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Theorem List for Metamath Proof Explorer - 12801-12900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfaccld 12801 Closure of the factorial function, deduction version of faccl 12800. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑁 ∈ ℕ0)       (𝜑 → (!‘𝑁) ∈ ℕ)
 
Theoremfacmapnn 12802 The factorial function restricted to positive integers is a mapping from the positive integers to the positive integers. (Contributed by AV, 8-Aug-2020.)
(𝑛 ∈ ℕ ↦ (!‘𝑛)) ∈ (ℕ ↑𝑚 ℕ)
 
Theoremfacne0 12803 The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.)
(𝑁 ∈ ℕ0 → (!‘𝑁) ≠ 0)
 
Theoremfacdiv 12804 A positive integer divides the factorial of an equal or larger number. (Contributed by NM, 2-May-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑁𝑀) → ((!‘𝑀) / 𝑁) ∈ ℕ)
 
Theoremfacndiv 12805 No positive integer (greater than one) divides the factorial plus one of an equal or larger number. (Contributed by NM, 3-May-2005.)
(((𝑀 ∈ ℕ0𝑁 ∈ ℕ) ∧ (1 < 𝑁𝑁𝑀)) → ¬ (((!‘𝑀) + 1) / 𝑁) ∈ ℤ)
 
Theoremfacwordi 12806 Ordering property of factorial. (Contributed by NM, 9-Dec-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (!‘𝑀) ≤ (!‘𝑁))
 
Theoremfaclbnd 12807 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀↑(𝑁 + 1)) ≤ ((𝑀𝑀) · (!‘𝑁)))
 
Theoremfaclbnd2 12808 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)
(𝑁 ∈ ℕ0 → ((2↑𝑁) / 2) ≤ (!‘𝑁))
 
Theoremfaclbnd3 12809 A lower bound for the factorial function. (Contributed by NM, 19-Dec-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝑁) ≤ ((𝑀𝑀) · (!‘𝑁)))
 
Theoremfaclbnd4lem1 12810 Lemma for faclbnd4 12814. Prepare the induction step. (Contributed by NM, 20-Dec-2005.)
𝑁 ∈ ℕ    &   𝐾 ∈ ℕ0    &   𝑀 ∈ ℕ0       ((((𝑁 − 1)↑𝐾) · (𝑀↑(𝑁 − 1))) ≤ (((2↑(𝐾↑2)) · (𝑀↑(𝑀 + 𝐾))) · (!‘(𝑁 − 1))) → ((𝑁↑(𝐾 + 1)) · (𝑀𝑁)) ≤ (((2↑((𝐾 + 1)↑2)) · (𝑀↑(𝑀 + (𝐾 + 1)))) · (!‘𝑁)))
 
Theoremfaclbnd4lem2 12811 Lemma for faclbnd4 12814. Use the weak deduction theorem to convert the hypotheses of faclbnd4lem1 12810 to antecedents. (Contributed by NM, 23-Dec-2005.)
((𝑀 ∈ ℕ0𝐾 ∈ ℕ0𝑁 ∈ ℕ) → ((((𝑁 − 1)↑𝐾) · (𝑀↑(𝑁 − 1))) ≤ (((2↑(𝐾↑2)) · (𝑀↑(𝑀 + 𝐾))) · (!‘(𝑁 − 1))) → ((𝑁↑(𝐾 + 1)) · (𝑀𝑁)) ≤ (((2↑((𝐾 + 1)↑2)) · (𝑀↑(𝑀 + (𝐾 + 1)))) · (!‘𝑁))))
 
Theoremfaclbnd4lem3 12812 Lemma for faclbnd4 12814. The 𝑁 = 0 case. (Contributed by NM, 23-Dec-2005.)
(((𝑀 ∈ ℕ0𝐾 ∈ ℕ0) ∧ 𝑁 = 0) → ((𝑁𝐾) · (𝑀𝑁)) ≤ (((2↑(𝐾↑2)) · (𝑀↑(𝑀 + 𝐾))) · (!‘𝑁)))
 
Theoremfaclbnd4lem4 12813 Lemma for faclbnd4 12814. Prove the 0 < 𝑁 case by induction on 𝐾. (Contributed by NM, 19-Dec-2005.)
((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℕ0𝑀 ∈ ℕ0) → ((𝑁𝐾) · (𝑀𝑁)) ≤ (((2↑(𝐾↑2)) · (𝑀↑(𝑀 + 𝐾))) · (!‘𝑁)))
 
Theoremfaclbnd4 12814 Variant of faclbnd5 12815 providing a non-strict lower bound. (Contributed by NM, 23-Dec-2005.)
((𝑁 ∈ ℕ0𝐾 ∈ ℕ0𝑀 ∈ ℕ0) → ((𝑁𝐾) · (𝑀𝑁)) ≤ (((2↑(𝐾↑2)) · (𝑀↑(𝑀 + 𝐾))) · (!‘𝑁)))
 
Theoremfaclbnd5 12815 The factorial function grows faster than powers and exponentiations. If we consider 𝐾 and 𝑀 to be constants, the right-hand side of the inequality is a constant times 𝑁-factorial. (Contributed by NM, 24-Dec-2005.)
((𝑁 ∈ ℕ0𝐾 ∈ ℕ0𝑀 ∈ ℕ) → ((𝑁𝐾) · (𝑀𝑁)) < ((2 · ((2↑(𝐾↑2)) · (𝑀↑(𝑀 + 𝐾)))) · (!‘𝑁)))
 
Theoremfaclbnd6 12816 Geometric lower bound for the factorial function, where N is usually held constant. (Contributed by Paul Chapman, 28-Dec-2007.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0) → ((!‘𝑁) · ((𝑁 + 1)↑𝑀)) ≤ (!‘(𝑁 + 𝑀)))
 
Theoremfacubnd 12817 An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.)
(𝑁 ∈ ℕ0 → (!‘𝑁) ≤ (𝑁𝑁))
 
Theoremfacavg 12818 The product of two factorials is greater than or equal to the factorial of (the floor of) their average. (Contributed by NM, 9-Dec-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (!‘(⌊‘((𝑀 + 𝑁) / 2))) ≤ ((!‘𝑀) · (!‘𝑁)))
 
5.6.10  The binomial coefficient operation
 
Syntaxcbc 12819 Extend class notation to include the binomial coefficient operation (combinatorial choose operation).
class C
 
Definitiondf-bc 12820* Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 26439).

In the literature, this function is often written as a column vector of the two arguments, or with the arguments as subscripts before and after the letter "C". (𝑁C𝐾) is read "𝑁 choose 𝐾." Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when 0 ≤ 𝑘𝑛 does not hold. (Contributed by NM, 10-Jul-2005.)

C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0))
 
Theorembcval 12821 Value of the binomial coefficient, 𝑁 choose 𝐾. Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when 0 ≤ 𝐾𝑁 does not hold. See bcval2 12822 for the value in the standard domain. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
((𝑁 ∈ ℕ0𝐾 ∈ ℤ) → (𝑁C𝐾) = if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))), 0))
 
Theorembcval2 12822 Value of the binomial coefficient, 𝑁 choose 𝐾, in its standard domain. (Contributed by NM, 9-Jun-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
(𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
 
Theorembcval3 12823 Value of the binomial coefficient, 𝑁 choose 𝐾, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
((𝑁 ∈ ℕ0𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0)
 
Theorembcval4 12824 Value of the binomial coefficient, 𝑁 choose 𝐾, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
((𝑁 ∈ ℕ0𝐾 ∈ ℤ ∧ (𝐾 < 0 ∨ 𝑁 < 𝐾)) → (𝑁C𝐾) = 0)
 
Theorembcrpcl 12825 Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 12840.) (Contributed by Mario Carneiro, 10-Mar-2014.)
(𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℝ+)
 
Theorembccmpl 12826 "Complementing" its second argument doesn't change a binary coefficient. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 5-Mar-2014.)
((𝑁 ∈ ℕ0𝐾 ∈ ℤ) → (𝑁C𝐾) = (𝑁C(𝑁𝐾)))
 
Theorembcn0 12827 𝑁 choose 0 is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
(𝑁 ∈ ℕ0 → (𝑁C0) = 1)
 
Theorembc0k 12828 The binomial coefficient " 0 choose 𝐾 " is 0 for a positive integer K. Note that (0C0) = 1 (see bcn0 12827). (Contributed by Alexander van der Vekens, 1-Jan-2018.)
(𝐾 ∈ ℕ → (0C𝐾) = 0)
 
Theorembcnn 12829 𝑁 choose 𝑁 is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
(𝑁 ∈ ℕ0 → (𝑁C𝑁) = 1)
 
Theorembcn1 12830 Binomial coefficient: 𝑁 choose 1. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
(𝑁 ∈ ℕ0 → (𝑁C1) = 𝑁)
 
Theorembcnp1n 12831 Binomial coefficient: 𝑁 + 1 choose 𝑁. (Contributed by NM, 20-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
(𝑁 ∈ ℕ0 → ((𝑁 + 1)C𝑁) = (𝑁 + 1))
 
Theorembcm1k 12832 The proportion of one binomial coefficient to another with 𝐾 decreased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)
(𝐾 ∈ (1...𝑁) → (𝑁C𝐾) = ((𝑁C(𝐾 − 1)) · ((𝑁 − (𝐾 − 1)) / 𝐾)))
 
Theorembcp1n 12833 The proportion of one binomial coefficient to another with 𝑁 increased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)
(𝐾 ∈ (0...𝑁) → ((𝑁 + 1)C𝐾) = ((𝑁C𝐾) · ((𝑁 + 1) / ((𝑁 + 1) − 𝐾))))
 
Theorembcp1nk 12834 The proportion of one binomial coefficient to another with 𝑁 and 𝐾 increased by 1. (Contributed by Mario Carneiro, 16-Jan-2015.)
(𝐾 ∈ (0...𝑁) → ((𝑁 + 1)C(𝐾 + 1)) = ((𝑁C𝐾) · ((𝑁 + 1) / (𝐾 + 1))))
 
Theorembcval5 12835 Write out the top and bottom parts of the binomial coefficient (𝑁C𝐾) = (𝑁 · (𝑁 − 1) · ... · ((𝑁𝐾) + 1)) / 𝐾! explicitly. In this form, it is valid even for 𝑁 < 𝐾, although it is no longer valid for nonpositive 𝐾. (Contributed by Mario Carneiro, 22-May-2014.)
((𝑁 ∈ ℕ0𝐾 ∈ ℕ) → (𝑁C𝐾) = ((seq((𝑁𝐾) + 1)( · , I )‘𝑁) / (!‘𝐾)))
 
Theorembcn2 12836 Binomial coefficient: 𝑁 choose 2. (Contributed by Mario Carneiro, 22-May-2014.)
(𝑁 ∈ ℕ0 → (𝑁C2) = ((𝑁 · (𝑁 − 1)) / 2))
 
Theorembcp1m1 12837 Compute the binomial coefficient of (𝑁 + 1) over (𝑁 − 1) (Contributed by Scott Fenton, 11-May-2014.) (Revised by Mario Carneiro, 22-May-2014.)
(𝑁 ∈ ℕ0 → ((𝑁 + 1)C(𝑁 − 1)) = (((𝑁 + 1) · 𝑁) / 2))
 
Theorembcpasc 12838 Pascal's rule for the binomial coefficient, generalized to all integers 𝐾. Equation 2 of [Gleason] p. 295. (Contributed by NM, 13-Jul-2005.) (Revised by Mario Carneiro, 10-Mar-2014.)
((𝑁 ∈ ℕ0𝐾 ∈ ℤ) → ((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾))
 
Theorembccl 12839 A binomial coefficient, in its extended domain, is a nonnegative integer. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 9-Nov-2013.)
((𝑁 ∈ ℕ0𝐾 ∈ ℤ) → (𝑁C𝐾) ∈ ℕ0)
 
Theorembccl2 12840 A binomial coefficient, in its standard domain, is a positive integer. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 10-Mar-2014.)
(𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℕ)
 
Theorembcn2m1 12841 Compute the binomial coefficient "𝑁 choose 2 " from "(𝑁 − 1) choose 2 ": (N-1) + ( (N-1) 2 ) = ( N 2 ). (Contributed by Alexander van der Vekens, 7-Jan-2018.)
(𝑁 ∈ ℕ → ((𝑁 − 1) + ((𝑁 − 1)C2)) = (𝑁C2))
 
Theorembcn2p1 12842 Compute the binomial coefficient "(𝑁 + 1) choose 2 " from "𝑁 choose 2 ": N + ( N 2 ) = ( (N+1) 2 ). (Contributed by Alexander van der Vekens, 8-Jan-2018.)
(𝑁 ∈ ℕ0 → (𝑁 + (𝑁C2)) = ((𝑁 + 1)C2))
 
Theorempermnn 12843 The number of permutations of 𝑁𝑅 objects from a collection of 𝑁 objects is a positive integer. (Contributed by Jason Orendorff, 24-Jan-2007.)
(𝑅 ∈ (0...𝑁) → ((!‘𝑁) / (!‘𝑅)) ∈ ℕ)
 
Theorembcnm1 12844 The binomial coefficent of (𝑁 − 1) is 𝑁. (Contributed by Scott Fenton, 16-May-2014.)
(𝑁 ∈ ℕ0 → (𝑁C(𝑁 − 1)) = 𝑁)
 
Theorem4bc3eq4 12845 The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)
(4C3) = 4
 
Theorem4bc2eq6 12846 The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)
(4C2) = 6
 
5.6.11  The ` # ` (set size) function
 
Syntaxchash 12847 Extend the definition of a class to include the set size function.
class #
 
Definitiondf-hash 12848 Define the set size function #, which gives the cardinality of a finite set as a member of 0, and assigns all infinite sets the value +∞. For example, (#‘{0, 1, 2}) = 3 (ex-hash 26440). (Contributed by Paul Chapman, 22-Jun-2011.)
# = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ∪ ((V ∖ Fin) × {+∞}))
 
Theoremhashkf 12849 The finite part of the size function maps all finite sets to their cardinality, as members of 0. (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)
𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)    &   𝐾 = (𝐺 ∘ card)       𝐾:Fin⟶ℕ0
 
Theoremhashgval 12850* The value of the # function in terms of the mapping 𝐺 from ω to 0. The proof avoids the use of ax-ac 9040. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 26-Dec-2014.)
𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)       (𝐴 ∈ Fin → (𝐺‘(card‘𝐴)) = (#‘𝐴))
 
Theoremhashginv 12851* 𝐺 maps the size function's value to card. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)
𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)       (𝐴 ∈ Fin → (𝐺‘(#‘𝐴)) = (card‘𝐴))
 
Theoremhashinf 12852 The value of the # function on an infinite set. (Contributed by Mario Carneiro, 13-Jul-2014.)
((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) = +∞)
 
Theoremhashbnd 12853 If 𝐴 has size bounded by an integer 𝐵, then 𝐴 is finite. (Contributed by Mario Carneiro, 14-Jun-2015.)
((𝐴𝑉𝐵 ∈ ℕ0 ∧ (#‘𝐴) ≤ 𝐵) → 𝐴 ∈ Fin)
 
Theoremhashf 12854 The size function maps all finite sets to their cardinality, as members of 0, and infinite sets to +∞. (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 13-Jul-2014.)
#:V⟶(ℕ0 ∪ {+∞})
 
Theoremhashresfn 12855 Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 31-Jan-2017.)
(# ↾ 𝐴) Fn 𝐴
 
Theoremdmhashres 12856 Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 12-Jan-2017.)
dom (# ↾ 𝐴) = 𝐴
 
Theoremhashnn0pnf 12857 The value of the hash function for a set is either a nonnegative integer or positive infinity. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
(𝑀𝑉 → ((#‘𝑀) ∈ ℕ0 ∨ (#‘𝑀) = +∞))
 
Theoremhashnnn0genn0 12858 If the size of a set is not a nonnegative integer, it is greater than or equal to any nonnegative integer. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
((𝑀𝑉 ∧ (#‘𝑀) ∉ ℕ0𝑁 ∈ ℕ0) → 𝑁 ≤ (#‘𝑀))
 
Theoremhashnemnf 12859 The size of a set is never minus infinity. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
(𝐴𝑉 → (#‘𝐴) ≠ -∞)
 
Theoremhashv01gt1 12860 The size of a set is either 0 or 1 or greater than 1. (Contributed by Alexander van der Vekens, 29-Dec-2017.)
(𝑀𝑉 → ((#‘𝑀) = 0 ∨ (#‘𝑀) = 1 ∨ 1 < (#‘𝑀)))
 
Theoremhashfz1 12861 The set (1...𝑁) has 𝑁 elements. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)
(𝑁 ∈ ℕ0 → (#‘(1...𝑁)) = 𝑁)
 
Theoremhashen 12862 Two finite sets have the same number of elements iff they are equinumerous. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) = (#‘𝐵) ↔ 𝐴𝐵))
 
Theoremhasheni 12863 Equinumerous sets have the same number of elements (even if they are not finite). (Contributed by Mario Carneiro, 15-Apr-2015.)
(𝐴𝐵 → (#‘𝐴) = (#‘𝐵))
 
Theoremhasheqf1o 12864* The size of two finite sets is equal if and only if there is a bijection mapping one of the sets onto the other. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) = (#‘𝐵) ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵))
 
Theoremfiinfnf1o 12865* There is no bijection between a finite set and an infinite set. (Contributed by Alexander van der Vekens, 25-Dec-2017.)
((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ¬ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
 
Theoremfocdmex 12866 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 12867* 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)))
 
Theoremhasheqf1oiOLD 12868* Obsolete version of hasheqf1oi 12867 as of 4-May-2021. (Contributed by Alexander van der Vekens, 25-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → (∃𝑓 𝑓:𝐴1-1-onto𝐵 → (#‘𝐴) = (#‘𝐵)))
 
Theoremhashf1rn 12869 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))
 
Theoremhashf1rnOLD 12870 Obsolete version of hashf1rn 12869 as of 4-May-2021. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by Alexander van der Vekens, 4-Feb-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → (𝐹:𝐴1-1𝐵 → (#‘𝐹) = (#‘ran 𝐹)))
 
Theoremhasheqf1od 12871 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 12872 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 12873 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 12874 Closure of the # function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.)
(𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0)
 
Theoremhashxrcl 12875 Extended real closure of the # function. (Contributed by Mario Carneiro, 22-Apr-2015.)
(𝐴𝑉 → (#‘𝐴) ∈ ℝ*)
 
Theoremhashclb 12876 Reverse closure of the # function. (Contributed by Mario Carneiro, 15-Jan-2015.)
(𝐴𝑉 → (𝐴 ∈ Fin ↔ (#‘𝐴) ∈ ℕ0))
 
Theoremhashvnfin 12877 A set of finite size is a finite set. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
((𝑆𝑉𝑁 ∈ ℕ0) → ((#‘𝑆) = 𝑁𝑆 ∈ Fin))
 
Theoremhashnfinnn0 12878 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 12879 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 12880 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 12881 Two ways of saying a set is not empty. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
(𝐴𝑉 → (0 < (#‘𝐴) ↔ 𝐴 ≠ ∅))
 
Theoremhashgt0n0 12882 If the size of a set is greater than 0, the set is not empty. (Contributed by AV, 5-Aug-2018.) (Prove shortened by AV, 18-Nov-2018.)
((𝐴𝑉 ∧ 0 < (#‘𝐴)) → 𝐴 ≠ ∅)
 
Theoremhashnncl 12883 Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)
(𝐴 ∈ Fin → ((#‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅))
 
Theoremhash0 12884 The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)
(#‘∅) = 0
 
Theoremhashsng 12885 The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
(𝐴𝑉 → (#‘{𝐴}) = 1)
 
Theoremhashen1 12886 A set with only one element is equinumerous to the ordinal number 1. (Contributed by AV, 14-Apr-2019.)
(𝐴𝑉 → ((#‘𝐴) = 1 ↔ 𝐴 ≈ 1𝑜))
 
Theoremhashrabrsn 12887* 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 12888* 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 12889* 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 12890 A function is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.)
(𝐹 Fn 𝐴 → (#‘𝐹) = (#‘𝐴))
 
Theoremfseq1hash 12891 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 12892 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 12893 A short expression for the 𝐺 function of hashgf1o 12500. (Contributed by Mario Carneiro, 24-Jan-2015.)
(# ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
 
Theoremhashdom 12894 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 12895 Non-strict order relation of the # function on the full cardinal poset. (Contributed by Stefan O'Rear, 12-Sep-2015.)
(𝐴𝐵 → (#‘𝐴) ≤ (#‘𝐵))
 
Theoremhashsdom 12896 Strict dominance relation for the size function. (Contributed by Mario Carneiro, 18-Aug-2014.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) < (#‘𝐵) ↔ 𝐴𝐵))
 
Theoremhashun 12897 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 12898 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 12899 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 12900 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)) = +∞)
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