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Theorem List for Intuitionistic Logic Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfunimass1 5001 A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)
((Fun 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))
 
Theoremfunimass2 5002 A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.)
((Fun 𝐹𝐴 ⊆ (𝐹𝐵)) → (𝐹𝐴) ⊆ 𝐵)
 
Theoremimadiflem 5003 One direction of imadif 5004. This direction does not require Fun 𝐹. (Contributed by Jim Kingdon, 25-Dec-2018.)
((𝐹𝐴) ∖ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵))
 
Theoremimadif 5004 The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
 
Theoremimainlem 5005 One direction of imain 5006. This direction does not require Fun 𝐹. (Contributed by Jim Kingdon, 25-Dec-2018.)
(𝐹 “ (𝐴𝐵)) ⊆ ((𝐹𝐴) ∩ (𝐹𝐵))
 
Theoremimain 5006 The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
 
Theoremfunimaexglem 5007 Lemma for funimaexg 5008. It constitutes the interesting part of funimaexg 5008, in which 𝐵 ⊆ dom 𝐴. (Contributed by Jim Kingdon, 27-Dec-2018.)
((Fun 𝐴𝐵𝐶𝐵 ⊆ dom 𝐴) → (𝐴𝐵) ∈ V)
 
Theoremfunimaexg 5008 Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 10-Sep-2006.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
 
Theoremfunimaex 5009 The image of a set under any function is also a set. Equivalent of Axiom of Replacement. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 17-Nov-2002.)
𝐵 ∈ V       (Fun 𝐴 → (𝐴𝐵) ∈ V)
 
Theoremisarep1 5010* Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by 𝜑(𝑥, 𝑦) i.e. the class ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴). If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
(𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥𝐴 [𝑏 / 𝑦]𝜑)
 
Theoremisarep2 5011* Part of a study of the Axiom of Replacement used by the Isabelle prover. In Isabelle, the sethood of PrimReplace is apparently postulated implicitly by its type signature "[ i, [ i, i ] => o ] => i", which automatically asserts that it is a set without using any axioms. To prove that it is a set in Metamath, we need the hypotheses of Isabelle's "Axiom of Replacement" as well as the Axiom of Replacement in the form funimaex 5009. (Contributed by NM, 26-Oct-2006.)
𝐴 ∈ V    &   𝑥𝐴𝑦𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)       𝑤 𝑤 = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)
 
Theoremfneq1 5012 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
 
Theoremfneq2 5013 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹 Fn 𝐴𝐹 Fn 𝐵))
 
Theoremfneq1d 5014 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐹 = 𝐺)       (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
 
Theoremfneq2d 5015 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹 Fn 𝐴𝐹 Fn 𝐵))
 
Theoremfneq12d 5016 Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐵))
 
Theoremfneq12 5017 Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)
((𝐹 = 𝐺𝐴 = 𝐵) → (𝐹 Fn 𝐴𝐺 Fn 𝐵))
 
Theoremfneq1i 5018 Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐹 = 𝐺       (𝐹 Fn 𝐴𝐺 Fn 𝐴)
 
Theoremfneq2i 5019 Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
𝐴 = 𝐵       (𝐹 Fn 𝐴𝐹 Fn 𝐵)
 
Theoremnffn 5020 Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
𝑥𝐹    &   𝑥𝐴       𝑥 𝐹 Fn 𝐴
 
Theoremfnfun 5021 A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
(𝐹 Fn 𝐴 → Fun 𝐹)
 
Theoremfnrel 5022 A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
(𝐹 Fn 𝐴 → Rel 𝐹)
 
Theoremfndm 5023 The domain of a function. (Contributed by NM, 2-Aug-1994.)
(𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
 
Theoremfunfni 5024 Inference to convert a function and domain antecedent. (Contributed by NM, 22-Apr-2004.)
((Fun 𝐹𝐵 ∈ dom 𝐹) → 𝜑)       ((𝐹 Fn 𝐴𝐵𝐴) → 𝜑)
 
Theoremfndmu 5025 A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
((𝐹 Fn 𝐴𝐹 Fn 𝐵) → 𝐴 = 𝐵)
 
Theoremfnbr 5026 The first argument of binary relation on a function belongs to the function's domain. (Contributed by NM, 7-May-2004.)
((𝐹 Fn 𝐴𝐵𝐹𝐶) → 𝐵𝐴)
 
Theoremfnop 5027 The first argument of an ordered pair in a function belongs to the function's domain. (Contributed by NM, 8-Aug-1994.)
((𝐹 Fn 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝐹) → 𝐵𝐴)
 
Theoremfneu 5028* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦 𝐵𝐹𝑦)
 
Theoremfneu2 5029* There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)
((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦𝐵, 𝑦⟩ ∈ 𝐹)
 
Theoremfnun 5030 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
(((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵))
 
Theoremfnunsn 5031 Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
(𝜑𝑋 ∈ V)    &   (𝜑𝑌 ∈ V)    &   (𝜑𝐹 Fn 𝐷)    &   𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})    &   𝐸 = (𝐷 ∪ {𝑋})    &   (𝜑 → ¬ 𝑋𝐷)       (𝜑𝐺 Fn 𝐸)
 
Theoremfnco 5032 Composition of two functions. (Contributed by NM, 22-May-2006.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn 𝐵)
 
Theoremfnresdm 5033 A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
(𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
 
Theoremfnresdisj 5034 A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)
(𝐹 Fn 𝐴 → ((𝐴𝐵) = ∅ ↔ (𝐹𝐵) = ∅))
 
Theorem2elresin 5035 Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴𝐵)))))
 
Theoremfnssresb 5036 Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)
(𝐹 Fn 𝐴 → ((𝐹𝐵) Fn 𝐵𝐵𝐴))
 
Theoremfnssres 5037 Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)
 
Theoremfnresin1 5038 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
(𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))
 
Theoremfnresin2 5039 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
(𝐹 Fn 𝐴 → (𝐹 ↾ (𝐵𝐴)) Fn (𝐵𝐴))
 
Theoremfnres 5040* An equivalence for functionality of a restriction. Compare dffun8 4954. (Contributed by Mario Carneiro, 20-May-2015.)
((𝐹𝐴) Fn 𝐴 ↔ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦)
 
Theoremfnresi 5041 Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
( I ↾ 𝐴) Fn 𝐴
 
Theoremfnima 5042 The image of a function's domain is its range. (Contributed by NM, 4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹 Fn 𝐴 → (𝐹𝐴) = ran 𝐹)
 
Theoremfn0 5043 A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹 Fn ∅ ↔ 𝐹 = ∅)
 
Theoremfnimadisj 5044 A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.)
((𝐹 Fn 𝐴 ∧ (𝐴𝐶) = ∅) → (𝐹𝐶) = ∅)
 
Theoremfnimaeq0 5045 Images under a function never map nonempty sets to empty sets. (Contributed by Stefan O'Rear, 21-Jan-2015.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = ∅ ↔ 𝐵 = ∅))
 
Theoremdfmpt3 5046 Alternate definition for the "maps to" notation df-mpt 3845. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝑥𝐴𝐵) = 𝑥𝐴 ({𝑥} × {𝐵})
 
Theoremfnopabg 5047* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}       (∀𝑥𝐴 ∃!𝑦𝜑𝐹 Fn 𝐴)
 
Theoremfnopab 5048* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.)
(𝑥𝐴 → ∃!𝑦𝜑)    &   𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}       𝐹 Fn 𝐴
 
Theoremmptfng 5049* The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.)
𝐹 = (𝑥𝐴𝐵)       (∀𝑥𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴)
 
Theoremfnmpt 5050* The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.)
𝐹 = (𝑥𝐴𝐵)       (∀𝑥𝐴 𝐵𝑉𝐹 Fn 𝐴)
 
Theoremmpt0 5051 A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝑥 ∈ ∅ ↦ 𝐴) = ∅
 
Theoremfnmpti 5052* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐵 ∈ V    &   𝐹 = (𝑥𝐴𝐵)       𝐹 Fn 𝐴
 
Theoremdmmpti 5053* Domain of an ordered-pair class abstraction that specifies a function. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐵 ∈ V    &   𝐹 = (𝑥𝐴𝐵)       dom 𝐹 = 𝐴
 
Theoremmptun 5054 Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝑥 ∈ (𝐴𝐵) ↦ 𝐶) = ((𝑥𝐴𝐶) ∪ (𝑥𝐵𝐶))
 
Theoremfeq1 5055 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹:𝐴𝐵𝐺:𝐴𝐵))
 
Theoremfeq2 5056 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹:𝐴𝐶𝐹:𝐵𝐶))
 
Theoremfeq3 5057 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹:𝐶𝐴𝐹:𝐶𝐵))
 
Theoremfeq23 5058 Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐴 = 𝐶𝐵 = 𝐷) → (𝐹:𝐴𝐵𝐹:𝐶𝐷))
 
Theoremfeq1d 5059 Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
(𝜑𝐹 = 𝐺)       (𝜑 → (𝐹:𝐴𝐵𝐺:𝐴𝐵))
 
Theoremfeq2d 5060 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹:𝐴𝐶𝐹:𝐵𝐶))
 
Theoremfeq12d 5061 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐶))
 
Theoremfeq123d 5062 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐷))
 
Theoremfeq123 5063 Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → (𝐹:𝐴𝐵𝐺:𝐶𝐷))
 
Theoremfeq1i 5064 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐹 = 𝐺       (𝐹:𝐴𝐵𝐺:𝐴𝐵)
 
Theoremfeq2i 5065 Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
𝐴 = 𝐵       (𝐹:𝐴𝐶𝐹:𝐵𝐶)
 
Theoremfeq23i 5066 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐴 = 𝐶    &   𝐵 = 𝐷       (𝐹:𝐴𝐵𝐹:𝐶𝐷)
 
Theoremfeq23d 5067 Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
(𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → (𝐹:𝐴𝐵𝐹:𝐶𝐷))
 
Theoremnff 5068 Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐹    &   𝑥𝐴    &   𝑥𝐵       𝑥 𝐹:𝐴𝐵
 
Theoremsbcfng 5069* Distribute proper substitution through the function predicate with a domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
(𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))
 
Theoremsbcfg 5070* Distribute proper substitution through the function predicate with domain and codomain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
(𝑋𝑉 → ([𝑋 / 𝑥]𝐹:𝐴𝐵𝑋 / 𝑥𝐹:𝑋 / 𝑥𝐴𝑋 / 𝑥𝐵))
 
Theoremffn 5071 A mapping is a function. (Contributed by NM, 2-Aug-1994.)
(𝐹:𝐴𝐵𝐹 Fn 𝐴)
 
Theoremdffn2 5072 Any function is a mapping into V. (Contributed by NM, 31-Oct-1995.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹 Fn 𝐴𝐹:𝐴⟶V)
 
Theoremffun 5073 A mapping is a function. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴𝐵 → Fun 𝐹)
 
Theoremfrel 5074 A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴𝐵 → Rel 𝐹)
 
Theoremfdm 5075 The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
(𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
 
Theoremfdmi 5076 The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
𝐹:𝐴𝐵       dom 𝐹 = 𝐴
 
Theoremfrn 5077 The range of a mapping. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴𝐵 → ran 𝐹𝐵)
 
Theoremdffn3 5078 A function maps to its range. (Contributed by NM, 1-Sep-1999.)
(𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
 
Theoremfss 5079 Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹:𝐴𝐵𝐵𝐶) → 𝐹:𝐴𝐶)
 
Theoremfssd 5080 Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐹:𝐴𝐶)
 
Theoremfco 5081 Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹:𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
 
Theoremfco2 5082 Functionality of a composition with weakened out of domain condition on the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.)
(((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
 
Theoremfssxp 5083 A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
 
Theoremfex2 5084 A function with bounded domain and range is a set. This version is proven without the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
 
Theoremfunssxp 5085 Two ways of specifying a partial function from 𝐴 to 𝐵. (Contributed by NM, 13-Nov-2007.)
((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
 
Theoremffdm 5086 A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
(𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
 
Theoremopelf 5087 The members of an ordered pair element of a mapping belong to the mapping's domain and codomain. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
((𝐹:𝐴𝐵 ∧ ⟨𝐶, 𝐷⟩ ∈ 𝐹) → (𝐶𝐴𝐷𝐵))
 
Theoremfun 5088 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
(((𝐹:𝐴𝐶𝐺:𝐵𝐷) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷))
 
Theoremfun2 5089 The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)
(((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
 
Theoremfnfco 5090 Composition of two functions. (Contributed by NM, 22-May-2006.)
((𝐹 Fn 𝐴𝐺:𝐵𝐴) → (𝐹𝐺) Fn 𝐵)
 
Theoremfssres 5091 Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)
((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
 
Theoremfssres2 5092 Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)
(((𝐹𝐴):𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
 
Theoremfresin 5093 An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)
(𝐹:𝐴𝐵 → (𝐹𝑋):(𝐴𝑋)⟶𝐵)
 
Theoremresasplitss 5094 If two functions agree on their common domain, their union contains a union of three functions with pairwise disjoint domains. If we assumed the law of the excluded middle, this would be equality rather than subset. (Contributed by Jim Kingdon, 28-Dec-2018.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)) ∪ ((𝐹 ↾ (𝐴𝐵)) ∪ (𝐺 ↾ (𝐵𝐴)))) ⊆ (𝐹𝐺))
 
Theoremfcoi1 5095 Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
 
Theoremfcoi2 5096 Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:𝐴𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)
 
Theoremfeu 5097* There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)
((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦𝐵𝐶, 𝑦⟩ ∈ 𝐹)
 
Theoremfcnvres 5098 The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
(𝐹:𝐴𝐵(𝐹𝐴) = (𝐹𝐵))
 
Theoremfimacnvdisj 5099 The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)
((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐹𝐶) = ∅)
 
Theoremfintm 5100* Function into an intersection. (Contributed by Jim Kingdon, 28-Dec-2018.)
𝑥 𝑥𝐵       (𝐹:𝐴 𝐵 ↔ ∀𝑥𝐵 𝐹:𝐴𝑥)
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