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Theorem List for Intuitionistic Logic Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremf0 5001 The empty function. (Contributed by NM, 14-Aug-1999.)
∅:∅⟶A
 
Theoremf00 5002 A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.)
(𝐹:A⟶∅ ↔ (𝐹 = ∅ A = ∅))
 
Theoremfconst 5003 A cross product with a singleton is a constant function. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
B V       (A × {B}):A⟶{B}
 
Theoremfconstg 5004 A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)
(B 𝑉 → (A × {B}):A⟶{B})
 
Theoremfnconstg 5005 A cross product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)
(B 𝑉 → (A × {B}) Fn A)
 
Theoremfconst6g 5006 Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(B 𝐶 → (A × {B}):A𝐶)
 
Theoremfconst6 5007 A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)
B 𝐶       (A × {B}):A𝐶
 
Theoremf1eq1 5008 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
(𝐹 = 𝐺 → (𝐹:A1-1B𝐺:A1-1B))
 
Theoremf1eq2 5009 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
(A = B → (𝐹:A1-1𝐶𝐹:B1-1𝐶))
 
Theoremf1eq3 5010 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
(A = B → (𝐹:𝐶1-1A𝐹:𝐶1-1B))
 
Theoremnff1 5011 Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)
x𝐹    &   xA    &   xB       x 𝐹:A1-1B
 
Theoremdff12 5012* Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)
(𝐹:A1-1B ↔ (𝐹:AB y∃*x x𝐹y))
 
Theoremf1f 5013 A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
(𝐹:A1-1B𝐹:AB)
 
Theoremf1fn 5014 A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.)
(𝐹:A1-1B𝐹 Fn A)
 
Theoremf1fun 5015 A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
(𝐹:A1-1B → Fun 𝐹)
 
Theoremf1rel 5016 A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
(𝐹:A1-1B → Rel 𝐹)
 
Theoremf1dm 5017 The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
(𝐹:A1-1B → dom 𝐹 = A)
 
Theoremf1ss 5018 A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Mario Carneiro, 12-Jan-2013.)
((𝐹:A1-1B B𝐶) → 𝐹:A1-1𝐶)
 
Theoremf1ssr 5019 Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)
((𝐹:A1-1B ran 𝐹𝐶) → 𝐹:A1-1𝐶)
 
Theoremf1ssres 5020 A function that is one-to-one is also one-to-one on some aubset of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.)
((𝐹:A1-1B 𝐶A) → (𝐹𝐶):𝐶1-1B)
 
Theoremf1cnvcnv 5021 Two ways to express that a set A (not necessarily a function) is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one. We do not introduce a separate notation since we rarely use it. (Contributed by NM, 13-Aug-2004.)
(A:dom A1-1→V ↔ (Fun A Fun A))
 
Theoremf1co 5022 Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.)
((𝐹:B1-1𝐶 𝐺:A1-1B) → (𝐹𝐺):A1-1𝐶)
 
Theoremfoeq1 5023 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹:AontoB𝐺:AontoB))
 
Theoremfoeq2 5024 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(A = B → (𝐹:Aonto𝐶𝐹:Bonto𝐶))
 
Theoremfoeq3 5025 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(A = B → (𝐹:𝐶ontoA𝐹:𝐶ontoB))
 
Theoremnffo 5026 Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)
x𝐹    &   xA    &   xB       x 𝐹:AontoB
 
Theoremfof 5027 An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
(𝐹:AontoB𝐹:AB)
 
Theoremfofun 5028 An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
(𝐹:AontoB → Fun 𝐹)
 
Theoremfofn 5029 An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)
(𝐹:AontoB𝐹 Fn A)
 
Theoremforn 5030 The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)
(𝐹:AontoB → ran 𝐹 = B)
 
Theoremdffo2 5031 Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
(𝐹:AontoB ↔ (𝐹:AB ran 𝐹 = B))
 
Theoremfoima 5032 The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
(𝐹:AontoB → (𝐹A) = B)
 
Theoremdffn4 5033 A function maps onto its range. (Contributed by NM, 10-May-1998.)
(𝐹 Fn A𝐹:Aonto→ran 𝐹)
 
Theoremfunforn 5034 A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)
(Fun AA:dom Aonto→ran A)
 
Theoremfodmrnu 5035 An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)
((𝐹:AontoB 𝐹:𝐶onto𝐷) → (A = 𝐶 B = 𝐷))
 
Theoremfores 5036 Restriction of a function. (Contributed by NM, 4-Mar-1997.)
((Fun 𝐹 A ⊆ dom 𝐹) → (𝐹A):Aonto→(𝐹A))
 
Theoremfoco 5037 Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
((𝐹:Bonto𝐶 𝐺:AontoB) → (𝐹𝐺):Aonto𝐶)
 
Theoremf1oeq1 5038 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐹 = 𝐺 → (𝐹:A1-1-ontoB𝐺:A1-1-ontoB))
 
Theoremf1oeq2 5039 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(A = B → (𝐹:A1-1-onto𝐶𝐹:B1-1-onto𝐶))
 
Theoremf1oeq3 5040 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(A = B → (𝐹:𝐶1-1-ontoA𝐹:𝐶1-1-ontoB))
 
Theoremf1oeq23 5041 Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)
((A = B 𝐶 = 𝐷) → (𝐹:A1-1-onto𝐶𝐹:B1-1-onto𝐷))
 
Theoremf1eq123d 5042 Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
(φ𝐹 = 𝐺)    &   (φA = B)    &   (φ𝐶 = 𝐷)       (φ → (𝐹:A1-1𝐶𝐺:B1-1𝐷))
 
Theoremfoeq123d 5043 Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
(φ𝐹 = 𝐺)    &   (φA = B)    &   (φ𝐶 = 𝐷)       (φ → (𝐹:Aonto𝐶𝐺:Bonto𝐷))
 
Theoremf1oeq123d 5044 Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
(φ𝐹 = 𝐺)    &   (φA = B)    &   (φ𝐶 = 𝐷)       (φ → (𝐹:A1-1-onto𝐶𝐺:B1-1-onto𝐷))
 
Theoremnff1o 5045 Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)
x𝐹    &   xA    &   xB       x 𝐹:A1-1-ontoB
 
Theoremf1of1 5046 A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)
(𝐹:A1-1-ontoB𝐹:A1-1B)
 
Theoremf1of 5047 A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)
(𝐹:A1-1-ontoB𝐹:AB)
 
Theoremf1ofn 5048 A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)
(𝐹:A1-1-ontoB𝐹 Fn A)
 
Theoremf1ofun 5049 A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)
(𝐹:A1-1-ontoB → Fun 𝐹)
 
Theoremf1orel 5050 A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)
(𝐹:A1-1-ontoB → Rel 𝐹)
 
Theoremf1odm 5051 The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)
(𝐹:A1-1-ontoB → dom 𝐹 = A)
 
Theoremdff1o2 5052 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐹:A1-1-ontoB ↔ (𝐹 Fn A Fun 𝐹 ran 𝐹 = B))
 
Theoremdff1o3 5053 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐹:A1-1-ontoB ↔ (𝐹:AontoB Fun 𝐹))
 
Theoremf1ofo 5054 A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)
(𝐹:A1-1-ontoB𝐹:AontoB)
 
Theoremdff1o4 5055 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐹:A1-1-ontoB ↔ (𝐹 Fn A 𝐹 Fn B))
 
Theoremdff1o5 5056 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐹:A1-1-ontoB ↔ (𝐹:A1-1B ran 𝐹 = B))
 
Theoremf1orn 5057 A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)
(𝐹:A1-1-onto→ran 𝐹 ↔ (𝐹 Fn A Fun 𝐹))
 
Theoremf1f1orn 5058 A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)
(𝐹:A1-1B𝐹:A1-1-onto→ran 𝐹)
 
Theoremf1oabexg 5059* The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)
𝐹 = {f ∣ (f:A1-1-ontoB φ)}       ((A 𝐶 B 𝐷) → 𝐹 V)
 
Theoremf1ocnv 5060 The converse of a one-to-one onto function is also one-to-one onto. (Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐹:A1-1-ontoB𝐹:B1-1-ontoA)
 
Theoremf1ocnvb 5061 A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by NM, 8-Dec-2003.)
(Rel 𝐹 → (𝐹:A1-1-ontoB𝐹:B1-1-ontoA))
 
Theoremf1ores 5062 The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)
((𝐹:A1-1B 𝐶A) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
 
Theoremf1orescnv 5063 The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)
((Fun 𝐹 (𝐹𝑅):𝑅1-1-onto𝑃) → (𝐹𝑃):𝑃1-1-onto𝑅)
 
Theoremf1imacnv 5064 Preimage of an image. (Contributed by NM, 30-Sep-2004.)
((𝐹:A1-1B 𝐶A) → (𝐹 “ (𝐹𝐶)) = 𝐶)
 
Theoremfoimacnv 5065 A reverse version of f1imacnv 5064. (Contributed by Jeff Hankins, 16-Jul-2009.)
((𝐹:AontoB 𝐶B) → (𝐹 “ (𝐹𝐶)) = 𝐶)
 
Theoremfoun 5066 The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)
(((𝐹:AontoB 𝐺:𝐶onto𝐷) (A𝐶) = ∅) → (𝐹𝐺):(A𝐶)–onto→(B𝐷))
 
Theoremf1oun 5067 The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)
(((𝐹:A1-1-ontoB 𝐺:𝐶1-1-onto𝐷) ((A𝐶) = ∅ (B𝐷) = ∅)) → (𝐹𝐺):(A𝐶)–1-1-onto→(B𝐷))
 
Theoremfun11iun 5068* The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by Mario Carneiro, 20-May-2013.) (Revised by Mario Carneiro, 24-Jun-2015.)
(x = yB = 𝐶)    &   B V       (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → x A B: x A 𝐷1-1𝑆)
 
Theoremresdif 5069 The restriction of a one-to-one onto function to a difference maps onto the difference of the images. (Contributed by Paul Chapman, 11-Apr-2009.)
((Fun 𝐹 (𝐹A):Aonto𝐶 (𝐹B):Bonto𝐷) → (𝐹 ↾ (AB)):(AB)–1-1-onto→(𝐶𝐷))
 
Theoremf1oco 5070 Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.)
((𝐹:B1-1-onto𝐶 𝐺:A1-1-ontoB) → (𝐹𝐺):A1-1-onto𝐶)
 
Theoremf1cnv 5071 The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)
(𝐹:A1-1B𝐹:ran 𝐹1-1-ontoA)
 
Theoremfuncocnv2 5072 Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
(Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
 
Theoremfococnv2 5073 The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
(𝐹:AontoB → (𝐹𝐹) = ( I ↾ B))
 
Theoremf1ococnv2 5074 The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Stefan O'Rear, 12-Feb-2015.)
(𝐹:A1-1-ontoB → (𝐹𝐹) = ( I ↾ B))
 
Theoremf1cocnv2 5075 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
(𝐹:A1-1B → (𝐹𝐹) = ( I ↾ ran 𝐹))
 
Theoremf1ococnv1 5076 The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain. (Contributed by NM, 13-Dec-2003.)
(𝐹:A1-1-ontoB → (𝐹𝐹) = ( I ↾ A))
 
Theoremf1cocnv1 5077 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
(𝐹:A1-1B → (𝐹𝐹) = ( I ↾ A))
 
Theoremfuncoeqres 5078 Re-express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.)
((Fun 𝐺 (𝐹𝐺) = 𝐻) → (𝐹 ↾ ran 𝐺) = (𝐻𝐺))
 
Theoremffoss 5079* Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)
𝐹 V       (𝐹:ABx(𝐹:Aontox xB))
 
Theoremf11o 5080* Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)
𝐹 V       (𝐹:A1-1Bx(𝐹:A1-1-ontox xB))
 
Theoremf10 5081 The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
∅:∅–1-1A
 
Theoremf1o00 5082 One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
(𝐹:∅–1-1-ontoA ↔ (𝐹 = ∅ A = ∅))
 
Theoremfo00 5083 Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
(𝐹:∅–ontoA ↔ (𝐹 = ∅ A = ∅))
 
Theoremf1o0 5084 One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
∅:∅–1-1-onto→∅
 
Theoremf1oi 5085 A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
( I ↾ A):A1-1-ontoA
 
Theoremf1ovi 5086 The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.)
I :V–1-1-onto→V
 
Theoremf1osn 5087 A singleton of an ordered pair is one-to-one onto function. (Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
A V    &   B V       {⟨A, B⟩}:{A}–1-1-onto→{B}
 
Theoremf1osng 5088 A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)
((A 𝑉 B 𝑊) → {⟨A, B⟩}:{A}–1-1-onto→{B})
 
Theoremf1oprg 5089 An unordered pair of ordered pairs with different elements is a one-to-one onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
(((A 𝑉 B 𝑊) (𝐶 𝑋 𝐷 𝑌)) → ((A𝐶 B𝐷) → {⟨A, B⟩, ⟨𝐶, 𝐷⟩}:{A, 𝐶}–1-1-onto→{B, 𝐷}))
 
Theoremtz6.12-2 5090* Function value when 𝐹 is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
∃!x A𝐹x → (𝐹A) = ∅)
 
Theoremfveu 5091* The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)
(∃!x A𝐹x → (𝐹A) = {xA𝐹x})
 
Theorembrprcneu 5092* If A is a proper class, then there is no unique binary relationship with A as the first element. (Contributed by Scott Fenton, 7-Oct-2017.)
A V → ¬ ∃!x A𝐹x)
 
Theoremfvprc 5093 A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)
A V → (𝐹A) = ∅)
 
Theoremfv2 5094* Alternate definition of function value. Definition 10.11 of [Quine] p. 68. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝐹A) = {xy(A𝐹yy = x)}
 
Theoremdffv3g 5095* A definition of function value in terms of iota. (Contributed by Jim Kingdon, 29-Dec-2018.)
(A 𝑉 → (𝐹A) = (℩xx (𝐹 “ {A})))
 
Theoremdffv4g 5096* The previous definition of function value, from before the operator was introduced. Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 4617), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
(A 𝑉 → (𝐹A) = {x ∣ (𝐹 “ {A}) = {x}})
 
Theoremelfv 5097* Membership in a function value. (Contributed by NM, 30-Apr-2004.)
(A (𝐹B) ↔ x(A x y(B𝐹yy = x)))
 
Theoremfveq1 5098 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
(𝐹 = 𝐺 → (𝐹A) = (𝐺A))
 
Theoremfveq2 5099 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
(A = B → (𝐹A) = (𝐹B))
 
Theoremfveq1i 5100 Equality inference for function value. (Contributed by NM, 2-Sep-2003.)
𝐹 = 𝐺       (𝐹A) = (𝐺A)
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