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Theorem List for Intuitionistic Logic Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfunin 5001 The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53. (Contributed by NM, 19-Mar-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(Fun 𝐹 → Fun (𝐹𝐺))

Theoremfunres11 5002 The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)
(Fun 𝐹 → Fun (𝐹𝐴))

Theoremfuncnvres 5003 The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)
(Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))

Theoremcnvresid 5004 Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
( I ↾ 𝐴) = ( I ↾ 𝐴)

Theoremfuncnvres2 5005 The converse of a restriction of the converse of a function equals the function restricted to the image of its converse. (Contributed by NM, 4-May-2005.)
(Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))

Theoremfunimacnv 5006 The image of the preimage of a function. (Contributed by NM, 25-May-2004.)
(Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = (𝐴 ∩ ran 𝐹))

Theoremfunimass1 5007 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 5008 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 5009 One direction of imadif 5010. This direction does not require Fun 𝐹. (Contributed by Jim Kingdon, 25-Dec-2018.)
((𝐹𝐴) ∖ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵))

Theoremimadif 5010 The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))

Theoremimainlem 5011 One direction of imain 5012. This direction does not require Fun 𝐹. (Contributed by Jim Kingdon, 25-Dec-2018.)
(𝐹 “ (𝐴𝐵)) ⊆ ((𝐹𝐴) ∩ (𝐹𝐵))

Theoremimain 5012 The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))

Theoremfunimaexglem 5013 Lemma for funimaexg 5014. It constitutes the interesting part of funimaexg 5014, in which 𝐵 ⊆ dom 𝐴. (Contributed by Jim Kingdon, 27-Dec-2018.)
((Fun 𝐴𝐵𝐶𝐵 ⊆ dom 𝐴) → (𝐴𝐵) ∈ V)

Theoremfunimaexg 5014 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 5015 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 5016* 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 5017* 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 5015. (Contributed by NM, 26-Oct-2006.)
𝐴 ∈ V    &   𝑥𝐴𝑦𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)       𝑤 𝑤 = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)

Theoremfneq1 5018 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))

Theoremfneq2 5019 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹 Fn 𝐴𝐹 Fn 𝐵))

Theoremfneq1d 5020 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐹 = 𝐺)       (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))

Theoremfneq2d 5021 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹 Fn 𝐴𝐹 Fn 𝐵))

Theoremfneq12d 5022 Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐵))

Theoremfneq12 5023 Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)
((𝐹 = 𝐺𝐴 = 𝐵) → (𝐹 Fn 𝐴𝐺 Fn 𝐵))

Theoremfneq1i 5024 Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐹 = 𝐺       (𝐹 Fn 𝐴𝐺 Fn 𝐴)

Theoremfneq2i 5025 Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
𝐴 = 𝐵       (𝐹 Fn 𝐴𝐹 Fn 𝐵)

Theoremnffn 5026 Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
𝑥𝐹    &   𝑥𝐴       𝑥 𝐹 Fn 𝐴

Theoremfnfun 5027 A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
(𝐹 Fn 𝐴 → Fun 𝐹)

Theoremfnrel 5028 A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
(𝐹 Fn 𝐴 → Rel 𝐹)

Theoremfndm 5029 The domain of a function. (Contributed by NM, 2-Aug-1994.)
(𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)

Theoremfunfni 5030 Inference to convert a function and domain antecedent. (Contributed by NM, 22-Apr-2004.)
((Fun 𝐹𝐵 ∈ dom 𝐹) → 𝜑)       ((𝐹 Fn 𝐴𝐵𝐴) → 𝜑)

Theoremfndmu 5031 A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
((𝐹 Fn 𝐴𝐹 Fn 𝐵) → 𝐴 = 𝐵)

Theoremfnbr 5032 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 5033 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 5034* 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 5035* There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)
((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦𝐵, 𝑦⟩ ∈ 𝐹)

Theoremfnun 5036 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
(((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵))

Theoremfnunsn 5037 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 5038 Composition of two functions. (Contributed by NM, 22-May-2006.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn 𝐵)

Theoremfnresdm 5039 A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
(𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)

Theoremfnresdisj 5040 A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)
(𝐹 Fn 𝐴 → ((𝐴𝐵) = ∅ ↔ (𝐹𝐵) = ∅))

Theorem2elresin 5041 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 5042 Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)
(𝐹 Fn 𝐴 → ((𝐹𝐵) Fn 𝐵𝐵𝐴))

Theoremfnssres 5043 Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)

Theoremfnresin1 5044 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
(𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))

Theoremfnresin2 5045 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
(𝐹 Fn 𝐴 → (𝐹 ↾ (𝐵𝐴)) Fn (𝐵𝐴))

Theoremfnres 5046* An equivalence for functionality of a restriction. Compare dffun8 4959. (Contributed by Mario Carneiro, 20-May-2015.)
((𝐹𝐴) Fn 𝐴 ↔ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦)

Theoremfnresi 5047 Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
( I ↾ 𝐴) Fn 𝐴

Theoremfnima 5048 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 5049 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 5050 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 5051 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 5052 Alternate definition for the "maps to" notation df-mpt 3849. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝑥𝐴𝐵) = 𝑥𝐴 ({𝑥} × {𝐵})

Theoremfnopabg 5053* 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 5054* 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 5055* 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 5056* The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.)
𝐹 = (𝑥𝐴𝐵)       (∀𝑥𝐴 𝐵𝑉𝐹 Fn 𝐴)

Theoremmpt0 5057 A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝑥 ∈ ∅ ↦ 𝐴) = ∅

Theoremfnmpti 5058* 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 5059* 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 5060 Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝑥 ∈ (𝐴𝐵) ↦ 𝐶) = ((𝑥𝐴𝐶) ∪ (𝑥𝐵𝐶))

Theoremfeq1 5061 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹:𝐴𝐵𝐺:𝐴𝐵))

Theoremfeq2 5062 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹:𝐴𝐶𝐹:𝐵𝐶))

Theoremfeq3 5063 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹:𝐶𝐴𝐹:𝐶𝐵))

Theoremfeq23 5064 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 5065 Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
(𝜑𝐹 = 𝐺)       (𝜑 → (𝐹:𝐴𝐵𝐺:𝐴𝐵))

Theoremfeq2d 5066 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹:𝐴𝐶𝐹:𝐵𝐶))

Theoremfeq12d 5067 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐶))

Theoremfeq123d 5068 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐷))

Theoremfeq123 5069 Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → (𝐹:𝐴𝐵𝐺:𝐶𝐷))

Theoremfeq1i 5070 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐹 = 𝐺       (𝐹:𝐴𝐵𝐺:𝐴𝐵)

Theoremfeq2i 5071 Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
𝐴 = 𝐵       (𝐹:𝐴𝐶𝐹:𝐵𝐶)

Theoremfeq23i 5072 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐴 = 𝐶    &   𝐵 = 𝐷       (𝐹:𝐴𝐵𝐹:𝐶𝐷)

Theoremfeq23d 5073 Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
(𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → (𝐹:𝐴𝐵𝐹:𝐶𝐷))

Theoremnff 5074 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 5075* 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 5076* 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 5077 A mapping is a function. (Contributed by NM, 2-Aug-1994.)
(𝐹:𝐴𝐵𝐹 Fn 𝐴)

Theoremdffn2 5078 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 5079 A mapping is a function. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴𝐵 → Fun 𝐹)

Theoremfrel 5080 A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴𝐵 → Rel 𝐹)

Theoremfdm 5081 The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
(𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)

Theoremfdmi 5082 The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
𝐹:𝐴𝐵       dom 𝐹 = 𝐴

Theoremfrn 5083 The range of a mapping. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴𝐵 → ran 𝐹𝐵)

Theoremdffn3 5084 A function maps to its range. (Contributed by NM, 1-Sep-1999.)
(𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)

Theoremfss 5085 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 5086 Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐹:𝐴𝐶)

Theoremfco 5087 Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹:𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)

Theoremfco2 5088 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 5089 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 5090 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 5091 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 5092 A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
(𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))

Theoremopelf 5093 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 5094 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
(((𝐹:𝐴𝐶𝐺:𝐵𝐷) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐷))

Theoremfun2 5095 The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)
(((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)

Theoremfnfco 5096 Composition of two functions. (Contributed by NM, 22-May-2006.)
((𝐹 Fn 𝐴𝐺:𝐵𝐴) → (𝐹𝐺) Fn 𝐵)

Theoremfssres 5097 Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)
((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)

Theoremfssres2 5098 Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)
(((𝐹𝐴):𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)

Theoremfresin 5099 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 5100 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))

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