P. 145 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 14401-14500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hashclb 14401	Reverse closure of the ♯ function. (Contributed by Mario Carneiro, 15-Jan-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ (♯‘𝐴) ∈ ℕ0))
Theorem	nfile 14402	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 14403	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 14404	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 14405	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 14406	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 14407	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 14408	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 14409	Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅))
Theorem	hash0 14410	The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)
⊢ (♯‘∅) = 0
Theorem	hashelne0d 14411	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 14412	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 14413	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 14414	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 14415*	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 14416*	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 14417*	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 14418	A function is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.)
⊢ (𝐹 Fn 𝐴 → (♯‘𝐹) = (♯‘𝐴))
Theorem	fseq1hash 14419	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 14420	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 14421	A short expression for the 𝐺 function of hashgf1o 14014. (Contributed by Mario Carneiro, 24-Jan-2015.)
⊢ (♯ ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
Theorem	hashdom 14422	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 14423	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 14424	Strict dominance relation for the size function. (Contributed by Mario Carneiro, 18-Aug-2014.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) < (♯‘𝐵) ↔ 𝐴 ≺ 𝐵))
Theorem	hashun 14425	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 14426	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 14427	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 14428	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 14429	The size of the union of disjoint sets is the result of the extended real addition of their sizes, analogous to hashun 14425. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +𝑒 (♯‘𝐵)))
Theorem	hashge0 14430	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 14431	The cardinality of a nonempty set is greater than zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → 0 < (♯‘𝐴))
Theorem	hashge1 14432	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 14433	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 14434	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 14435	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 14436	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 14435. (Contributed by BTernaryTau, 9-Sep-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝐵 ∈ 𝐴 → (♯‘(𝐴 ∪ {𝐵})) = ((♯‘𝐴) +𝑒 1)))
Theorem	hashunsnggt 14437	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 14438	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 14439	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 14440	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 14441	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 14442	There is (at least) one set with two different elements: the unordered pair containing 0 and 1. In contrast to pr0hash2ex 14451, 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 14443	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 14444*	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 14445*	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 14446	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 14447	Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ (♯‘1o) = 1
Theorem	hash2 14448	Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ (♯‘2o) = 2
Theorem	hash3 14449	Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ (♯‘3o) = 3
Theorem	hash4 14450	Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ (♯‘4o) = 4
Theorem	pr0hash2ex 14451	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 14452	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	hashpss 14453	The size of a proper subset is less than the size of its finite superset. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴) → (♯‘𝐵) < (♯‘𝐴))
Theorem	prsshashgt1 14454	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 14455	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 14456	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 14457	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 14458	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 14459	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 14460	The size of a singleton is either 0 or 1. (Contributed by AV, 23-Feb-2021.)
⊢ ((♯‘{𝐴}) = 0 ∨ (♯‘{𝐴}) = 1)
Theorem	hashsnle1 14461	The size of a singleton is less than or equal to 1. (Contributed by AV, 23-Feb-2021.)
⊢ (♯‘{𝐴}) ≤ 1
Theorem	hashsnlei 14462	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 14463*	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 14464*	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 14465	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 14466*	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 14467*	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 14468*	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 14469	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 14470	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 14471	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 14472	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 14473	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 14474	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 14475	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 14476	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 14477	Lemma for hashxp 14478. (Contributed by Paul Chapman, 30-Nov-2012.)
⊢ 𝐵 ∈ Fin    ⇒   ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))
Theorem	hashxp 14478	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 14479	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 14480	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 14481	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 14482	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 14483	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 14484	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 14485	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 14486	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 14487	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 14488	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 14489	TODO-AV: Revise using 𝐹 ∈ Word dom 𝐼? Formerly part of proof of eupth2lem3 30598: 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 14490	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 14491	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 14492	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 14493	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 14494	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 14495	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 14496*	Lemma for hashbc 14497: 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 14497*	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 14498*	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 14499*	Lemma for hashf1 14501. (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 14500*	Lemma for hashf1 14501. (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})))