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Theorem List for Intuitionistic Logic Explorer - 4901-5000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremimadif 4901 The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.)
(Fun 𝐹 → (𝐹 “ (AB)) = ((𝐹A) ∖ (𝐹B)))
 
Theoremimainlem 4902 One direction of imain 4903. This direction does not require Fun 𝐹. (Contributed by Jim Kingdon, 25-Dec-2018.)
(𝐹 “ (AB)) ⊆ ((𝐹A) ∩ (𝐹B))
 
Theoremimain 4903 The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)
(Fun 𝐹 → (𝐹 “ (AB)) = ((𝐹A) ∩ (𝐹B)))
 
Theoremfunimaexglem 4904 Lemma for funimaexg 4905. It constitutes the interesting part of funimaexg 4905, in which B ⊆ dom A. (Contributed by Jim Kingdon, 27-Dec-2018.)
((Fun A B 𝐶 B ⊆ dom A) → (AB) V)
 
Theoremfunimaexg 4905 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 4906 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 4907* 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 4908* 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 4906. (Contributed by NM, 26-Oct-2006.)
A V    &   x A yz((φ [z / y]φ) → y = z)       w w = ({⟨x, y⟩ ∣ φ} “ A)
 
Theoremfneq1 4909 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹 Fn A𝐺 Fn A))
 
Theoremfneq2 4910 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
(A = B → (𝐹 Fn A𝐹 Fn B))
 
Theoremfneq1d 4911 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
(φ𝐹 = 𝐺)       (φ → (𝐹 Fn A𝐺 Fn A))
 
Theoremfneq2d 4912 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
(φA = B)       (φ → (𝐹 Fn A𝐹 Fn B))
 
Theoremfneq12d 4913 Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.)
(φ𝐹 = 𝐺)    &   (φA = B)       (φ → (𝐹 Fn A𝐺 Fn B))
 
Theoremfneq12 4914 Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)
((𝐹 = 𝐺 A = B) → (𝐹 Fn A𝐺 Fn B))
 
Theoremfneq1i 4915 Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐹 = 𝐺       (𝐹 Fn A𝐺 Fn A)
 
Theoremfneq2i 4916 Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
A = B       (𝐹 Fn A𝐹 Fn B)
 
Theoremnffn 4917 Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
x𝐹    &   xA       x 𝐹 Fn A
 
Theoremfnfun 4918 A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
(𝐹 Fn A → Fun 𝐹)
 
Theoremfnrel 4919 A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
(𝐹 Fn A → Rel 𝐹)
 
Theoremfndm 4920 The domain of a function. (Contributed by NM, 2-Aug-1994.)
(𝐹 Fn A → dom 𝐹 = A)
 
Theoremfunfni 4921 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 4922 A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
((𝐹 Fn A 𝐹 Fn B) → A = B)
 
Theoremfnbr 4923 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 4924 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 4925* 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 4926* There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)
((𝐹 Fn A B A) → ∃!yB, y 𝐹)
 
Theoremfnun 4927 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 4928 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 4929 Composition of two functions. (Contributed by NM, 22-May-2006.)
((𝐹 Fn A 𝐺 Fn B ran 𝐺A) → (𝐹𝐺) Fn B)
 
Theoremfnresdm 4930 A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
(𝐹 Fn A → (𝐹A) = 𝐹)
 
Theoremfnresdisj 4931 A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)
(𝐹 Fn A → ((AB) = ∅ ↔ (𝐹B) = ∅))
 
Theorem2elresin 4932 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 4933 Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)
(𝐹 Fn A → ((𝐹B) Fn BBA))
 
Theoremfnssres 4934 Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)
((𝐹 Fn A BA) → (𝐹B) Fn B)
 
Theoremfnresin1 4935 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
(𝐹 Fn A → (𝐹 ↾ (AB)) Fn (AB))
 
Theoremfnresin2 4936 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
(𝐹 Fn A → (𝐹 ↾ (BA)) Fn (BA))
 
Theoremfnres 4937* An equivalence for functionality of a restriction. Compare dffun8 4851. (Contributed by Mario Carneiro, 20-May-2015.)
((𝐹A) Fn Ax A ∃!y x𝐹y)
 
Theoremfnresi 4938 Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
( I ↾ A) Fn A
 
Theoremfnima 4939 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 4940 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 4941 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 4942 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 4943 Alternate definition for the "maps to" notation df-mpt 3790. (Contributed by Mario Carneiro, 30-Dec-2016.)
(x AB) = x A ({x} × {B})
 
Theoremfnopabg 4944* 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 4945* 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 4946* 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 4947* 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 4948 A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
(x ∅ ↦ A) = ∅
 
Theoremfnmpti 4949* 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 4950* 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 4951 Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)
(x (AB) ↦ 𝐶) = ((x A𝐶) ∪ (x B𝐶))
 
Theoremfeq1 4952 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(𝐹 = 𝐺 → (𝐹:AB𝐺:AB))
 
Theoremfeq2 4953 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(A = B → (𝐹:A𝐶𝐹:B𝐶))
 
Theoremfeq3 4954 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
(A = B → (𝐹:𝐶A𝐹:𝐶B))
 
Theoremfeq23 4955 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 4956 Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
(φ𝐹 = 𝐺)       (φ → (𝐹:AB𝐺:AB))
 
Theoremfeq2d 4957 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(φA = B)       (φ → (𝐹:A𝐶𝐹:B𝐶))
 
Theoremfeq12d 4958 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(φ𝐹 = 𝐺)    &   (φA = B)       (φ → (𝐹:A𝐶𝐺:B𝐶))
 
Theoremfeq123d 4959 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
(φ𝐹 = 𝐺)    &   (φA = B)    &   (φ𝐶 = 𝐷)       (φ → (𝐹:A𝐶𝐺:B𝐷))
 
Theoremfeq123 4960 Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
((𝐹 = 𝐺 A = 𝐶 B = 𝐷) → (𝐹:AB𝐺:𝐶𝐷))
 
Theoremfeq1i 4961 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐹 = 𝐺       (𝐹:AB𝐺:AB)
 
Theoremfeq2i 4962 Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
A = B       (𝐹:A𝐶𝐹:B𝐶)
 
Theoremfeq23i 4963 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
A = 𝐶    &   B = 𝐷       (𝐹:AB𝐹:𝐶𝐷)
 
Theoremfeq23d 4964 Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
(φA = 𝐶)    &   (φB = 𝐷)       (φ → (𝐹:AB𝐹:𝐶𝐷))
 
Theoremnff 4965 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 4966* 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 4967* 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 4968 A mapping is a function. (Contributed by NM, 2-Aug-1994.)
(𝐹:AB𝐹 Fn A)
 
Theoremdffn2 4969 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 4970 A mapping is a function. (Contributed by NM, 3-Aug-1994.)
(𝐹:AB → Fun 𝐹)
 
Theoremfrel 4971 A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
(𝐹:AB → Rel 𝐹)
 
Theoremfdm 4972 The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
(𝐹:AB → dom 𝐹 = A)
 
Theoremfdmi 4973 The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
𝐹:AB       dom 𝐹 = A
 
Theoremfrn 4974 The range of a mapping. (Contributed by NM, 3-Aug-1994.)
(𝐹:AB → ran 𝐹B)
 
Theoremdffn3 4975 A function maps to its range. (Contributed by NM, 1-Sep-1999.)
(𝐹 Fn A𝐹:A⟶ran 𝐹)
 
Theoremfss 4976 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)
 
Theoremfco 4977 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 4978 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)
 
Theoremfssxp 4979 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 4980 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:AB A 𝑉 B 𝑊) → 𝐹 V)
 
Theoremfunssxp 4981 Two ways of specifying a partial function from A to B. (Contributed by NM, 13-Nov-2007.)
((Fun 𝐹 𝐹 ⊆ (A × B)) ↔ (𝐹:dom 𝐹B dom 𝐹A))
 
Theoremffdm 4982 A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
(𝐹:AB → (𝐹:dom 𝐹B dom 𝐹A))
 
Theoremopelf 4983 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) → (𝐶 A 𝐷 B))
 
Theoremfun 4984 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
(((𝐹:A𝐶 𝐺:B𝐷) (AB) = ∅) → (𝐹𝐺):(AB)⟶(𝐶𝐷))
 
Theoremfun2 4985 The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)
(((𝐹:A𝐶 𝐺:B𝐶) (AB) = ∅) → (𝐹𝐺):(AB)⟶𝐶)
 
Theoremfnfco 4986 Composition of two functions. (Contributed by NM, 22-May-2006.)
((𝐹 Fn A 𝐺:BA) → (𝐹𝐺) Fn B)
 
Theoremfssres 4987 Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)
((𝐹:AB 𝐶A) → (𝐹𝐶):𝐶B)
 
Theoremfssres2 4988 Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)
(((𝐹A):AB 𝐶A) → (𝐹𝐶):𝐶B)
 
Theoremfresin 4989 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 4990 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 A 𝐺 Fn B (𝐹 ↾ (AB)) = (𝐺 ↾ (AB))) → ((𝐹 ↾ (AB)) ∪ ((𝐹 ↾ (AB)) ∪ (𝐺 ↾ (BA)))) ⊆ (𝐹𝐺))
 
Theoremfcoi1 4991 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 4992 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 4993* There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)
((𝐹:AB 𝐶 A) → ∃!y B𝐶, y 𝐹)
 
Theoremfcnvres 4994 The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
(𝐹:AB(𝐹A) = (𝐹B))
 
Theoremfimacnvdisj 4995 The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)
((𝐹:AB (B𝐶) = ∅) → (𝐹𝐶) = ∅)
 
Theoremfintm 4996* Function into an intersection. (Contributed by Jim Kingdon, 28-Dec-2018.)
x x B       (𝐹:A Bx B 𝐹:Ax)
 
Theoremfin 4997 Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝐹:A⟶(B𝐶) ↔ (𝐹:AB 𝐹:A𝐶))
 
Theoremfabexg 4998* Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)
𝐹 = {x ∣ (x:AB φ)}       ((A 𝐶 B 𝐷) → 𝐹 V)
 
Theoremfabex 4999* Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
A V    &   B V    &   𝐹 = {x ∣ (x:AB φ)}       𝐹 V
 
Theoremdmfex 5000 If a mapping is a set, its domain is a set. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹 𝐶 𝐹:AB) → A V)
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