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Theorem List for Intuitionistic Logic Explorer - 5301-5400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremffvelrni 5301 A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)
𝐹:𝐴𝐵       (𝐶𝐴 → (𝐹𝐶) ∈ 𝐵)

Theoremffvelrnda 5302 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐹:𝐴𝐵)       ((𝜑𝐶𝐴) → (𝐹𝐶) ∈ 𝐵)

Theoremffvelrnd 5303 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → (𝐹𝐶) ∈ 𝐵)

Theoremrexrn 5304* Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
(𝑥 = (𝐹𝑦) → (𝜑𝜓))       (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦𝐴 𝜓))

Theoremralrn 5305* Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
(𝑥 = (𝐹𝑦) → (𝜑𝜓))       (𝐹 Fn 𝐴 → (∀𝑥 ∈ ran 𝐹𝜑 ↔ ∀𝑦𝐴 𝜓))

Theoremelrnrexdm 5306* For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
(Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))

Theoremelrnrexdmb 5307* For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
(Fun 𝐹 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))

Theoremeldmrexrn 5308* For any element in the domain of a function there is an element in the range of the function which is the function value for the element of the domain. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
(Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))

Theoremralrnmpt 5309* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑦 = 𝐵 → (𝜓𝜒))       (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 𝜒))

Theoremrexrnmpt 5310* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑦 = 𝐵 → (𝜓𝜒))       (∀𝑥𝐴 𝐵𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥𝐴 𝜒))

Theoremdff2 5311 Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)))

Theoremdff3im 5312* Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)
(𝐹:𝐴𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))

Theoremdff4im 5313* Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)
(𝐹:𝐴𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝐹𝑦))

Theoremdffo3 5314* An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)
(𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))

Theoremdffo4 5315* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
(𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))

Theoremdffo5 5316* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
(𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦))

Theoremfoelrn 5317* Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.)
((𝐹:𝐴onto𝐵𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))

Theoremfoco2 5318 If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.)
((𝐹:𝐵𝐶𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) → 𝐹:𝐵onto𝐶)

Theoremfmpt 5319* Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐹 = (𝑥𝐴𝐶)       (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)

Theoremf1ompt 5320* Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.)
𝐹 = (𝑥𝐴𝐶)       (𝐹:𝐴1-1-onto𝐵 ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶))

Theoremfmpti 5321* Functionality of the mapping operation. (Contributed by NM, 19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
𝐹 = (𝑥𝐴𝐶)    &   (𝑥𝐴𝐶𝐵)       𝐹:𝐴𝐵

Theoremfmptd 5322* Domain and codomain of the mapping operation; deduction form. (Contributed by Mario Carneiro, 13-Jan-2013.)
((𝜑𝑥𝐴) → 𝐵𝐶)    &   𝐹 = (𝑥𝐴𝐵)       (𝜑𝐹:𝐴𝐶)

Theoremffnfv 5323* A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))

Theoremffnfvf 5324 A function maps to a class to which all values belong. This version of ffnfv 5323 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝐹       (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))

Theoremfnfvrnss 5325* An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)
((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → ran 𝐹𝐵)

Theoremrnmptss 5326* The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.)
𝐹 = (𝑥𝐴𝐵)       (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)

Theoremfmpt2d 5327* Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑𝐹 = (𝑥𝐴𝐵))    &   ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ 𝐶)       (𝜑𝐹:𝐴𝐶)

Theoremffvresb 5328* A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)
(Fun 𝐹 → ((𝐹𝐴):𝐴𝐵 ↔ ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)))

Theoremf1oresrab 5329* Build a bijection between restricted abstract builders, given a bijection between the base classes, deduction version. (Contributed by Thierry Arnoux, 17-Aug-2018.)
𝐹 = (𝑥𝐴𝐶)    &   (𝜑𝐹:𝐴1-1-onto𝐵)    &   ((𝜑𝑥𝐴𝑦 = 𝐶) → (𝜒𝜓))       (𝜑 → (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})

Theoremfmptco 5330* Composition of two functions expressed as ordered-pair class abstractions. If 𝐹 has the equation ( x + 2 ) and 𝐺 the equation ( 3 * z ) then (𝐺𝐹) has the equation ( 3 * ( x + 2 ) ) . (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.)
((𝜑𝑥𝐴) → 𝑅𝐵)    &   (𝜑𝐹 = (𝑥𝐴𝑅))    &   (𝜑𝐺 = (𝑦𝐵𝑆))    &   (𝑦 = 𝑅𝑆 = 𝑇)       (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))

Theoremfmptcof 5331* Version of fmptco 5330 where 𝜑 needn't be distinct from 𝑥. (Contributed by NM, 27-Dec-2014.)
(𝜑 → ∀𝑥𝐴 𝑅𝐵)    &   (𝜑𝐹 = (𝑥𝐴𝑅))    &   (𝜑𝐺 = (𝑦𝐵𝑆))    &   (𝑦 = 𝑅𝑆 = 𝑇)       (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))

Theoremfmptcos 5332* Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝜑 → ∀𝑥𝐴 𝑅𝐵)    &   (𝜑𝐹 = (𝑥𝐴𝑅))    &   (𝜑𝐺 = (𝑦𝐵𝑆))       (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑅 / 𝑦𝑆))

Theoremfcompt 5333* Express composition of two functions as a maps-to applying both in sequence. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (𝐴‘(𝐵𝑥))))

Theoremfcoconst 5334 Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
((𝐹 Fn 𝑋𝑌𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹𝑌)}))

Theoremfsn 5335 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩})

Theoremfsng 5336 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)
((𝐴𝐶𝐵𝐷) → (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩}))

Theoremfsn2 5337 A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.)
𝐴 ∈ V       (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))

Theoremxpsng 5338 The cross product of two singletons. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝐴𝑉𝐵𝑊) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})

Theoremxpsn 5339 The cross product of two singletons. (Contributed by NM, 4-Nov-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩}

Theoremdfmpt 5340 Alternate definition for the "maps to" notation df-mpt 3820 (although it requires that 𝐵 be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.)
𝐵 ∈ V       (𝑥𝐴𝐵) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}

Theoremfnasrn 5341 A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
𝐵 ∈ V       (𝑥𝐴𝐵) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩)

Theoremdfmptg 5342 Alternate definition for the "maps to" notation df-mpt 3820 (which requires that 𝐵 be a set). (Contributed by Jim Kingdon, 9-Jan-2019.)
(∀𝑥𝐴 𝐵𝑉 → (𝑥𝐴𝐵) = 𝑥𝐴 {⟨𝑥, 𝐵⟩})

Theoremfnasrng 5343 A function expressed as the range of another function. (Contributed by Jim Kingdon, 9-Jan-2019.)
(∀𝑥𝐴 𝐵𝑉 → (𝑥𝐴𝐵) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩))

Theoremressnop0 5344 If 𝐴 is not in 𝐶, then the restriction of a singleton of 𝐴, 𝐵 to 𝐶 is null. (Contributed by Scott Fenton, 15-Apr-2011.)
𝐴𝐶 → ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅)

Theoremfpr 5345 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       (𝐴𝐵 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}⟶{𝐶, 𝐷})

Theoremfprg 5346 A function with a domain of two elements. (Contributed by FL, 2-Feb-2014.)
(((𝐴𝐸𝐵𝐹) ∧ (𝐶𝐺𝐷𝐻) ∧ 𝐴𝐵) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}⟶{𝐶, 𝐷})

Theoremftpg 5347 A function with a domain of three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
(((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴𝐹𝐵𝐺𝐶𝐻) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩}:{𝑋, 𝑌, 𝑍}⟶{𝐴, 𝐵, 𝐶})

Theoremftp 5348 A function with a domain of three elements. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by Alexander van der Vekens, 23-Jan-2018.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝑋 ∈ V    &   𝑌 ∈ V    &   𝑍 ∈ V    &   𝐴𝐵    &   𝐴𝐶    &   𝐵𝐶       {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩, ⟨𝐶, 𝑍⟩}:{𝐴, 𝐵, 𝐶}⟶{𝑋, 𝑌, 𝑍}

Theoremfnressn 5349 A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩})

Theoremfressnfv 5350 The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹𝐵) ∈ 𝐶))

Theoremfvconst 5351 The value of a constant function. (Contributed by NM, 30-May-1999.)
((𝐹:𝐴⟶{𝐵} ∧ 𝐶𝐴) → (𝐹𝐶) = 𝐵)

Theoremfmptsn 5352* Express a singleton function in maps-to notation. (Contributed by NM, 6-Jun-2006.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 28-Feb-2015.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))

Theoremfmptap 5353* Append an additional value to a function. (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑅 ∪ {𝐴}) = 𝑆    &   (𝑥 = 𝐴𝐶 = 𝐵)       ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶)

Theoremfmptapd 5354* Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)    &   ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐵)       (𝜑 → ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶))

Theoremfmptpr 5355* Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)    &   ((𝜑𝑥 = 𝐴) → 𝐸 = 𝐶)    &   ((𝜑𝑥 = 𝐵) → 𝐸 = 𝐷)       (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸))

Theoremfvresi 5356 The value of a restricted identity function. (Contributed by NM, 19-May-2004.)
(𝐵𝐴 → (( I ↾ 𝐴)‘𝐵) = 𝐵)

Theoremfvunsng 5357 Remove an ordered pair not participating in a function value. (Contributed by Jim Kingdon, 7-Jan-2019.)
((𝐷𝑉𝐵𝐷) → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = (𝐴𝐷))

Theoremfvsn 5358 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.)
𝐴 ∈ V    &   𝐵 ∈ V       ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵

Theoremfvsng 5359 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.)
((𝐴𝑉𝐵𝑊) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)

Theoremfvsnun1 5360 The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5361. (Contributed by NM, 23-Sep-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))       (𝐺𝐴) = 𝐵

Theoremfvsnun2 5361 The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5360. (Contributed by NM, 23-Sep-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))       (𝐷 ∈ (𝐶 ∖ {𝐴}) → (𝐺𝐷) = (𝐹𝐷))

Theoremfsnunf 5362 Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇)

Theoremfsnunfv 5363 Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.)
((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋) = 𝑌)

Theoremfsnunres 5364 Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = 𝐹)

Theoremfvpr1 5365 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
𝐴 ∈ V    &   𝐶 ∈ V       (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)

Theoremfvpr2 5366 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
𝐵 ∈ V    &   𝐷 ∈ V       (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷)

Theoremfvpr1g 5367 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
((𝐴𝑉𝐶𝑊𝐴𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)

Theoremfvpr2g 5368 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
((𝐵𝑉𝐷𝑊𝐴𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷)

Theoremfvtp1g 5369 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
(((𝐴𝑉𝐷𝑊) ∧ (𝐴𝐵𝐴𝐶)) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐴) = 𝐷)

Theoremfvtp2g 5370 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
(((𝐵𝑉𝐸𝑊) ∧ (𝐴𝐵𝐵𝐶)) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐵) = 𝐸)

Theoremfvtp3g 5371 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
(((𝐶𝑉𝐹𝑊) ∧ (𝐴𝐶𝐵𝐶)) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐶) = 𝐹)

Theoremfvtp1 5372 The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
𝐴 ∈ V    &   𝐷 ∈ V       ((𝐴𝐵𝐴𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐴) = 𝐷)

Theoremfvtp2 5373 The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
𝐵 ∈ V    &   𝐸 ∈ V       ((𝐴𝐵𝐵𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐵) = 𝐸)

Theoremfvtp3 5374 The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
𝐶 ∈ V    &   𝐹 ∈ V       ((𝐴𝐶𝐵𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐶) = 𝐹)

Theoremfvconst2g 5375 The value of a constant function. (Contributed by NM, 20-Aug-2005.)
((𝐵𝐷𝐶𝐴) → ((𝐴 × {𝐵})‘𝐶) = 𝐵)

Theoremfconst2g 5376 A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.)
(𝐵𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))

Theoremfvconst2 5377 The value of a constant function. (Contributed by NM, 16-Apr-2005.)
𝐵 ∈ V       (𝐶𝐴 → ((𝐴 × {𝐵})‘𝐶) = 𝐵)

Theoremfconst2 5378 A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.)
𝐵 ∈ V       (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵}))

Theoremfconstfvm 5379* A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5378. (Contributed by Jim Kingdon, 8-Jan-2019.)
(∃𝑦 𝑦𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵)))

Theoremfconst3m 5380* Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)
(∃𝑥 𝑥𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵}))))

Theoremfconst4m 5381* Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
(∃𝑥 𝑥𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (𝐹 “ {𝐵}) = 𝐴)))

Theoremresfunexg 5382 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)

Theoremfnex 5383 If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 5382. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)

Theoremfunex 5384 If the domain of a function exists, so the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 5383. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995.)
((Fun 𝐹 ∧ dom 𝐹𝐵) → 𝐹 ∈ V)

Theoremopabex 5385* Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)
𝐴 ∈ V    &   (𝑥𝐴 → ∃*𝑦𝜑)       {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V

Theoremmptexg 5386* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)

Theoremmptex 5387* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.)
𝐴 ∈ V       (𝑥𝐴𝐵) ∈ V

Theoremfex 5388 If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)
((𝐹:𝐴𝐵𝐴𝐶) → 𝐹 ∈ V)

Theoremeufnfv 5389* A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       ∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵)

Theoremfunfvima 5390 A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)
((Fun 𝐹𝐵 ∈ dom 𝐹) → (𝐵𝐴 → (𝐹𝐵) ∈ (𝐹𝐴)))

Theoremfunfvima2 5391 A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐵𝐴 → (𝐹𝐵) ∈ (𝐹𝐴)))

Theoremfunfvima3 5392 A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.)
((Fun 𝐹𝐹𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ (𝐺 “ {𝐴})))

Theoremfnfvima 5393 The function value of an operand in a set is contained in the image of that set, using the Fn abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.)
((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → (𝐹𝑋) ∈ (𝐹𝑆))

Theoremrexima 5394* Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
(𝑥 = (𝐹𝑦) → (𝜑𝜓))       ((𝐹 Fn 𝐴𝐵𝐴) → (∃𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∃𝑦𝐵 𝜓))

Theoremralima 5395* Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
(𝑥 = (𝐹𝑦) → (𝜑𝜓))       ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∀𝑦𝐵 𝜓))

Theoremidref 5396* TODO: This is the same as issref 4707 (which has a much longer proof). Should we replace issref 4707 with this one? - NM 9-May-2016.

Two ways to state a relation is reflexive. (Adapted from Tarski.) (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Proof modification is discouraged.)

(( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥𝐴 𝑥𝑅𝑥)

Theoremelabrex 5397* Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)
𝐵 ∈ V       (𝑥𝐴𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})

Theoremabrexco 5398* Composition of two image maps 𝐶(𝑦) and 𝐵(𝑤). (Contributed by NM, 27-May-2013.)
𝐵 ∈ V    &   (𝑦 = 𝐵𝐶 = 𝐷)       {𝑥 ∣ ∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶} = {𝑥 ∣ ∃𝑤𝐴 𝑥 = 𝐷}

Theoremimaiun 5399* The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝐴 𝑥𝐵 𝐶) = 𝑥𝐵 (𝐴𝐶)

Theoremimauni 5400* The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
(𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥)

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