HomeHome Metamath Proof Explorer
Theorem List (p. 62 of 419)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-27663)
  Hilbert Space Explorer  Hilbert Space Explorer
(27664-29188)
  Users' Mathboxes  Users' Mathboxes
(29189-41884)
 

Theorem List for Metamath Proof Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremf1oeq3d 6101 Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹:𝐶1-1-onto𝐴𝐹:𝐶1-1-onto𝐵))
 
Theoremnff1o 6102 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 6103 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 6104 A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
 
Theoremf1ofn 6105 A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵𝐹 Fn 𝐴)
 
Theoremf1ofun 6106 A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
 
Theoremf1orel 6107 A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵 → Rel 𝐹)
 
Theoremf1odm 6108 The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴1-1-onto𝐵 → dom 𝐹 = 𝐴)
 
Theoremdff1o2 6109 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 6110 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 6111 A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)
(𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
 
Theoremdff1o4 6112 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 6113 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 6114 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 6115 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)
 
Theoremf1ocnv 6116 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 6117 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 6118 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 6119 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 6120 Preimage of an image. (Contributed by NM, 30-Sep-2004.)
((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹 “ (𝐹𝐶)) = 𝐶)
 
Theoremfoimacnv 6121 A reverse version of f1imacnv 6120. (Contributed by Jeff Hankins, 16-Jul-2009.)
((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)
 
Theoremfoun 6122 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 6123 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))
 
Theoremresdif 6124 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))
 
Theoremresin 6125 The restriction of a one-to-one onto function to an intersection maps onto the intersection of the images. (Contributed by Paul Chapman, 11-Apr-2009.)
((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷))
 
Theoremf1oco 6126 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 6127 The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)
(𝐹:𝐴1-1𝐵𝐹:ran 𝐹1-1-onto𝐴)
 
Theoremfuncocnv2 6128 Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
(Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
 
Theoremfococnv2 6129 The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
(𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))
 
Theoremf1ococnv2 6130 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 6131 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
(𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ ran 𝐹))
 
Theoremf1ococnv1 6132 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 6133 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
(𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
 
Theoremfuncoeqres 6134 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))
 
Theoremf1ssf1 6135 A subset of an injective function is injective. (Contributed by AV, 20-Nov-2020.)
((Fun 𝐹 ∧ Fun 𝐹𝐺𝐹) → Fun 𝐺)
 
Theoremf10 6136 The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
∅:∅–1-1𝐴
 
Theoremf10d 6137 The empty set maps one-to-one into any class, deduction version. (Contributed by AV, 25-Nov-2020.)
(𝜑𝐹 = ∅)       (𝜑𝐹:dom 𝐹1-1𝐴)
 
Theoremf1o00 6138 One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
(𝐹:∅–1-1-onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
 
Theoremfo00 6139 Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
(𝐹:∅–onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
 
Theoremf1o0 6140 One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
∅:∅–1-1-onto→∅
 
Theoremf1oi 6141 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 6142 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 6143 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 6144 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})
 
Theoremf1sng 6145 A singleton of an ordered pair is a one-to-one function. (Contributed by AV, 17-Apr-2021.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1𝑊)
 
Theoremfsnd 6146 A singleton of an ordered pair is a function. (Contributed by AV, 17-Apr-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → {⟨𝐴, 𝐵⟩}:{𝐴}⟶𝑊)
 
Theoremf1oprswap 6147 A two-element swap is a bijection on a pair. (Contributed by Mario Carneiro, 23-Jan-2015.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵})
 
Theoremf1oprg 6148 An unordered pair of ordered pairs with different elements is a one-to-one onto function, analogous to f1oprswap 6147. (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 6149* 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 6150* The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)
(∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = {𝑥𝐴𝐹𝑥})
 
Theorembrprcneu 6151* 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 6152 A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)
𝐴 ∈ V → (𝐹𝐴) = ∅)
 
Theoremfv2 6153* 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)}
 
Theoremdffv3 6154* A definition of function value in terms of iota. (Contributed by Scott Fenton, 19-Feb-2013.)
(𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))
 
Theoremdffv4 6155* 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 5462), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
(𝐹𝐴) = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
 
Theoremelfv 6156* Membership in a function value. (Contributed by NM, 30-Apr-2004.)
(𝐴 ∈ (𝐹𝐵) ↔ ∃𝑥(𝐴𝑥 ∧ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)))
 
Theoremfveq1 6157 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
(𝐹 = 𝐺 → (𝐹𝐴) = (𝐺𝐴))
 
Theoremfveq2 6158 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
(𝐴 = 𝐵 → (𝐹𝐴) = (𝐹𝐵))
 
Theoremfveq1i 6159 Equality inference for function value. (Contributed by NM, 2-Sep-2003.)
𝐹 = 𝐺       (𝐹𝐴) = (𝐺𝐴)
 
Theoremfveq1d 6160 Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)
(𝜑𝐹 = 𝐺)       (𝜑 → (𝐹𝐴) = (𝐺𝐴))
 
Theoremfveq2i 6161 Equality inference for function value. (Contributed by NM, 28-Jul-1999.)
𝐴 = 𝐵       (𝐹𝐴) = (𝐹𝐵)
 
Theoremfveq2d 6162 Equality deduction for function value. (Contributed by NM, 29-May-1999.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹𝐴) = (𝐹𝐵))
 
Theoremfveq12i 6163 Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)
𝐹 = 𝐺    &   𝐴 = 𝐵       (𝐹𝐴) = (𝐺𝐵)
 
Theoremfveq12d 6164 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹𝐴) = (𝐺𝐵))
 
Theoremnffv 6165 Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐹    &   𝑥𝐴       𝑥(𝐹𝐴)
 
Theoremnffvmpt1 6166* Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝑥((𝑥𝐴𝐵)‘𝐶)
 
Theoremnffvd 6167 Deduction version of bound-variable hypothesis builder nffv 6165. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝑥𝐹)    &   (𝜑𝑥𝐴)       (𝜑𝑥(𝐹𝐴))
 
Theoremfvex 6168 The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by NM, 30-Dec-1996.)
(𝐹𝐴) ∈ V
 
Theoremfvexi 6169 The value of a class exists. Inference form of fvex 6168. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝐴 = (𝐹𝐵)       𝐴 ∈ V
 
Theoremfvexd 6170 The value of a class exists (as consequent of anything). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑 → (𝐹𝐴) ∈ V)
 
Theoremfvif 6171 Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹𝐴), (𝐹𝐵))
 
Theoremiffv 6172 Move a conditional outside of a function. (Contributed by Thierry Arnoux, 28-Sep-2018.)
(if(𝜑, 𝐹, 𝐺)‘𝐴) = if(𝜑, (𝐹𝐴), (𝐺𝐴))
 
Theoremfv3 6173* Alternate definition of the value of a function. Definition 6.11 of [TakeutiZaring] p. 26. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝐹𝐴) = {𝑥 ∣ (∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)}
 
Theoremfvres 6174 The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
(𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
 
Theoremfvresd 6175 The value of a restricted function, deduction version of fvres 6174. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴𝐵)       (𝜑 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
 
Theoremfunssfv 6176 The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.)
((Fun 𝐹𝐺𝐹𝐴 ∈ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
 
Theoremtz6.12-1 6177* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
 
Theoremtz6.12 6178* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.)
((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)
 
Theoremtz6.12f 6179* Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)
𝑦𝐹       ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)
 
Theoremtz6.12c 6180* Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
(∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
 
Theoremtz6.12i 6181 Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by Mario Carneiro, 17-Nov-2014.)
(𝐵 ≠ ∅ → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))
 
Theoremfvbr0 6182 Two possibilities for the behavior of a function value. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
(𝑋𝐹(𝐹𝑋) ∨ (𝐹𝑋) = ∅)
 
Theoremfvrn0 6183 A function value is a member of the range plus null. (Contributed by Scott Fenton, 8-Jun-2011.) (Revised by Stefan O'Rear, 3-Jan-2015.)
(𝐹𝑋) ∈ (ran 𝐹 ∪ {∅})
 
Theoremfvssunirn 6184 The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝐹𝑋) ⊆ ran 𝐹
 
Theoremndmfv 6185 The value of a class outside its domain is the empty set. (Contributed by NM, 24-Aug-1995.)
𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
 
Theoremndmfvrcl 6186 Reverse closure law for function with the empty set not in its domain. (Contributed by NM, 26-Apr-1996.)
dom 𝐹 = 𝑆    &    ¬ ∅ ∈ 𝑆       ((𝐹𝐴) ∈ 𝑆𝐴𝑆)
 
Theoremelfvdm 6187 If a function value has a member, the argument belongs to the domain. (Contributed by NM, 12-Feb-2007.)
(𝐴 ∈ (𝐹𝐵) → 𝐵 ∈ dom 𝐹)
 
Theoremelfvex 6188 If a function value has a member, the argument is a set. (Contributed by Mario Carneiro, 6-Nov-2015.)
(𝐴 ∈ (𝐹𝐵) → 𝐵 ∈ V)
 
Theoremelfvexd 6189 If a function value is nonempty, its argument is a set. Deduction form of elfvex 6188. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (𝐵𝐶))       (𝜑𝐶 ∈ V)
 
Theoremeliman0 6190 A non-nul function value is an element of the image of the function. (Contributed by Thierry Arnoux, 25-Jun-2019.)
((𝐴𝐵 ∧ ¬ (𝐹𝐴) = ∅) → (𝐹𝐴) ∈ (𝐹𝐵))
 
Theoremnfvres 6191 The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)
𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)
 
Theoremnfunsn 6192 If the restriction of a class to a singleton is not a function, its value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ∅)
 
Theoremfvfundmfvn0 6193 If a class' value at an argument is not the empty set, the argument is contained in the domain of the class, and the class restricted to the argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐹𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
 
Theorem0fv 6194 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
(∅‘𝐴) = ∅
 
Theoremfv2prc 6195 A function's value at a function's value at a proper class is the empty set. (Contributed by AV, 8-Apr-2021.)
𝐴 ∈ V → ((𝐹𝐴)‘𝐵) = ∅)
 
Theoremelfv2ex 6196 If a function value of a function value has a member, the first argument is a set. (Contributed by AV, 8-Apr-2021.)
(𝐴 ∈ ((𝐹𝐵)‘𝐶) → 𝐵 ∈ V)
 
Theoremfveqres 6197 Equal values imply equal values in a restriction. (Contributed by NM, 13-Nov-1995.)
((𝐹𝐴) = (𝐺𝐴) → ((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴))
 
Theoremcsbfv12 6198 Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 20-Aug-2018.)
𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
 
Theoremcsbfv2g 6199* Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)
(𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐹𝐴 / 𝑥𝐵))
 
Theoremcsbfv 6200* Substitution for a function value. (Contributed by NM, 1-Jan-2006.) (Revised by NM, 20-Aug-2018.)
𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41884
  Copyright terms: Public domain < Previous  Next >