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Theorem List for Metamath Proof Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsvrelfun 6101 A single-valued relation is a function. (See fun2cnv 6100 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ Fun 𝐴))
 
Theoremfncnv 6102* Single-rootedness (see funcnv 6098) of a class cut down by a Cartesian product. (Contributed by NM, 5-Mar-2007.)
((𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑥𝑅𝑦)
 
Theoremfun11 6103* Two ways of stating that 𝐴 is one-to-one (but not necessarily a function). Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one (but not necessarily a function). (Contributed by NM, 17-Jan-2006.)
((Fun 𝐴 ∧ Fun 𝐴) ↔ ∀𝑥𝑦𝑧𝑤((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)))
 
Theoremfununi 6104* The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.)
(∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Fun 𝐴)
 
Theoremfunin 6105 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 6106 The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)
(Fun 𝐹 → Fun (𝐹𝐴))
 
Theoremfuncnvres 6107 The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)
(Fun 𝐹(𝐹𝐴) = (𝐹 ↾ (𝐹𝐴)))
 
Theoremcnvresid 6108 Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
( I ↾ 𝐴) = ( I ↾ 𝐴)
 
Theoremfuncnvres2 6109 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 6110 The image of the preimage of a function. (Contributed by NM, 25-May-2004.)
(Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = (𝐴 ∩ ran 𝐹))
 
Theoremfunimass1 6111 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 6112 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)
 
Theoremimadif 6113 The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
 
Theoremimain 6114 The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
 
Theoremfunimaexg 6115 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 6116 The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 4902. 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 6117* 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 6118* 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 6116. (Contributed by NM, 26-Oct-2006.)
𝐴 ∈ V    &   𝑥𝐴𝑦𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)       𝑤 𝑤 = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)
 
Theoremfneq1 6119 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
 
Theoremfneq2 6120 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹 Fn 𝐴𝐹 Fn 𝐵))
 
Theoremfneq1d 6121 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐹 = 𝐺)       (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
 
Theoremfneq2d 6122 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹 Fn 𝐴𝐹 Fn 𝐵))
 
Theoremfneq12d 6123 Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐵))
 
Theoremfneq12 6124 Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)
((𝐹 = 𝐺𝐴 = 𝐵) → (𝐹 Fn 𝐴𝐺 Fn 𝐵))
 
Theoremfneq1i 6125 Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐹 = 𝐺       (𝐹 Fn 𝐴𝐺 Fn 𝐴)
 
Theoremfneq2i 6126 Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
𝐴 = 𝐵       (𝐹 Fn 𝐴𝐹 Fn 𝐵)
 
Theoremnffn 6127 Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
𝑥𝐹    &   𝑥𝐴       𝑥 𝐹 Fn 𝐴
 
Theoremfnfun 6128 A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
(𝐹 Fn 𝐴 → Fun 𝐹)
 
Theoremfnrel 6129 A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
(𝐹 Fn 𝐴 → Rel 𝐹)
 
Theoremfndm 6130 The domain of a function. (Contributed by NM, 2-Aug-1994.)
(𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
 
Theoremfunfni 6131 Inference to convert a function and domain antecedent. (Contributed by NM, 22-Apr-2004.)
((Fun 𝐹𝐵 ∈ dom 𝐹) → 𝜑)       ((𝐹 Fn 𝐴𝐵𝐴) → 𝜑)
 
Theoremfndmu 6132 A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
((𝐹 Fn 𝐴𝐹 Fn 𝐵) → 𝐴 = 𝐵)
 
Theoremfnbr 6133 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 6134 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 6135* 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 6136* There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)
((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦𝐵, 𝑦⟩ ∈ 𝐹)
 
Theoremfnun 6137 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
(((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵))
 
Theoremfnunsn 6138 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 6139 Composition of two functions. (Contributed by NM, 22-May-2006.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn 𝐵)
 
Theoremfnresdm 6140 A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
(𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
 
Theoremfnresdisj 6141 A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)
(𝐹 Fn 𝐴 → ((𝐴𝐵) = ∅ ↔ (𝐹𝐵) = ∅))
 
Theorem2elresin 6142 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 6143 Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)
(𝐹 Fn 𝐴 → ((𝐹𝐵) Fn 𝐵𝐵𝐴))
 
Theoremfnssres 6144 Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)
 
Theoremfnresin1 6145 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
(𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))
 
Theoremfnresin2 6146 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
(𝐹 Fn 𝐴 → (𝐹 ↾ (𝐵𝐴)) Fn (𝐵𝐴))
 
Theoremfnres 6147* An equivalence for functionality of a restriction. Compare dffun8 6059. (Contributed by Mario Carneiro, 20-May-2015.)
((𝐹𝐴) Fn 𝐴 ↔ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦)
 
Theoremfnresi 6148 Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
( I ↾ 𝐴) Fn 𝐴
 
Theoremidssxp 6149 A diagonal set as a subset of a Cartesian product. (Contributed by Thierry Arnoux, 29-Dec-2019.)
( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
 
Theoremfnima 6150 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 6151 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 6152 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 6153 Images under a function never map nonempty sets to empty sets. EDITORIAL: usable in fnwe2lem2 38140. (Contributed by Stefan O'Rear, 21-Jan-2015.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = ∅ ↔ 𝐵 = ∅))
 
Theoremdfmpt3 6154 Alternate definition for the "maps to" notation df-mpt 4862. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝑥𝐴𝐵) = 𝑥𝐴 ({𝑥} × {𝐵})
 
Theoremmptfnf 6155 The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Thierry Arnoux, 10-May-2017.)
𝑥𝐴       (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
 
Theoremfnmptf 6156 The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)
𝑥𝐴       (∀𝑥𝐴 𝐵𝑉 → (𝑥𝐴𝐵) Fn 𝐴)
 
Theoremfnopabg 6157* 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 6158* 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 6159* 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 6160* The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.)
𝐹 = (𝑥𝐴𝐵)       (∀𝑥𝐴 𝐵𝑉𝐹 Fn 𝐴)
 
Theoremmpt0 6161 A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝑥 ∈ ∅ ↦ 𝐴) = ∅
 
Theoremfnmpti 6162* 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 6163* Domain of the mapping operation. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐵 ∈ V    &   𝐹 = (𝑥𝐴𝐵)       dom 𝐹 = 𝐴
 
Theoremdmmptd 6164* The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴 = (𝑥𝐵𝐶)    &   ((𝜑𝑥𝐵) → 𝐶𝑉)       (𝜑 → dom 𝐴 = 𝐵)
 
Theoremmptun 6165 Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝑥 ∈ (𝐴𝐵) ↦ 𝐶) = ((𝑥𝐴𝐶) ∪ (𝑥𝐵𝐶))
 
Theoremfeq1 6166 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹:𝐴𝐵𝐺:𝐴𝐵))
 
Theoremfeq2 6167 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹:𝐴𝐶𝐹:𝐵𝐶))
 
Theoremfeq3 6168 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (𝐹:𝐶𝐴𝐹:𝐶𝐵))
 
Theoremfeq23 6169 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 6170 Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
(𝜑𝐹 = 𝐺)       (𝜑 → (𝐹:𝐴𝐵𝐺:𝐴𝐵))
 
Theoremfeq2d 6171 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹:𝐴𝐶𝐹:𝐵𝐶))
 
Theoremfeq3d 6172 Equality deduction for functions. (Contributed by AV, 1-Jan-2020.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹:𝑋𝐴𝐹:𝑋𝐵))
 
Theoremfeq12d 6173 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐶))
 
Theoremfeq123d 6174 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐷))
 
Theoremfeq123 6175 Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
((𝐹 = 𝐺𝐴 = 𝐶𝐵 = 𝐷) → (𝐹:𝐴𝐵𝐺:𝐶𝐷))
 
Theoremfeq1i 6176 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐹 = 𝐺       (𝐹:𝐴𝐵𝐺:𝐴𝐵)
 
Theoremfeq2i 6177 Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
𝐴 = 𝐵       (𝐹:𝐴𝐶𝐹:𝐵𝐶)
 
Theoremfeq12i 6178 Equality inference for functions. (Contributed by AV, 7-Feb-2021.)
𝐹 = 𝐺    &   𝐴 = 𝐵       (𝐹:𝐴𝐶𝐺:𝐵𝐶)
 
Theoremfeq23i 6179 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐴 = 𝐶    &   𝐵 = 𝐷       (𝐹:𝐴𝐵𝐹:𝐶𝐷)
 
Theoremfeq23d 6180 Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
(𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → (𝐹:𝐴𝐵𝐹:𝐶𝐷))
 
Theoremnff 6181 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 6182* 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 6183* 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))
 
Theoremelimf 6184 Eliminate a mapping hypothesis for the weak deduction theorem dedth 4276, when a special case 𝐺:𝐴𝐵 is provable, in order to convert 𝐹:𝐴𝐵 from a hypothesis to an antecedent. (Contributed by NM, 24-Aug-2006.)
𝐺:𝐴𝐵       if(𝐹:𝐴𝐵, 𝐹, 𝐺):𝐴𝐵
 
Theoremffn 6185 A mapping is a function with domain. (Contributed by NM, 2-Aug-1994.)
(𝐹:𝐴𝐵𝐹 Fn 𝐴)
 
Theoremffnd 6186 A mapping is a function with domain, deduction form. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐹:𝐴𝐵)       (𝜑𝐹 Fn 𝐴)
 
Theoremdffn2 6187 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 6188 A mapping is a function. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴𝐵 → Fun 𝐹)
 
Theoremffund 6189 A mapping is a function, deduction version. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → Fun 𝐹)
 
Theoremfrel 6190 A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴𝐵 → Rel 𝐹)
 
Theoremfdm 6191 The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
(𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
 
Theoremfdmi 6192 The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
𝐹:𝐴𝐵       dom 𝐹 = 𝐴
 
Theoremfrn 6193 The range of a mapping. (Contributed by NM, 3-Aug-1994.)
(𝐹:𝐴𝐵 → ran 𝐹𝐵)
 
Theoremdffn3 6194 A function maps to its range. (Contributed by NM, 1-Sep-1999.)
(𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
 
Theoremffrn 6195 A function maps to its range. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝐹:𝐴𝐵𝐹:𝐴⟶ran 𝐹)
 
Theoremfss 6196 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 6197 Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐹:𝐴𝐶)
 
Theoremfco 6198 Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹:𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
 
Theoremfco2 6199 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 6200 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))
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