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Theorem List for Intuitionistic Logic Explorer - 5101-5200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfimacnvdisj 5101 The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)
((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐹𝐶) = ∅)
 
Theoremfintm 5102* Function into an intersection. (Contributed by Jim Kingdon, 28-Dec-2018.)
𝑥 𝑥𝐵       (𝐹:𝐴 𝐵 ↔ ∀𝑥𝐵 𝐹:𝐴𝑥)
 
Theoremfin 5103 Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴⟶(𝐵𝐶) ↔ (𝐹:𝐴𝐵𝐹:𝐴𝐶))
 
Theoremfabexg 5104* Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)
𝐹 = {𝑥 ∣ (𝑥:𝐴𝐵𝜑)}       ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)
 
Theoremfabex 5105* Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐹 = {𝑥 ∣ (𝑥:𝐴𝐵𝜑)}       𝐹 ∈ V
 
Theoremdmfex 5106 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.)
((𝐹𝐶𝐹:𝐴𝐵) → 𝐴 ∈ V)
 
Theoremf0 5107 The empty function. (Contributed by NM, 14-Aug-1999.)
∅:∅⟶𝐴
 
Theoremf00 5108 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 5109 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 5110 A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)
(𝐵𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
 
Theoremfnconstg 5111 A cross product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)
(𝐵𝑉 → (𝐴 × {𝐵}) Fn 𝐴)
 
Theoremfconst6g 5112 Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐵𝐶 → (𝐴 × {𝐵}):𝐴𝐶)
 
Theoremfconst6 5113 A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)
𝐵𝐶       (𝐴 × {𝐵}):𝐴𝐶
 
Theoremf1eq1 5114 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
(𝐹 = 𝐺 → (𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵))
 
Theoremf1eq2 5115 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐵 → (𝐹:𝐴1-1𝐶𝐹:𝐵1-1𝐶))
 
Theoremf1eq3 5116 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐵 → (𝐹:𝐶1-1𝐴𝐹:𝐶1-1𝐵))
 
Theoremnff1 5117 Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)
𝑥𝐹    &   𝑥𝐴    &   𝑥𝐵       𝑥 𝐹:𝐴1-1𝐵
 
Theoremdff12 5118* Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)
(𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦))
 
Theoremf1f 5119 A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
(𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
 
Theoremf1fn 5120 A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
 
Theoremf1fun 5121 A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴1-1𝐵 → Fun 𝐹)
 
Theoremf1rel 5122 A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴1-1𝐵 → Rel 𝐹)
 
Theoremf1dm 5123 The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴1-1𝐵 → dom 𝐹 = 𝐴)
 
Theoremf1ss 5124 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:𝐴1-1𝐵𝐵𝐶) → 𝐹:𝐴1-1𝐶)
 
Theoremf1ssr 5125 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 5126 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:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1𝐵)
 
Theoremf1cnvcnv 5127 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 5128 Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.)
((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → (𝐹𝐺):𝐴1-1𝐶)
 
Theoremfoeq1 5129 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵))
 
Theoremfoeq2 5130 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹:𝐴onto𝐶𝐹:𝐵onto𝐶))
 
Theoremfoeq3 5131 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹:𝐶onto𝐴𝐹:𝐶onto𝐵))
 
Theoremnffo 5132 Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)
𝑥𝐹    &   𝑥𝐴    &   𝑥𝐵       𝑥 𝐹:𝐴onto𝐵
 
Theoremfof 5133 An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
 
Theoremfofun 5134 An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
(𝐹:𝐴onto𝐵 → Fun 𝐹)
 
Theoremfofn 5135 An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)
(𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
 
Theoremforn 5136 The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
 
Theoremdffo2 5137 Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
(𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
 
Theoremfoima 5138 The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
(𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
 
Theoremdffn4 5139 A function maps onto its range. (Contributed by NM, 10-May-1998.)
(𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
 
Theoremfunforn 5140 A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)
(Fun 𝐴𝐴:dom 𝐴onto→ran 𝐴)
 
Theoremfodmrnu 5141 An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)
((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremfores 5142 Restriction of a function. (Contributed by NM, 4-Mar-1997.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴):𝐴onto→(𝐹𝐴))
 
Theoremfoco 5143 Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → (𝐹𝐺):𝐴onto𝐶)
 
Theoremf1oeq1 5144 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐹 = 𝐺 → (𝐹:𝐴1-1-onto𝐵𝐺:𝐴1-1-onto𝐵))
 
Theoremf1oeq2 5145 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐵 → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐶))
 
Theoremf1oeq3 5146 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐵 → (𝐹:𝐶1-1-onto𝐴𝐹:𝐶1-1-onto𝐵))
 
Theoremf1oeq23 5147 Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐷))
 
Theoremf1eq123d 5148 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 5149 Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐹:𝐴onto𝐶𝐺:𝐵onto𝐷))
 
Theoremf1oeq123d 5150 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 5151 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 5152 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 5153 A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
 
Theoremf1ofn 5154 A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵𝐹 Fn 𝐴)
 
Theoremf1ofun 5155 A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
 
Theoremf1orel 5156 A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵 → Rel 𝐹)
 
Theoremf1odm 5157 The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴1-1-onto𝐵 → dom 𝐹 = 𝐴)
 
Theoremdff1o2 5158 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 5159 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 5160 A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)
(𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
 
Theoremdff1o4 5161 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 𝐴𝐹 Fn 𝐵))
 
Theoremdff1o5 5162 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 5163 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 5164 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 5165* 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-onto𝐵𝜑)}       ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)
 
Theoremf1ocnv 5166 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 5167 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 5168 The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)
((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
 
Theoremf1orescnv 5169 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 5170 Preimage of an image. (Contributed by NM, 30-Sep-2004.)
((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹 “ (𝐹𝐶)) = 𝐶)
 
Theoremfoimacnv 5171 A reverse version of f1imacnv 5170. (Contributed by Jeff Hankins, 16-Jul-2009.)
((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)
 
Theoremfoun 5172 The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)
(((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺):(𝐴𝐶)–onto→(𝐵𝐷))
 
Theoremf1oun 5173 The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)
(((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝐹𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷))
 
Theoremfun11iun 5174* 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       (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵: 𝑥𝐴 𝐷1-1𝑆)
 
Theoremresdif 5175 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 5176 Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.)
((𝐹:𝐵1-1-onto𝐶𝐺:𝐴1-1-onto𝐵) → (𝐹𝐺):𝐴1-1-onto𝐶)
 
Theoremf1cnv 5177 The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)
(𝐹:𝐴1-1𝐵𝐹:ran 𝐹1-1-onto𝐴)
 
Theoremfuncocnv2 5178 Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
(Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
 
Theoremfococnv2 5179 The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
(𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))
 
Theoremf1ococnv2 5180 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 5181 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
(𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ ran 𝐹))
 
Theoremf1ococnv1 5182 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 5183 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
(𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
 
Theoremfuncoeqres 5184 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 5185* Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)
𝐹 ∈ V       (𝐹:𝐴𝐵 ↔ ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
 
Theoremf11o 5186* Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)
𝐹 ∈ V       (𝐹:𝐴1-1𝐵 ↔ ∃𝑥(𝐹:𝐴1-1-onto𝑥𝑥𝐵))
 
Theoremf10 5187 The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
∅:∅–1-1𝐴
 
Theoremf1o00 5188 One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
(𝐹:∅–1-1-onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
 
Theoremfo00 5189 Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
(𝐹:∅–onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
 
Theoremf1o0 5190 One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
∅:∅–1-1-onto→∅
 
Theoremf1oi 5191 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 5192 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 5193 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 5194 A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})
 
Theoremf1oprg 5195 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.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ((𝐴𝐶𝐵𝐷) → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}:{𝐴, 𝐶}–1-1-onto→{𝐵, 𝐷}))
 
Theoremtz6.12-2 5196* 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 5197* The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)
(∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = {𝑥𝐴𝐹𝑥})
 
Theorembrprcneu 5198* 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 5199 A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)
𝐴 ∈ V → (𝐹𝐴) = ∅)
 
Theoremfv2 5200* 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) = {𝑥 ∣ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥)}
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