴𝐵)) = (((♯‘𝐴) + (♯‘𝐵)) − (♯‘(𝐴𝐵))))
 
Theoremhashinfxadd 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.)
((𝐴𝑉𝐵𝑊 ∧ (♯‘𝐴) ∉ ℕ0) → ((♯‘𝐴) +𝑒 (♯‘𝐵)) = +∞)
 
Theoremhashunx 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.)
((𝐴𝑉𝐵𝑊 ∧ (𝐴𝐵) = ∅) → (♯‘(𝐴𝐵)) = ((♯‘𝐴) +𝑒 (♯‘𝐵)))
 
Theoremhashge0 13738 The cardinality of a set is greater than or equal to zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)
(𝐴𝑉 → 0 ≤ (♯‘𝐴))
 
Theoremhashgt0 13739 The cardinality of a nonempty set is greater than zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)
((𝐴𝑉𝐴 ≠ ∅) → 0 < (♯‘𝐴))
 
Theoremhashge1 13740 The cardinality of a nonempty set is greater than or equal to one. (Contributed by Thierry Arnoux, 20-Jun-2017.)
((𝐴𝑉𝐴 ≠ ∅) → 1 ≤ (♯‘𝐴))
 
Theorem1elfz0hash 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)))
 
Theoremhashnn0n0nn 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.)
(((𝑉𝑊𝑌 ∈ ℕ0) ∧ ((♯‘𝑉) = 𝑌𝑁𝑉)) → 𝑌 ∈ ℕ)
 
Theoremhashunsng 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.)
(𝐵𝑉 → ((𝐴 ∈ Fin ∧ ¬ 𝐵𝐴) → (♯‘(𝐴 ∪ {𝐵})) = ((♯‘𝐴) + 1)))
 
Theoremhashunsngx 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.)
((𝐴𝑉𝐵𝑊) → (¬ 𝐵𝐴 → (♯‘(𝐴 ∪ {𝐵})) = ((♯‘𝐴) +𝑒 1)))
 
Theoremhashunsnggt 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.)
(((𝐴𝑉𝐵𝑊𝑁 ∈ ℕ0) ∧ ¬ 𝐵𝐴) → (𝑁 < (♯‘𝐴) ↔ (𝑁 + 1) < (♯‘(𝐴 ∪ {𝐵}))))
 
Theoremhashprg 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.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2))
 
Theoremelprchashprn2 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)
 
Theoremhashprb 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 ∧ 𝑀𝑁) ↔ (♯‘{𝑀, 𝑁}) = 2)
 
Theoremhashprdifel 13749 The elements of an unordered pair of size 2 are different sets. (Contributed by AV, 27-Jan-2020.)
𝑆 = {𝐴, 𝐵}       ((♯‘𝑆) = 2 → (𝐴𝑆𝐵𝑆𝐴𝐵))
 
Theoremprhash2ex 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
 
Theoremhashle00 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.)
(𝑉𝑊 → ((♯‘𝑉) ≤ 0 ↔ 𝑉 = ∅))
 
Theoremhashgt0elex 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.)
((𝑉𝑊 ∧ 0 < (♯‘𝑉)) → ∃𝑥 𝑥𝑉)
 
Theoremhashgt0elexb 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.)
(𝑉𝑊 → (0 < (♯‘𝑉) ↔ ∃𝑥 𝑥𝑉))
 
Theoremhashp1i 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
 
Theoremhash1 13755 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(♯‘1o) = 1
 
Theoremhash2 13756 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(♯‘2o) = 2
 
Theoremhash3 13757 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(♯‘3o) = 3
 
Theoremhash4 13758 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(♯‘4o) = 4
 
Theorempr0hash2ex 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
 
Theoremhashss 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.)
((𝐴𝑉𝐵𝐴) → (♯‘𝐵) ≤ (♯‘𝐴))
 
Theoremprsshashgt1 13761 The size of a superset of a proper unordered pair is greater than 1. (Contributed by AV, 6-Feb-2021.)
(((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ 𝐶𝑈) → ({𝐴, 𝐵} ⊆ 𝐶 → 2 ≤ (♯‘𝐶)))
 
Theoremhashin 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.)
(𝐴𝑉 → (♯‘(𝐴𝐵)) ≤ (♯‘𝐴))
 
Theoremhashssdif 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 ∧ 𝐵𝐴) → (♯‘(𝐴𝐵)) = ((♯‘𝐴) − (♯‘𝐵)))
 
Theoremhashdif 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 → (♯‘(𝐴𝐵)) = ((♯‘𝐴) − (♯‘(𝐴𝐵))))
 
Theoremhashdifsn 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 ∧ 𝐵𝐴) → (♯‘(𝐴 ∖ {𝐵})) = ((♯‘𝐴) − 1))
 
Theoremhashdifpr 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 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → (♯‘(𝐴 ∖ {𝐵, 𝐶})) = ((♯‘𝐴) − 2))
 
Theoremhashsn01 13767 The size of a singleton is either 0 or 1. (Contributed by AV, 23-Feb-2021.)
((♯‘{𝐴}) = 0 ∨ (♯‘{𝐴}) = 1)
 
Theoremhashsnle1 13768 The size of a singleton is less than or equal to 1. (Contributed by AV, 23-Feb-2021.)
(♯‘{𝐴}) ≤ 1
 
Theoremhashsnlei 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)
 
Theoremhash1snb 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.)
(𝑉𝑊 → ((♯‘𝑉) = 1 ↔ ∃𝑎 𝑉 = {𝑎}))
 
Theoremeuhash1 13771* The size of a set is 1 in terms of existential uniqueness. (Contributed by Alexander van der Vekens, 8-Feb-2018.)
(𝑉𝑊 → ((♯‘𝑉) = 1 ↔ ∃!𝑎 𝑎𝑉))
 
Theoremhash1n0 13772 If the size of a set is 1 the set is not empty. (Contributed by AV, 23-Dec-2020.)
((𝐴𝑉 ∧ (♯‘𝐴) = 1) → 𝐴 ≠ ∅)
 
Theoremhashgt12el 13773* In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.)
((𝑉𝑊 ∧ 1 < (♯‘𝑉)) → ∃𝑎𝑉𝑏𝑉 𝑎𝑏)
 
Theoremhashgt12el2 13774* In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.)
((𝑉𝑊 ∧ 1 < (♯‘𝑉) ∧ 𝐴𝑉) → ∃𝑏𝑉 𝐴𝑏)
 
Theoremhashgt23el 13775* A set with more than two elements has at least three different elements. (Contributed by BTernaryTau, 21-Sep-2023.)
((𝑉𝑊 ∧ 2 < (♯‘𝑉)) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑎𝑏𝑎𝑐𝑏𝑐))
 
Theoremhashunlei 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 = (𝐴𝐵)    &   (𝐴 ∈ Fin ∧ (♯‘𝐴) ≤ 𝐾)    &   (𝐵 ∈ Fin ∧ (♯‘𝐵) ≤ 𝑀)    &   𝐾 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   (𝐾 + 𝑀) = 𝑁       (𝐶 ∈ Fin ∧ (♯‘𝐶) ≤ 𝑁)
 
Theoremhashsslei 13777 Get an upper bound on a concretely specified finite set. Transfer boundedness to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐵𝐴    &   (𝐴 ∈ Fin ∧ (♯‘𝐴) ≤ 𝑁)    &   𝑁 ∈ ℕ0       (𝐵 ∈ Fin ∧ (♯‘𝐵) ≤ 𝑁)
 
Theoremhashfz 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 ∈ (ℤ𝐴) → (♯‘(𝐴...𝐵)) = ((𝐵𝐴) + 1))
 
Theoremfzsdom2 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 ∈ (ℤ𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐴...𝐵) ≺ (𝐴...𝐶))
 
Theoremhashfzo 13780 Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝐵 ∈ (ℤ𝐴) → (♯‘(𝐴..^𝐵)) = (𝐵𝐴))
 
Theoremhashfzo0 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)
 
Theoremhashfzp1 13782 Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.)
(𝐵 ∈ (ℤ𝐴) → (♯‘((𝐴 + 1)...𝐵)) = (𝐵𝐴))
 
Theoremhashfz0 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))
 
Theoremhashxplem 13784 Lemma for hashxp 13785. (Contributed by Paul Chapman, 30-Nov-2012.)
𝐵 ∈ Fin       (𝐴 ∈ Fin → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))
 
Theoremhashxp 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)))
 
Theoremhashmap 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) → (♯‘(𝐴m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵)))
 
Theoremhashpw 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)))
 
Theoremhashfun 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)))
 
Theoremhashres 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 𝐴𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴) → (♯‘(𝐴𝐵)) = (♯‘𝐵))
 
Theoremhashreshashfun 13790 The number of elements of a finite function expressed by a restriction. (Contributed by AV, 15-Dec-2021.)
((Fun 𝐴𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴) → (♯‘𝐴) = ((♯‘(𝐴𝐵)) + (♯‘(dom 𝐴𝐵))))
 
Theoremhashimarn 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 𝐸1-1→ran 𝐸𝐸𝑉) → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → (♯‘(𝐸 “ ran 𝐹)) = (♯‘𝐹)))
 
Theoremhashimarni 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 𝐸1-1→ran 𝐸𝐸𝑉) → ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸𝑃 = (𝐸 “ ran 𝐹) ∧ (♯‘𝑃) = 𝑁) → (♯‘𝐹) = 𝑁))
 
Theoremresunimafz0 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)    &   (𝜑𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)    &   (𝜑𝑁 ∈ (0..^(♯‘𝐹)))       (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))
 
Theoremfnfz0hash 13794 The size of a function on a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 25-Jun-2018.)
((𝑁 ∈ ℕ0𝐹 Fn (0...𝑁)) → (♯‘𝐹) = (𝑁 + 1))
 
Theoremffz0hash 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 ∈ ℕ0𝐹:(0...𝑁)⟶𝐵) → (♯‘𝐹) = (𝑁 + 1))
 
Theoremfnfz0hashnn0 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)
 
Theoremffzo0hash 13797 The size of a function on a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
((𝑁 ∈ ℕ0𝐹 Fn (0..^𝑁)) → (♯‘𝐹) = 𝑁)
 
Theoremfnfzo0hash 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 ∈ ℕ0𝐹:(0..^𝑁)⟶𝐵) → (♯‘𝐹) = 𝑁)
 
Theoremfnfzo0hashnn0 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)
 
Theoremhashbclem 13800* Lemma for hashbc 13801: inductive step. (Contributed by Mario Carneiro, 13-Jul-2014.)
(𝜑𝐴 ∈ Fin)    &   (𝜑 → ¬ 𝑧𝐴)    &   (𝜑 → ∀𝑗 ∈ ℤ ((♯‘𝐴)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}))    &   (𝜑𝐾 ∈ ℤ)       (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾}))
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