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Theorem List for Intuitionistic Logic Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfssxp 5001 A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:AB𝐹 ⊆ (A × B))

Theoremfex2 5002 A function with bounded domain and range is a set. This version is proven without the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
((𝐹:AB A 𝑉 B 𝑊) → 𝐹 V)

Theoremfunssxp 5003 Two ways of specifying a partial function from A to B. (Contributed by NM, 13-Nov-2007.)
((Fun 𝐹 𝐹 ⊆ (A × B)) ↔ (𝐹:dom 𝐹B dom 𝐹A))

Theoremffdm 5004 A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
(𝐹:AB → (𝐹:dom 𝐹B dom 𝐹A))

Theoremopelf 5005 The members of an ordered pair element of a mapping belong to the mapping's domain and codomain. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
((𝐹:AB 𝐶, 𝐷 𝐹) → (𝐶 A 𝐷 B))

Theoremfun 5006 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
(((𝐹:A𝐶 𝐺:B𝐷) (AB) = ∅) → (𝐹𝐺):(AB)⟶(𝐶𝐷))

Theoremfun2 5007 The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)
(((𝐹:A𝐶 𝐺:B𝐶) (AB) = ∅) → (𝐹𝐺):(AB)⟶𝐶)

Theoremfnfco 5008 Composition of two functions. (Contributed by NM, 22-May-2006.)
((𝐹 Fn A 𝐺:BA) → (𝐹𝐺) Fn B)

Theoremfssres 5009 Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)
((𝐹:AB 𝐶A) → (𝐹𝐶):𝐶B)

Theoremfssres2 5010 Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)
(((𝐹A):AB 𝐶A) → (𝐹𝐶):𝐶B)

Theoremfresin 5011 An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)
(𝐹:AB → (𝐹𝑋):(A𝑋)⟶B)

Theoremresasplitss 5012 If two functions agree on their common domain, their union contains a union of three functions with pairwise disjoint domains. If we assumed the law of the excluded middle, this would be equality rather than subset. (Contributed by Jim Kingdon, 28-Dec-2018.)
((𝐹 Fn A 𝐺 Fn B (𝐹 ↾ (AB)) = (𝐺 ↾ (AB))) → ((𝐹 ↾ (AB)) ∪ ((𝐹 ↾ (AB)) ∪ (𝐺 ↾ (BA)))) ⊆ (𝐹𝐺))

Theoremfcoi1 5013 Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:AB → (𝐹 ∘ ( I ↾ A)) = 𝐹)

Theoremfcoi2 5014 Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:AB → (( I ↾ B) ∘ 𝐹) = 𝐹)

Theoremfeu 5015* There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)
((𝐹:AB 𝐶 A) → ∃!y B𝐶, y 𝐹)

Theoremfcnvres 5016 The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
(𝐹:AB(𝐹A) = (𝐹B))

Theoremfimacnvdisj 5017 The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)
((𝐹:AB (B𝐶) = ∅) → (𝐹𝐶) = ∅)

Theoremfintm 5018* Function into an intersection. (Contributed by Jim Kingdon, 28-Dec-2018.)
x x B       (𝐹:A Bx B 𝐹:Ax)

Theoremfin 5019 Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:A⟶(B𝐶) ↔ (𝐹:AB 𝐹:A𝐶))

Theoremfabexg 5020* Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)
𝐹 = {x ∣ (x:AB φ)}       ((A 𝐶 B 𝐷) → 𝐹 V)

Theoremfabex 5021* Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
A V    &   B V    &   𝐹 = {x ∣ (x:AB φ)}       𝐹 V

Theoremdmfex 5022 If a mapping is a set, its domain is a set. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹 𝐶 𝐹:AB) → A V)

Theoremf0 5023 The empty function. (Contributed by NM, 14-Aug-1999.)
∅:∅⟶A

Theoremf00 5024 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 5025 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 5026 A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)
(B 𝑉 → (A × {B}):A⟶{B})

Theoremfnconstg 5027 A cross product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)
(B 𝑉 → (A × {B}) Fn A)

Theoremfconst6g 5028 Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(B 𝐶 → (A × {B}):A𝐶)

Theoremfconst6 5029 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 5030 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
(𝐹 = 𝐺 → (𝐹:A1-1B𝐺:A1-1B))

Theoremf1eq2 5031 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
(A = B → (𝐹:A1-1𝐶𝐹:B1-1𝐶))

Theoremf1eq3 5032 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
(A = B → (𝐹:𝐶1-1A𝐹:𝐶1-1B))

Theoremnff1 5033 Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)
x𝐹    &   xA    &   xB       x 𝐹:A1-1B

Theoremdff12 5034* Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)
(𝐹:A1-1B ↔ (𝐹:AB y∃*x x𝐹y))

Theoremf1f 5035 A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
(𝐹:A1-1B𝐹:AB)

Theoremf1fn 5036 A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.)
(𝐹:A1-1B𝐹 Fn A)

Theoremf1fun 5037 A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
(𝐹:A1-1B → Fun 𝐹)

Theoremf1rel 5038 A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
(𝐹:A1-1B → Rel 𝐹)

Theoremf1dm 5039 The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
(𝐹:A1-1B → dom 𝐹 = A)

Theoremf1ss 5040 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 5041 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 5042 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 5043 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 5044 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 5045 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹:AontoB𝐺:AontoB))

Theoremfoeq2 5046 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(A = B → (𝐹:Aonto𝐶𝐹:Bonto𝐶))

Theoremfoeq3 5047 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(A = B → (𝐹:𝐶ontoA𝐹:𝐶ontoB))

Theoremnffo 5048 Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)
x𝐹    &   xA    &   xB       x 𝐹:AontoB

Theoremfof 5049 An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
(𝐹:AontoB𝐹:AB)

Theoremfofun 5050 An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
(𝐹:AontoB → Fun 𝐹)

Theoremfofn 5051 An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)
(𝐹:AontoB𝐹 Fn A)

Theoremforn 5052 The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)
(𝐹:AontoB → ran 𝐹 = B)

Theoremdffo2 5053 Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
(𝐹:AontoB ↔ (𝐹:AB ran 𝐹 = B))

Theoremfoima 5054 The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
(𝐹:AontoB → (𝐹A) = B)

Theoremdffn4 5055 A function maps onto its range. (Contributed by NM, 10-May-1998.)
(𝐹 Fn A𝐹:Aonto→ran 𝐹)

Theoremfunforn 5056 A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)
(Fun AA:dom Aonto→ran A)

Theoremfodmrnu 5057 An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)
((𝐹:AontoB 𝐹:𝐶onto𝐷) → (A = 𝐶 B = 𝐷))

Theoremfores 5058 Restriction of a function. (Contributed by NM, 4-Mar-1997.)
((Fun 𝐹 A ⊆ dom 𝐹) → (𝐹A):Aonto→(𝐹A))

Theoremfoco 5059 Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
((𝐹:Bonto𝐶 𝐺:AontoB) → (𝐹𝐺):Aonto𝐶)

Theoremf1oeq1 5060 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐹 = 𝐺 → (𝐹:A1-1-ontoB𝐺:A1-1-ontoB))

Theoremf1oeq2 5061 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(A = B → (𝐹:A1-1-onto𝐶𝐹:B1-1-onto𝐶))

Theoremf1oeq3 5062 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(A = B → (𝐹:𝐶1-1-ontoA𝐹:𝐶1-1-ontoB))

Theoremf1oeq23 5063 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 5064 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 5065 Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
(φ𝐹 = 𝐺)    &   (φA = B)    &   (φ𝐶 = 𝐷)       (φ → (𝐹:Aonto𝐶𝐺:Bonto𝐷))

Theoremf1oeq123d 5066 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 5067 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 5068 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 5069 A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)
(𝐹:A1-1-ontoB𝐹:AB)

Theoremf1ofn 5070 A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)
(𝐹:A1-1-ontoB𝐹 Fn A)

Theoremf1ofun 5071 A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)
(𝐹:A1-1-ontoB → Fun 𝐹)

Theoremf1orel 5072 A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)
(𝐹:A1-1-ontoB → Rel 𝐹)

Theoremf1odm 5073 The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)
(𝐹:A1-1-ontoB → dom 𝐹 = A)

Theoremdff1o2 5074 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 5075 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 5076 A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)
(𝐹:A1-1-ontoB𝐹:AontoB)

Theoremdff1o4 5077 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 5078 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 5079 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 5080 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 5081* 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 5082 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 5083 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 5084 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 5085 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 5086 Preimage of an image. (Contributed by NM, 30-Sep-2004.)
((𝐹:A1-1B 𝐶A) → (𝐹 “ (𝐹𝐶)) = 𝐶)

Theoremfoimacnv 5087 A reverse version of f1imacnv 5086. (Contributed by Jeff Hankins, 16-Jul-2009.)
((𝐹:AontoB 𝐶B) → (𝐹 “ (𝐹𝐶)) = 𝐶)

Theoremfoun 5088 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 5089 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 5090* 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 5091 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 5092 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 5093 The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)
(𝐹:A1-1B𝐹:ran 𝐹1-1-ontoA)

Theoremfuncocnv2 5094 Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
(Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))

Theoremfococnv2 5095 The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
(𝐹:AontoB → (𝐹𝐹) = ( I ↾ B))

Theoremf1ococnv2 5096 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 5097 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
(𝐹:A1-1B → (𝐹𝐹) = ( I ↾ ran 𝐹))

Theoremf1ococnv1 5098 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 5099 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
(𝐹:A1-1B → (𝐹𝐹) = ( I ↾ A))

Theoremfuncoeqres 5100 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))

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