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Theorem List for Metamath Proof Explorer - 13201-13300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhashrabsn1 13201* 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 13202 A function is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.)
(𝐹 Fn 𝐴 → (#‘𝐹) = (#‘𝐴))

Theoremfseq1hash 13203 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 13204 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 13205 A short expression for the 𝐺 function of hashgf1o 12810. (Contributed by Mario Carneiro, 24-Jan-2015.)
(# ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)

Theoremhashdom 13206 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 13207 Non-strict order relation of the # function on the full cardinal poset. (Contributed by Stefan O'Rear, 12-Sep-2015.)
(𝐴𝐵 → (#‘𝐴) ≤ (#‘𝐵))

Theoremhashsdom 13208 Strict dominance relation for the size function. (Contributed by Mario Carneiro, 18-Aug-2014.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) < (#‘𝐵) ↔ 𝐴𝐵))

Theoremhashun 13209 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 13210 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 13211 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 13212 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 13213 The size of the union of disjoint sets is the result of the extended real addition of their sizes, analogous to hashun 13209. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
((𝐴𝑉𝐵𝑊 ∧ (𝐴𝐵) = ∅) → (#‘(𝐴𝐵)) = ((#‘𝐴) +𝑒 (#‘𝐵)))

Theoremhashge0 13214 The cardinality of a set is greater than or equal to zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)
(𝐴𝑉 → 0 ≤ (#‘𝐴))

Theoremhashgt0 13215 The cardinality of a nonempty set is greater than zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)
((𝐴𝑉𝐴 ≠ ∅) → 0 < (#‘𝐴))

Theoremhashge1 13216 The cardinality of a nonempty set is greater or equal to one. (Contributed by Thierry Arnoux, 20-Jun-2017.)
((𝐴𝑉𝐴 ≠ ∅) → 1 ≤ (#‘𝐴))

Theorem1elfz0hash 13217 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 13218 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 13219 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)))

Theoremhashprg 13220 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))

TheoremhashprgOLD 13221 Obsolete version of hashprg 13220 as of 18-Sep-2021. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝑉𝐵𝑉) → (𝐴𝐵 ↔ (#‘{𝐴, 𝐵}) = 2))

Theoremelprchashprn2 13222 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 13223 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 13224 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 13225 There is (at least) one set with two different elements: the unordered pair containing 0 and 1. In contrast to pr0hash2ex 13234, numbers are used instead of sets because their representation is shorter (and more comprehensive). (Contributed by AV, 29-Jan-2020.)
(#‘{0, 1}) = 2

Theoremhashle00 13226 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 13227* If the size of a set is greater than zero, the set must contain at least one element. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
((𝑉𝑊 ∧ 0 < (#‘𝑉)) → ∃𝑥 𝑥𝑉)

Theoremhashgt0elexb 13228* 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 13229 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 13230 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(#‘1𝑜) = 1

Theoremhash2 13231 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(#‘2𝑜) = 2

Theoremhash3 13232 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(#‘3𝑜) = 3

Theoremhash4 13233 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(#‘4𝑜) = 4

Theorempr0hash2ex 13234 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 13235 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 13236 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 13237 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 13238 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 13239 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 13240 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 13241 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 13242 The size of a singleton is either 0 or 1. (Contributed by AV, 23-Feb-2021.)
((#‘{𝐴}) = 0 ∨ (#‘{𝐴}) = 1)

Theoremhashsnle1 13243 The size of a singleton is less than or equal to 1. (Contributed by AV, 23-Feb-2021.)
(#‘{𝐴}) ≤ 1

Theoremhashsnlei 13244 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 13245* 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 13246* 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 13247 If the size of a set is 1 the set is not empty. (Contributed by AV, 23-Dec-2020.)
((𝐴𝑉 ∧ (#‘𝐴) = 1) → 𝐴 ≠ ∅)

Theoremhashgt12el 13248* 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 13249* 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)

Theoremhashunlei 13250 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 13251 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 13252 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 13253 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 13254 Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝐵 ∈ (ℤ𝐴) → (#‘(𝐴..^𝐵)) = (𝐵𝐴))

Theoremhashfzo0 13255 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 13256 Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.)
(𝐵 ∈ (ℤ𝐴) → (#‘((𝐴 + 1)...𝐵)) = (𝐵𝐴))

Theoremhashfz0 13257 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 13258 Lemma for hashxp 13259. (Contributed by Paul Chapman, 30-Nov-2012.)
𝐵 ∈ Fin       (𝐴 ∈ Fin → (#‘(𝐴 × 𝐵)) = ((#‘𝐴) · (#‘𝐵)))

Theoremhashxp 13259 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 13260 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.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐴𝑚 𝐵)) = ((#‘𝐴)↑(#‘𝐵)))

Theoremhashpw 13261 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 13262 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 13263 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 13264 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 13265 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 13266 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 13267 TODO-AV: Revise using 𝐹 ∈ Word dom 𝐼? Formerly part of proof of eupth2lem3 27214: 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 13268 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 13269 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 13270 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 13271 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 13272 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 13273 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 13274* Lemma for hashbc 13275: 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}))

Theoremhashbc 13275* 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}))

Theoremhashfacen 13276* The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.)
((𝐴𝐵𝐶𝐷) → {𝑓𝑓:𝐴1-1-onto𝐶} ≈ {𝑓𝑓:𝐵1-1-onto𝐷})

Theoremhashf1lem1 13277* Lemma for hashf1 13279. (Contributed by Mario Carneiro, 17-Apr-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → ¬ 𝑧𝐴)    &   (𝜑 → ((#‘𝐴) + 1) ≤ (#‘𝐵))    &   (𝜑𝐹:𝐴1-1𝐵)       (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ≈ (𝐵 ∖ ran 𝐹))

Theoremhashf1lem2 13278* Lemma for hashf1 13279. (Contributed by Mario Carneiro, 17-Apr-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → ¬ 𝑧𝐴)    &   (𝜑 → ((#‘𝐴) + 1) ≤ (#‘𝐵))       (𝜑 → (#‘{𝑓𝑓:(𝐴 ∪ {𝑧})–1-1𝐵}) = (((#‘𝐵) − (#‘𝐴)) · (#‘{𝑓𝑓:𝐴1-1𝐵})))

Theoremhashf1 13279* 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) → (#‘{𝑓𝑓:𝐴1-1𝐵}) = ((!‘(#‘𝐴)) · ((#‘𝐵)C(#‘𝐴))))

Theoremhashfac 13280* 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 → (#‘{𝑓𝑓:𝐴1-1-onto𝐴}) = (!‘(#‘𝐴)))

Theoremleiso 13281 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)))

Theoremleisorel 13282 Version of isorel 6616 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*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝐷 ↔ (𝐹𝐶) ≤ (𝐹𝐷)))

Theoremfz1isolem 13283* Lemma for fz1iso 13284. (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 𝐴𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , 𝑅 ((1...(#‘𝐴)), 𝐴))

Theoremfz1iso 13284* Any finite ordered set has an order isometry to a one-based finite sequence. (Contributed by Mario Carneiro, 2-Apr-2014.)
((𝑅 Or 𝐴𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , 𝑅 ((1...(#‘𝐴)), 𝐴))

Theoremishashinf 13285* Any set that is not finite contains subsets of arbitrarily large finite cardinality. Cf. isinf 8214. (Contributed by Thierry Arnoux, 5-Jul-2017.)
𝐴 ∈ Fin → ∀𝑛 ∈ ℕ ∃𝑥 ∈ 𝒫 𝐴(#‘𝑥) = 𝑛)

Theoremseqcoll 13286* The function 𝐹 contains a sparse set of nonzero values to be summed. The function 𝐺 is an order isomorphism from the set of nonzero values of 𝐹 to a 1-based finite sequence, and 𝐻 collects these nonzero values together. Under these conditions, the sum over the values in 𝐻 yields the same result as the sum over the original set 𝐹. (Contributed by Mario Carneiro, 2-Apr-2014.)
((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)    &   ((𝜑𝑘𝑆) → (𝑘 + 𝑍) = 𝑘)    &   ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)    &   (𝜑𝑍𝑆)    &   (𝜑𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))    &   (𝜑𝑁 ∈ (1...(#‘𝐴)))    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴)))) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)    &   ((𝜑𝑛 ∈ (1...(#‘𝐴))) → (𝐻𝑛) = (𝐹‘(𝐺𝑛)))       (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺𝑁)) = (seq1( + , 𝐻)‘𝑁))

Theoremseqcoll2 13287* The function 𝐹 contains a sparse set of nonzero values to be summed. The function 𝐺 is an order isomorphism from the set of nonzero values of 𝐹 to a 1-based finite sequence, and 𝐻 collects these nonzero values together. Under these conditions, the sum over the values in 𝐻 yields the same result as the sum over the original set 𝐹. (Contributed by Mario Carneiro, 13-Dec-2014.)
((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)    &   ((𝜑𝑘𝑆) → (𝑘 + 𝑍) = 𝑘)    &   ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)    &   (𝜑𝑍𝑆)    &   (𝜑𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))    &   (𝜑𝐴 ≠ ∅)    &   (𝜑𝐴 ⊆ (𝑀...𝑁))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)    &   ((𝜑𝑛 ∈ (1...(#‘𝐴))) → (𝐻𝑛) = (𝐹‘(𝐺𝑛)))       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(#‘𝐴)))

5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)

Theoremhashprlei 13288 An unordered pair has at most two elements. (Contributed by Mario Carneiro, 11-Feb-2015.)
({𝐴, 𝐵} ∈ Fin ∧ (#‘{𝐴, 𝐵}) ≤ 2)

Theoremhash2pr 13289* A set of size two is an unordered pair. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
((𝑉𝑊 ∧ (#‘𝑉) = 2) → ∃𝑎𝑏 𝑉 = {𝑎, 𝑏})

Theoremhash2prde 13290* A set of size two is an unordered pair of two different elements. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
((𝑉𝑊 ∧ (#‘𝑉) = 2) → ∃𝑎𝑏(𝑎𝑏𝑉 = {𝑎, 𝑏}))

Theoremhash2exprb 13291* A set of size two is an unordered pair if and only if it contains two different elements. (Contributed by Alexander van der Vekens, 14-Jan-2018.)
(𝑉𝑊 → ((#‘𝑉) = 2 ↔ ∃𝑎𝑏(𝑎𝑏𝑉 = {𝑎, 𝑏})))

Theoremhash2prb 13292* A set of size two is a proper unordered pair. (Contributed by AV, 1-Nov-2020.)
(𝑉𝑊 → ((#‘𝑉) = 2 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏})))

Theoremprprrab 13293 The set of proper pairs of elements of a given set expressed in two ways. (Contributed by AV, 24-Nov-2020.)
{𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ∣ (#‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 2}

Theoremnehash2 13294 The cardinality of a set with two distinct elements. (Contributed by Thierry Arnoux, 27-Aug-2019.)
(𝜑𝑃𝑉)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝐵)       (𝜑 → 2 ≤ (#‘𝑃))

Theoremhash2prd 13295 A set of size two is an unordered pair if it contains two different elements. (Contributed by Alexander van der Vekens, 9-Dec-2018.) (Proof shortened by AV, 1-Nov-2020.)
((𝑃𝑉 ∧ (#‘𝑃) = 2) → ((𝑋𝑃𝑌𝑃𝑋𝑌) → 𝑃 = {𝑋, 𝑌}))

Theoremhash2pwpr 13296 If the size of a subset of an unordered pair is 2, the subset is the pair itself. (Contributed by Alexander van der Vekens, 9-Dec-2018.)
(((#‘𝑃) = 2 ∧ 𝑃 ∈ 𝒫 {𝑋, 𝑌}) → 𝑃 = {𝑋, 𝑌})

Theoremhashle2pr 13297* A nonempty set of size less than or equal to two is an unordered pair of sets. (Contributed by AV, 24-Nov-2021.)
((𝑃𝑉𝑃 ≠ ∅) → ((#‘𝑃) ≤ 2 ↔ ∃𝑎𝑏 𝑃 = {𝑎, 𝑏}))

Theoremhashle2prv 13298* A nonempty subset of a powerset of a class 𝑉 has size less than or equal to two iff it is an unordered pair of elements of 𝑉. (Contributed by AV, 24-Nov-2021.)
(𝑃 ∈ (𝒫 𝑉 ∖ {∅}) → ((#‘𝑃) ≤ 2 ↔ ∃𝑎𝑉𝑏𝑉 𝑃 = {𝑎, 𝑏}))

Theorempr2pwpr 13299* The set of subsets of a pair having length 2 is the set of the pair as singleton. (Contributed by AV, 9-Dec-2018.)
((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝑝 ∈ 𝒫 {𝐴, 𝐵} ∣ 𝑝 ≈ 2𝑜} = {{𝐴, 𝐵}})

Theoremhashge2el2dif 13300* A set with size at least 2 has at least 2 different elements. (Contributed by AV, 18-Mar-2019.)
((𝐷𝑉 ∧ 2 ≤ (#‘𝐷)) → ∃𝑥𝐷𝑦𝐷 𝑥𝑦)

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