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Theorem List for Intuitionistic Logic Explorer - 5301-5400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrelelfvdm 5301 If a function value has a member, the argument belongs to the domain. (Contributed by Jim Kingdon, 22-Jan-2019.)
((Rel 𝐹𝐴 ∈ (𝐹𝐵)) → 𝐵 ∈ dom 𝐹)
 
Theoremnfvres 5302 The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)
𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)
 
Theoremnfunsn 5303 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) = ∅)
 
Theorem0fv 5304 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
(∅‘𝐴) = ∅
 
Theoremcsbfv12g 5305 Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)
(𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
 
Theoremcsbfv2g 5306* Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)
(𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐹𝐴 / 𝑥𝐵))
 
Theoremcsbfvg 5307* Substitution for a function value. (Contributed by NM, 1-Jan-2006.)
(𝐴𝐶𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴))
 
Theoremfunbrfv 5308 The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
(Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹𝐴) = 𝐵))
 
Theoremfunopfv 5309 The second element in an ordered pair member of a function is the function's value. (Contributed by NM, 19-Jul-1996.)
(Fun 𝐹 → (⟨𝐴, 𝐵⟩ ∈ 𝐹 → (𝐹𝐴) = 𝐵))
 
Theoremfnbrfvb 5310 Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶𝐵𝐹𝐶))
 
Theoremfnopfvb 5311 Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))
 
Theoremfunbrfvb 5312 Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))
 
Theoremfunopfvb 5313 Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
 
Theoremfunbrfv2b 5314 Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)
(Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹𝐴) = 𝐵)))
 
Theoremdffn5im 5315* Representation of a function in terms of its values. The converse holds given the law of the excluded middle; as it is we have most of the converse via funmpt 5019 and dmmptss 4895. (Contributed by Jim Kingdon, 31-Dec-2018.)
(𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
 
Theoremfnrnfv 5316* The range of a function expressed as a collection of the function's values. (Contributed by NM, 20-Oct-2005.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
(𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
 
Theoremfvelrnb 5317* A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)
(𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵))
 
Theoremdfimafn 5318* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦})
 
Theoremdfimafn2 5319* Alternate definition of the image of a function as an indexed union of singletons of function values. (Contributed by Raph Levien, 20-Nov-2006.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹𝑥)})
 
Theoremfunimass4 5320* Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremfvelima 5321* Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42. (Contributed by NM, 29-Apr-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → ∃𝑥𝐵 (𝐹𝑥) = 𝐴)
 
Theoremfeqmptd 5322* Deduction form of dffn5im 5315. (Contributed by Mario Carneiro, 8-Jan-2015.)
(𝜑𝐹:𝐴𝐵)       (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
 
Theoremfeqresmpt 5323* Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ (𝐹𝑥)))
 
Theoremdffn5imf 5324* Representation of a function in terms of its values. (Contributed by Jim Kingdon, 31-Dec-2018.)
𝑥𝐹       (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
 
Theoremfvelimab 5325* Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
 
Theoremfvi 5326 The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐴𝑉 → ( I ‘𝐴) = 𝐴)
 
Theoremfniinfv 5327* The indexed intersection of a function's values is the intersection of its range. (Contributed by NM, 20-Oct-2005.)
(𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)
 
Theoremfnsnfv 5328 Singleton of function value. (Contributed by NM, 22-May-1998.)
((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))
 
Theoremfnimapr 5329 The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011.)
((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹𝐵), (𝐹𝐶)})
 
Theoremssimaex 5330* The existence of a subimage. (Contributed by NM, 8-Apr-2007.)
𝐴 ∈ V       ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
 
Theoremssimaexg 5331* The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
((𝐴𝐶 ∧ Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
 
Theoremfunfvdm 5332 A simplified expression for the value of a function when we know it's a function. (Contributed by Jim Kingdon, 1-Jan-2019.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = (𝐹 “ {𝐴}))
 
Theoremfunfvdm2 5333* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by Jim Kingdon, 1-Jan-2019.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
 
Theoremfunfvdm2f 5334 The value of a function. Version of funfvdm2 5333 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 1-Jan-2019.)
𝑦𝐴    &   𝑦𝐹       ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
 
Theoremfvun1 5335 The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))
 
Theoremfvun2 5336 The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐵)) → ((𝐹𝐺)‘𝑋) = (𝐺𝑋))
 
Theoremdmfco 5337 Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺𝐴) ∈ dom 𝐹))
 
Theoremfvco2 5338 Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (Contributed by NM, 9-Oct-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 16-Oct-2014.)
((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
 
Theoremfvco 5339 Value of a function composition. Similar to Exercise 5 of [TakeutiZaring] p. 28. (Contributed by NM, 22-Apr-2006.) (Proof shortened by Mario Carneiro, 26-Dec-2014.)
((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
 
Theoremfvco3 5340 Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.)
((𝐺:𝐴𝐵𝐶𝐴) → ((𝐹𝐺)‘𝐶) = (𝐹‘(𝐺𝐶)))
 
Theoremfvco4 5341 Value of a composition. (Contributed by BJ, 7-Jul-2022.)
(((𝐾:𝐴𝑋 ∧ (𝐻𝐾) = 𝐹) ∧ (𝑢𝐴𝑥 = (𝐾𝑢))) → (𝐻𝑥) = (𝐹𝑢))
 
Theoremfvopab3g 5342* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑥𝐶 → ∃!𝑦𝜑)    &   𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}       ((𝐴𝐶𝐵𝐷) → ((𝐹𝐴) = 𝐵𝜒))
 
Theoremfvopab3ig 5343* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑥𝐶 → ∃*𝑦𝜑)    &   𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}       ((𝐴𝐶𝐵𝐷) → (𝜒 → (𝐹𝐴) = 𝐵))
 
Theoremfvmptss2 5344* A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)
(𝑥 = 𝐷𝐵 = 𝐶)    &   𝐹 = (𝑥𝐴𝐵)       (𝐹𝐷) ⊆ 𝐶
 
Theoremfvmptg 5345* Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)       ((𝐴𝐷𝐶𝑅) → (𝐹𝐴) = 𝐶)
 
Theoremfvmpt 5346* Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)
(𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)    &   𝐶 ∈ V       (𝐴𝐷 → (𝐹𝐴) = 𝐶)
 
Theoremfvmpts 5347* Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐹 = (𝑥𝐶𝐵)       ((𝐴𝐶𝐴 / 𝑥𝐵𝑉) → (𝐹𝐴) = 𝐴 / 𝑥𝐵)
 
Theoremfvmpt3 5348* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)    &   (𝑥𝐷𝐵𝑉)       (𝐴𝐷 → (𝐹𝐴) = 𝐶)
 
Theoremfvmpt3i 5349* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)    &   𝐵 ∈ V       (𝐴𝐷 → (𝐹𝐴) = 𝐶)
 
Theoremfvmptd 5350* Deduction version of fvmpt 5346. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝜑𝐹 = (𝑥𝐷𝐵))    &   ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)    &   (𝜑𝐴𝐷)    &   (𝜑𝐶𝑉)       (𝜑 → (𝐹𝐴) = 𝐶)
 
Theoremfvmpt2 5351* Value of a function given by the maps-to notation. (Contributed by FL, 21-Jun-2010.)
𝐹 = (𝑥𝐴𝐵)       ((𝑥𝐴𝐵𝐶) → (𝐹𝑥) = 𝐵)
 
Theoremfvmptssdm 5352* If all the values of the mapping are subsets of a class 𝐶, then so is any evaluation of the mapping at a value in the domain of the mapping. (Contributed by Jim Kingdon, 3-Jan-2018.)
𝐹 = (𝑥𝐴𝐵)       ((𝐷 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝐷) ⊆ 𝐶)
 
Theoremmptfvex 5353* Sufficient condition for a maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
𝐹 = (𝑥𝐴𝐵)       ((∀𝑥 𝐵𝑉𝐶𝑊) → (𝐹𝐶) ∈ V)
 
Theoremfvmpt2d 5354* Deduction version of fvmpt2 5351. (Contributed by Thierry Arnoux, 8-Dec-2016.)
(𝜑𝐹 = (𝑥𝐴𝐵))    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
 
Theoremfvmptdf 5355* Alternate deduction version of fvmpt 5346, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))    &   𝑥𝐹    &   𝑥𝜓       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
 
Theoremfvmptdv 5356* Alternate deduction version of fvmpt 5346, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
 
Theoremfvmptdv2 5357* Alternate deduction version of fvmpt 5346, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = 𝐶))
 
Theoremmpteqb 5358* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5362. (Contributed by Mario Carneiro, 14-Nov-2014.)
(∀𝑥𝐴 𝐵𝑉 → ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ↔ ∀𝑥𝐴 𝐵 = 𝐶))
 
Theoremfvmptt 5359* Closed theorem form of fvmpt 5346. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)
((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → (𝐹𝐴) = 𝐶)
 
Theoremfvmptf 5360* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5345 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴    &   𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)       ((𝐴𝐷𝐶𝑉) → (𝐹𝐴) = 𝐶)
 
Theoremfvopab6 5361* Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐵)}    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐴𝐵 = 𝐶)       ((𝐴𝐷𝐶𝑅𝜓) → (𝐹𝐴) = 𝐶)
 
Theoremeqfnfv 5362* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
 
Theoremeqfnfv2 5363* Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
 
Theoremeqfnfv3 5364* Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥)))))
 
Theoremeqfnfvd 5365* Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐴)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))       (𝜑𝐹 = 𝐺)
 
Theoremeqfnfv2f 5366* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 5362 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)
𝑥𝐹    &   𝑥𝐺       ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
 
Theoremeqfunfv 5367* Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)
((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 = 𝐺 ↔ (dom 𝐹 = dom 𝐺 ∧ ∀𝑥 ∈ dom 𝐹(𝐹𝑥) = (𝐺𝑥))))
 
Theoremfvreseq 5368* Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)
(((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐵𝐴) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
 
Theoremfndmdif 5369* Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
 
Theoremfndmdifcom 5370 The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = dom (𝐺𝐹))
 
Theoremfndmin 5371* Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐺𝑥)})
 
Theoremfneqeql 5372 Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ dom (𝐹𝐺) = 𝐴))
 
Theoremfneqeql2 5373 Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺𝐴 ⊆ dom (𝐹𝐺)))
 
Theoremfnreseql 5374 Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = (𝐺𝑋) ↔ 𝑋 ⊆ dom (𝐹𝐺)))
 
Theoremchfnrn 5375* The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain. (Contributed by NM, 31-Aug-1999.)
((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → ran 𝐹 𝐴)
 
Theoremfunfvop 5376 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
 
Theoremfunfvbrb 5377 Two ways to say that 𝐴 is in the domain of 𝐹. (Contributed by Mario Carneiro, 1-May-2014.)
(Fun 𝐹 → (𝐴 ∈ dom 𝐹𝐴𝐹(𝐹𝐴)))
 
Theoremfvimacnvi 5378 A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)
((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → (𝐹𝐴) ∈ 𝐵)
 
Theoremfvimacnv 5379 The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 5059 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
 
Theoremfunimass3 5380 A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv 5379 would be the special case of 𝐴 being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))
 
Theoremfunimass5 5381* A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐴 ⊆ (𝐹𝐵) ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremfunconstss 5382* Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∀𝑥𝐴 (𝐹𝑥) = 𝐵𝐴 ⊆ (𝐹 “ {𝐵})))
 
Theoremelpreima 5383 Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐹 Fn 𝐴 → (𝐵 ∈ (𝐹𝐶) ↔ (𝐵𝐴 ∧ (𝐹𝐵) ∈ 𝐶)))
 
Theoremfniniseg 5384 Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
(𝐹 Fn 𝐴 → (𝐶 ∈ (𝐹 “ {𝐵}) ↔ (𝐶𝐴 ∧ (𝐹𝐶) = 𝐵)))
 
Theoremfncnvima2 5385* Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐹 Fn 𝐴 → (𝐹𝐵) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ 𝐵})
 
Theoremfniniseg2 5386* Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) = {𝑥𝐴 ∣ (𝐹𝑥) = 𝐵})
 
Theoremfnniniseg2 5387* Support sets of functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐹 Fn 𝐴 → (𝐹 “ (V ∖ {𝐵})) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝐵})
 
Theoremrexsupp 5388* Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.)
(𝐹 Fn 𝐴 → (∃𝑥 ∈ (𝐹 “ (V ∖ {𝑍}))𝜑 ↔ ∃𝑥𝐴 ((𝐹𝑥) ≠ 𝑍𝜑)))
 
Theoremunpreima 5389 Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∪ (𝐹𝐵)))
 
Theoreminpreima 5390 Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
 
Theoremdifpreima 5391 Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
 
Theoremrespreima 5392 The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)
(Fun 𝐹 → ((𝐹𝐵) “ 𝐴) = ((𝐹𝐴) ∩ 𝐵))
 
Theoremfimacnv 5393 The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)
(𝐹:𝐴𝐵 → (𝐹𝐵) = 𝐴)
 
Theoremfnopfv 5394 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.)
((𝐹 Fn 𝐴𝐵𝐴) → ⟨𝐵, (𝐹𝐵)⟩ ∈ 𝐹)
 
Theoremfvelrn 5395 A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)
 
Theoremfnfvelrn 5396 A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) ∈ ran 𝐹)
 
Theoremffvelrn 5397 A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)
((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐵)
 
Theoremffvelrni 5398 A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)
𝐹:𝐴𝐵       (𝐶𝐴 → (𝐹𝐶) ∈ 𝐵)
 
Theoremffvelrnda 5399 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐹:𝐴𝐵)       ((𝜑𝐶𝐴) → (𝐹𝐶) ∈ 𝐵)
 
Theoremffvelrnd 5400 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → (𝐹𝐶) ∈ 𝐵)
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