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Theorem List for Intuitionistic Logic Explorer - 4901-5000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfuncnvcnv 4901 The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
(Fun A → Fun A)
 
Theoremfuncnv2 4902* A simpler equivalence for single-rooted (see funcnv 4903). (Contributed by NM, 9-Aug-2004.)
(Fun Ay∃*x xAy)
 
Theoremfuncnv 4903* The converse of a class is a function iff the class is single-rooted, which means that for any y in the range of A there is at most one x such that xAy. Definition of single-rooted in [Enderton] p. 43. See funcnv2 4902 for a simpler version. (Contributed by NM, 13-Aug-2004.)
(Fun Ay ran A∃*x xAy)
 
Theoremfuncnv3 4904* A condition showing a class is single-rooted. (See funcnv 4903). (Contributed by NM, 26-May-2006.)
(Fun Ay ran A∃!x dom A xAy)
 
Theoremfuncnveq 4905* Another way of expressing that a class is single-rooted. Counterpart to dffun2 4855. (Contributed by Jim Kingdon, 24-Dec-2018.)
(Fun Axyz((xAy zAy) → x = z))
 
Theoremfun2cnv 4906* The double converse of a class is a function iff the class is single-valued. Each side is equivalent to Definition 6.4(2) of [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued. Note that A is not necessarily a function. (Contributed by NM, 13-Aug-2004.)
(Fun Ax∃*y xAy)
 
Theoremsvrelfun 4907 A single-valued relation is a function. (See fun2cnv 4906 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)
(Fun A ↔ (Rel A Fun A))
 
Theoremfncnv 4908* Single-rootedness (see funcnv 4903) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.)
((𝑅 ∩ (A × B)) Fn By B ∃!x A x𝑅y)
 
Theoremfun11 4909* 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((xAy zAw) → (x = zy = w)))
 
Theoremfununi 4910* The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.)
(f A (Fun f g A (fg gf)) → Fun A)
 
Theoremfuncnvuni 4911* The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 4903 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)
(f A (Fun f g A (fg gf)) → Fun A)
 
Theoremfun11uni 4912* The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by NM, 11-Aug-2004.)
(f A ((Fun f Fun f) g A (fg gf)) → (Fun A Fun A))
 
Theoremfunin 4913 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 4914 The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)
(Fun 𝐹 → Fun (𝐹A))
 
Theoremfuncnvres 4915 The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)
(Fun 𝐹(𝐹A) = (𝐹 ↾ (𝐹A)))
 
Theoremcnvresid 4916 Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
( I ↾ A) = ( I ↾ A)
 
Theoremfuncnvres2 4917 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 4918 The image of the preimage of a function. (Contributed by NM, 25-May-2004.)
(Fun 𝐹 → (𝐹 “ (𝐹A)) = (A ∩ ran 𝐹))
 
Theoremfunimass1 4919 A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)
((Fun 𝐹 A ⊆ ran 𝐹) → ((𝐹A) ⊆ BA ⊆ (𝐹B)))
 
Theoremfunimass2 4920 A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.)
((Fun 𝐹 A ⊆ (𝐹B)) → (𝐹A) ⊆ B)
 
Theoremimadiflem 4921 One direction of imadif 4922. This direction does not require Fun 𝐹. (Contributed by Jim Kingdon, 25-Dec-2018.)
((𝐹A) ∖ (𝐹B)) ⊆ (𝐹 “ (AB))
 
Theoremimadif 4922 The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.)
(Fun 𝐹 → (𝐹 “ (AB)) = ((𝐹A) ∖ (𝐹B)))
 
Theoremimainlem 4923 One direction of imain 4924. This direction does not require Fun 𝐹. (Contributed by Jim Kingdon, 25-Dec-2018.)
(𝐹 “ (AB)) ⊆ ((𝐹A) ∩ (𝐹B))
 
Theoremimain 4924 The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)
(Fun 𝐹 → (𝐹 “ (AB)) = ((𝐹A) ∩ (𝐹B)))
 
Theoremfunimaexglem 4925 Lemma for funimaexg 4926. It constitutes the interesting part of funimaexg 4926, in which B ⊆ dom A. (Contributed by Jim Kingdon, 27-Dec-2018.)
((Fun A B 𝐶 B ⊆ dom A) → (AB) V)
 
Theoremfunimaexg 4926 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 A B 𝐶) → (AB) V)
 
Theoremfunimaex 4927 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 4928* 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) ↔ x A [𝑏 / y]φ)
 
Theoremisarep2 4929* 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 4927. (Contributed by NM, 26-Oct-2006.)
A V    &   x A yz((φ [z / y]φ) → y = z)       w w = ({⟨x, y⟩ ∣ φ} “ A)
 
Theoremfneq1 4930 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹 Fn A𝐺 Fn A))
 
Theoremfneq2 4931 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
(A = B → (𝐹 Fn A𝐹 Fn B))
 
Theoremfneq1d 4932 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
(φ𝐹 = 𝐺)       (φ → (𝐹 Fn A𝐺 Fn A))
 
Theoremfneq2d 4933 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
(φA = B)       (φ → (𝐹 Fn A𝐹 Fn B))
 
Theoremfneq12d 4934 Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.)
(φ𝐹 = 𝐺)    &   (φA = B)       (φ → (𝐹 Fn A𝐺 Fn B))
 
Theoremfneq12 4935 Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)
((𝐹 = 𝐺 A = B) → (𝐹 Fn A𝐺 Fn B))
 
Theoremfneq1i 4936 Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐹 = 𝐺       (𝐹 Fn A𝐺 Fn A)
 
Theoremfneq2i 4937 Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
A = B       (𝐹 Fn A𝐹 Fn B)
 
Theoremnffn 4938 Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
x𝐹    &   xA       x 𝐹 Fn A
 
Theoremfnfun 4939 A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
(𝐹 Fn A → Fun 𝐹)
 
Theoremfnrel 4940 A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
(𝐹 Fn A → Rel 𝐹)
 
Theoremfndm 4941 The domain of a function. (Contributed by NM, 2-Aug-1994.)
(𝐹 Fn A → dom 𝐹 = A)
 
Theoremfunfni 4942 Inference to convert a function and domain antecedent. (Contributed by NM, 22-Apr-2004.)
((Fun 𝐹 B dom 𝐹) → φ)       ((𝐹 Fn A B A) → φ)
 
Theoremfndmu 4943 A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
((𝐹 Fn A 𝐹 Fn B) → A = B)
 
Theoremfnbr 4944 The first argument of binary relation on a function belongs to the function's domain. (Contributed by NM, 7-May-2004.)
((𝐹 Fn A B𝐹𝐶) → B A)
 
Theoremfnop 4945 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) → B A)
 
Theoremfneu 4946* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹 Fn A B A) → ∃!y B𝐹y)
 
Theoremfneu2 4947* There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)
((𝐹 Fn A B A) → ∃!yB, y 𝐹)
 
Theoremfnun 4948 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
(((𝐹 Fn A 𝐺 Fn B) (AB) = ∅) → (𝐹𝐺) Fn (AB))
 
Theoremfnunsn 4949 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 → ¬ 𝑋 𝐷)       (φ𝐺 Fn 𝐸)
 
Theoremfnco 4950 Composition of two functions. (Contributed by NM, 22-May-2006.)
((𝐹 Fn A 𝐺 Fn B ran 𝐺A) → (𝐹𝐺) Fn B)
 
Theoremfnresdm 4951 A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
(𝐹 Fn A → (𝐹A) = 𝐹)
 
Theoremfnresdisj 4952 A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)
(𝐹 Fn A → ((AB) = ∅ ↔ (𝐹B) = ∅))
 
Theorem2elresin 4953 Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.)
((𝐹 Fn A 𝐺 Fn B) → ((⟨x, y 𝐹 x, z 𝐺) ↔ (⟨x, y (𝐹 ↾ (AB)) x, z (𝐺 ↾ (AB)))))
 
Theoremfnssresb 4954 Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)
(𝐹 Fn A → ((𝐹B) Fn BBA))
 
Theoremfnssres 4955 Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)
((𝐹 Fn A BA) → (𝐹B) Fn B)
 
Theoremfnresin1 4956 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
(𝐹 Fn A → (𝐹 ↾ (AB)) Fn (AB))
 
Theoremfnresin2 4957 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
(𝐹 Fn A → (𝐹 ↾ (BA)) Fn (BA))
 
Theoremfnres 4958* An equivalence for functionality of a restriction. Compare dffun8 4872. (Contributed by Mario Carneiro, 20-May-2015.)
((𝐹A) Fn Ax A ∃!y x𝐹y)
 
Theoremfnresi 4959 Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
( I ↾ A) Fn A
 
Theoremfnima 4960 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 4961 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 4962 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 4963 Images under a function never map nonempty sets to empty sets. (Contributed by Stefan O'Rear, 21-Jan-2015.)
((𝐹 Fn A BA) → ((𝐹B) = ∅ ↔ B = ∅))
 
Theoremdfmpt3 4964 Alternate definition for the "maps to" notation df-mpt 3811. (Contributed by Mario Carneiro, 30-Dec-2016.)
(x AB) = x A ({x} × {B})
 
Theoremfnopabg 4965* 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⟩ ∣ (x A φ)}       (x A ∃!yφ𝐹 Fn A)
 
Theoremfnopab 4966* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.)
(x A∃!yφ)    &   𝐹 = {⟨x, y⟩ ∣ (x A φ)}       𝐹 Fn A
 
Theoremmptfng 4967* The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.)
𝐹 = (x AB)       (x A B V ↔ 𝐹 Fn A)
 
Theoremfnmpt 4968* The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.)
𝐹 = (x AB)       (x A B 𝑉𝐹 Fn A)
 
Theoremmpt0 4969 A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
(x ∅ ↦ A) = ∅
 
Theoremfnmpti 4970* 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 = (x AB)       𝐹 Fn A
 
Theoremdmmpti 4971* 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 = (x AB)       dom 𝐹 = A
 
Theoremmptun 4972 Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)
(x (AB) ↦ 𝐶) = ((x A𝐶) ∪ (x B𝐶))
 
Theoremfeq1 4973 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹:AB𝐺:AB))
 
Theoremfeq2 4974 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(A = B → (𝐹:A𝐶𝐹:B𝐶))
 
Theoremfeq3 4975 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(A = B → (𝐹:𝐶A𝐹:𝐶B))
 
Theoremfeq23 4976 Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((A = 𝐶 B = 𝐷) → (𝐹:AB𝐹:𝐶𝐷))
 
Theoremfeq1d 4977 Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
(φ𝐹 = 𝐺)       (φ → (𝐹:AB𝐺:AB))
 
Theoremfeq2d 4978 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(φA = B)       (φ → (𝐹:A𝐶𝐹:B𝐶))
 
Theoremfeq12d 4979 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(φ𝐹 = 𝐺)    &   (φA = B)       (φ → (𝐹:A𝐶𝐺:B𝐶))
 
Theoremfeq123d 4980 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(φ𝐹 = 𝐺)    &   (φA = B)    &   (φ𝐶 = 𝐷)       (φ → (𝐹:A𝐶𝐺:B𝐷))
 
Theoremfeq123 4981 Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
((𝐹 = 𝐺 A = 𝐶 B = 𝐷) → (𝐹:AB𝐺:𝐶𝐷))
 
Theoremfeq1i 4982 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐹 = 𝐺       (𝐹:AB𝐺:AB)
 
Theoremfeq2i 4983 Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
A = B       (𝐹:A𝐶𝐹:B𝐶)
 
Theoremfeq23i 4984 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
A = 𝐶    &   B = 𝐷       (𝐹:AB𝐹:𝐶𝐷)
 
Theoremfeq23d 4985 Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
(φA = 𝐶)    &   (φB = 𝐷)       (φ → (𝐹:AB𝐹:𝐶𝐷))
 
Theoremnff 4986 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 4987* Distribute proper substitution through the function predicate with a domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
(𝑋 𝑉 → ([𝑋 / x]𝐹 Fn A𝑋 / x𝐹 Fn 𝑋 / xA))
 
Theoremsbcfg 4988* Distribute proper substitution through the function predicate with domain and codomain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
(𝑋 𝑉 → ([𝑋 / x]𝐹:AB𝑋 / x𝐹:𝑋 / xA𝑋 / xB))
 
Theoremffn 4989 A mapping is a function. (Contributed by NM, 2-Aug-1994.)
(𝐹:AB𝐹 Fn A)
 
Theoremdffn2 4990 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 4991 A mapping is a function. (Contributed by NM, 3-Aug-1994.)
(𝐹:AB → Fun 𝐹)
 
Theoremfrel 4992 A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
(𝐹:AB → Rel 𝐹)
 
Theoremfdm 4993 The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
(𝐹:AB → dom 𝐹 = A)
 
Theoremfdmi 4994 The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
𝐹:AB       dom 𝐹 = A
 
Theoremfrn 4995 The range of a mapping. (Contributed by NM, 3-Aug-1994.)
(𝐹:AB → ran 𝐹B)
 
Theoremdffn3 4996 A function maps to its range. (Contributed by NM, 1-Sep-1999.)
(𝐹 Fn A𝐹:A⟶ran 𝐹)
 
Theoremfss 4997 Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹:AB B𝐶) → 𝐹:A𝐶)
 
Theoremfssd 4998 Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(φ𝐹:AB)    &   (φB𝐶)       (φ𝐹:A𝐶)
 
Theoremfco 4999 Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹:B𝐶 𝐺:AB) → (𝐹𝐺):A𝐶)
 
Theoremfco2 5000 Functionality of a composition with weakened out of domain condition on the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.)
(((𝐹B):B𝐶 𝐺:AB) → (𝐹𝐺):A𝐶)
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