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Theorem List for Metamath Proof Explorer - 5901-6000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremf1dm 5901 The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴1-1𝐵 → dom 𝐹 = 𝐴)
 
Theoremf1ss 5902 A function that is one-to-one is also one-to-one on some superset of its codomain. (Contributed by Mario Carneiro, 12-Jan-2013.)
((𝐹:𝐴1-1𝐵𝐵𝐶) → 𝐹:𝐴1-1𝐶)
 
Theoremf1ssr 5903 A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Stefan O'Rear, 20-Feb-2015.)
((𝐹:𝐴1-1𝐵 ∧ ran 𝐹𝐶) → 𝐹:𝐴1-1𝐶)
 
Theoremf1ssres 5904 A function that is one-to-one is also one-to-one on some subset of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.)
((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1𝐵)
 
Theoremf1cnvcnv 5905 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 5906 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 5907 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵))
 
Theoremfoeq2 5908 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹:𝐴onto𝐶𝐹:𝐵onto𝐶))
 
Theoremfoeq3 5909 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹:𝐶onto𝐴𝐹:𝐶onto𝐵))
 
Theoremnffo 5910 Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)
𝑥𝐹    &   𝑥𝐴    &   𝑥𝐵       𝑥 𝐹:𝐴onto𝐵
 
Theoremfof 5911 An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
 
Theoremfofun 5912 An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
(𝐹:𝐴onto𝐵 → Fun 𝐹)
 
Theoremfofn 5913 An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)
(𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
 
Theoremforn 5914 The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
 
Theoremdffo2 5915 Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
(𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
 
Theoremfoima 5916 The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
(𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
 
Theoremdffn4 5917 A function maps onto its range. (Contributed by NM, 10-May-1998.)
(𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
 
Theoremfunforn 5918 A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)
(Fun 𝐴𝐴:dom 𝐴onto→ran 𝐴)
 
Theoremfodmrnu 5919 An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)
((𝐹:𝐴onto𝐵𝐹:𝐶onto𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremfores 5920 Restriction of an onto function. (Contributed by NM, 4-Mar-1997.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴):𝐴onto→(𝐹𝐴))
 
Theoremfoco 5921 Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → (𝐹𝐺):𝐴onto𝐶)
 
Theoremfoconst 5922 A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007.)
((𝐹:𝐴⟶{𝐵} ∧ 𝐹 ≠ ∅) → 𝐹:𝐴onto→{𝐵})
 
Theoremf1oeq1 5923 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐹 = 𝐺 → (𝐹:𝐴1-1-onto𝐵𝐺:𝐴1-1-onto𝐵))
 
Theoremf1oeq2 5924 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐵 → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐶))
 
Theoremf1oeq3 5925 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
(𝐴 = 𝐵 → (𝐹:𝐶1-1-onto𝐴𝐹:𝐶1-1-onto𝐵))
 
Theoremf1oeq23 5926 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 5927 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 5928 Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐹:𝐴onto𝐶𝐺:𝐵onto𝐷))
 
Theoremf1oeq123d 5929 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))
 
Theoremf1oeq3d 5930 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 5931 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 5932 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 5933 A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
 
Theoremf1ofn 5934 A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵𝐹 Fn 𝐴)
 
Theoremf1ofun 5935 A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
 
Theoremf1orel 5936 A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵 → Rel 𝐹)
 
Theoremf1odm 5937 The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴1-1-onto𝐵 → dom 𝐹 = 𝐴)
 
Theoremdff1o2 5938 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 5939 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 5940 A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)
(𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
 
Theoremdff1o4 5941 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 5942 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 5943 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 5944 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 5945 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 5946 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 5947 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 5948 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 5949 Preimage of an image. (Contributed by NM, 30-Sep-2004.)
((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹 “ (𝐹𝐶)) = 𝐶)
 
Theoremfoimacnv 5950 A reverse version of f1imacnv 5949. (Contributed by Jeff Hankins, 16-Jul-2009.)
((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)
 
Theoremfoun 5951 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 5952 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 5953 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 5954 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 5955 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 5956 The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)
(𝐹:𝐴1-1𝐵𝐹:ran 𝐹1-1-onto𝐴)
 
Theoremfuncocnv2 5957 Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
(Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
 
Theoremfococnv2 5958 The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
(𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))
 
Theoremf1ococnv2 5959 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 5960 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
(𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ ran 𝐹))
 
Theoremf1ococnv1 5961 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 5962 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
(𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
 
Theoremfuncoeqres 5963 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))
 
Theoremf10 5964 The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
∅:∅–1-1𝐴
 
Theoremf10d 5965 The empty set maps one-to-one into any class, deduction version. (Contributed by AV, 25-Nov-2020.)
(𝜑𝐹 = ∅)       (𝜑𝐹:dom 𝐹1-1𝐴)
 
Theoremf1o00 5966 One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
(𝐹:∅–1-1-onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
 
Theoremfo00 5967 Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
(𝐹:∅–onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
 
Theoremf1o0 5968 One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
∅:∅–1-1-onto→∅
 
Theoremf1oi 5969 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 5970 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 5971 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 5972 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 5973 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 5974 A singleton of an ordered pair is a function. (Contributed by AV, 17-Apr-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → {⟨𝐴, 𝐵⟩}:{𝐴}⟶𝑊)
 
Theoremf1oprswap 5975 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 5976 An unordered pair of ordered pairs with different elements is a one-to-one onto function, analogous to f1oprswap 5975. (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 5977* 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 5978* The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)
(∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = {𝑥𝐴𝐹𝑥})
 
Theorembrprcneu 5979* 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 5980 A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)
𝐴 ∈ V → (𝐹𝐴) = ∅)
 
Theoremfv2 5981* 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 5982* A definition of function value in terms of iota. (Contributed by Scott Fenton, 19-Feb-2013.)
(𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))
 
Theoremdffv4 5983* 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 5303), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
(𝐹𝐴) = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
 
Theoremelfv 5984* Membership in a function value. (Contributed by NM, 30-Apr-2004.)
(𝐴 ∈ (𝐹𝐵) ↔ ∃𝑥(𝐴𝑥 ∧ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)))
 
Theoremfveq1 5985 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
(𝐹 = 𝐺 → (𝐹𝐴) = (𝐺𝐴))
 
Theoremfveq2 5986 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
(𝐴 = 𝐵 → (𝐹𝐴) = (𝐹𝐵))
 
Theoremfveq1i 5987 Equality inference for function value. (Contributed by NM, 2-Sep-2003.)
𝐹 = 𝐺       (𝐹𝐴) = (𝐺𝐴)
 
Theoremfveq1d 5988 Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)
(𝜑𝐹 = 𝐺)       (𝜑 → (𝐹𝐴) = (𝐺𝐴))
 
Theoremfveq2i 5989 Equality inference for function value. (Contributed by NM, 28-Jul-1999.)
𝐴 = 𝐵       (𝐹𝐴) = (𝐹𝐵)
 
Theoremfveq2d 5990 Equality deduction for function value. (Contributed by NM, 29-May-1999.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹𝐴) = (𝐹𝐵))
 
Theoremfveq12i 5991 Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)
𝐹 = 𝐺    &   𝐴 = 𝐵       (𝐹𝐴) = (𝐺𝐵)
 
Theoremfveq12d 5992 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹𝐴) = (𝐺𝐵))
 
Theoremnffv 5993 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 5994* Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝑥((𝑥𝐴𝐵)‘𝐶)
 
Theoremnffvd 5995 Deduction version of bound-variable hypothesis builder nffv 5993. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝑥𝐹)    &   (𝜑𝑥𝐴)       (𝜑𝑥(𝐹𝐴))
 
Theoremfvex 5996 The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by NM, 30-Dec-1996.)
(𝐹𝐴) ∈ V
 
Theoremfvif 5997 Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹𝐴), (𝐹𝐵))
 
Theoremiffv 5998 Move a conditional outside of a function. (Contributed by Thierry Arnoux, 28-Sep-2018.)
(if(𝜑, 𝐹, 𝐺)‘𝐴) = if(𝜑, (𝐹𝐴), (𝐺𝐴))
 
Theoremfv3 5999* 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 6000 The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
(𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
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