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Theorem List for New Foundations Explorer - 4901-5000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdfdm3 4901* Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by set.mm contributors, 28-Dec-1996.)
dom A = {x yx, y A}

Theoremdfrn2 4902* Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by set.mm contributors, 27-Dec-1996.)
ran A = {y x xAy}

Theoremdfrn3 4903* Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by set.mm contributors, 28-Dec-1996.)
ran A = {y xx, y A}

Theoremdfrn4 4904 Alternate definition of range. (Contributed by set.mm contributors, 5-Feb-2015.)
ran A = dom A

Theoremdfdmf 4905* Definition of domain, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
xA    &   yA       dom A = {x y xAy}

Theoremdmss 4906 Subset theorem for domain. (Contributed by set.mm contributors, 11-Aug-1994.)
(A B → dom A dom B)

Theoremdmeq 4907 Equality theorem for domain. (Contributed by set.mm contributors, 11-Aug-1994.)
(A = B → dom A = dom B)

Theoremdmeqi 4908 Equality inference for domain. (Contributed by set.mm contributors, 4-Mar-2004.)
A = B       dom A = dom B

Theoremdmeqd 4909 Equality deduction for domain. (Contributed by set.mm contributors, 4-Mar-2004.)
(φA = B)       (φ → dom A = dom B)

Theoremopeldm 4910 Membership of first of an ordered pair in a domain. (Contributed by set.mm contributors, 30-Jul-1995.)
(A, B CA dom C)

Theorembreldm 4911 Membership of first of a binary relation in a domain. (Contributed by set.mm contributors, 8-Jan-2015.)
(ARBA dom R)

Theoremdmun 4912 The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 12-Aug-1994.) (Revised by set.mm contributors, 27-Aug-2011.)
dom (AB) = (dom A ∪ dom B)

Theoremdmin 4913 The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60. (Contributed by set.mm contributors, 15-Sep-2004.)
dom (AB) (dom A ∩ dom B)

Theoremdmuni 4914* The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by set.mm contributors, 3-Feb-2004.)
dom A = x A dom x

Theoremdmopab 4915* The domain of a class of ordered pairs. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
dom {x, y φ} = {x yφ}

Theoremdmopabss 4916* Upper bound for the domain of a restricted class of ordered pairs. (Contributed by set.mm contributors, 31-Jan-2004.)
dom {x, y (x A φ)} A

Theoremdmopab3 4917* The domain of a restricted class of ordered pairs. (Contributed by set.mm contributors, 31-Jan-2004.)
(x A yφ ↔ dom {x, y (x A φ)} = A)

Theoremdm0 4918 The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 4-Jul-1994.) (Revised by set.mm contributors, 27-Aug-2011.)
dom =

Theoremdmi 4919 The domain of the identity relation is the universe. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 30-Apr-1998.) (Revised by set.mm contributors, 27-Aug-2011.)
dom I = V

Theoremdmv 4920 The domain of the universe is the universe. (Contributed by set.mm contributors, 8-Aug-2003.)
dom V = V

Theoremdm0rn0 4921 An empty domain implies an empty range. (Contributed by set.mm contributors, 21-May-1998.)
(dom A = ↔ ran A = )

Theoremdmeq0 4922 A class is empty iff its domain is empty. (Contributed by set.mm contributors, 15-Sep-2004.) (Revised by Scott Fenton, 17-Apr-2021.)
(A = ↔ dom A = )

Theoremdmxp 4923 The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 28-Jul-1995.) (Revised by set.mm contributors, 27-Aug-2011.)
(B → dom (A × B) = A)

Theoremdmxpid 4924 The domain of a square cross product. (Contributed by set.mm contributors, 28-Jul-1995.)
dom (A × A) = A

Theoremdmxpin 4925 The domain of the intersection of two square cross products. Unlike dmin 4913, equality holds. (Contributed by set.mm contributors, 29-Jan-2008.)
dom ((A × A) ∩ (B × B)) = (AB)

Theoremxpid11 4926 The cross product of a class with itself is one-to-one. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 5-Nov-2006.) (Revised by set.mm contributors, 27-Aug-2011.)
((A × A) = (B × B) ↔ A = B)

Theoremproj1eldm 4927 The first member of an ordered pair in a class belongs to the domain of the class. (Contributed by set.mm contributors, 28-Jul-2004.) (Revised by Scott Fenton, 18-Apr-2021.)
(B A Proj1 B dom A)

Theoremreseq1 4928 Equality theorem for restrictions. (Contributed by set.mm contributors, 7-Aug-1994.)
(A = B → (A C) = (B C))

Theoremreseq2 4929 Equality theorem for restrictions. (Contributed by set.mm contributors, 8-Aug-1994.)
(A = B → (C A) = (C B))

Theoremreseq1i 4930 Equality inference for restrictions. (Contributed by set.mm contributors, 21-Oct-2014.)
A = B       (A C) = (B C)

Theoremreseq2i 4931 Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
A = B       (C A) = (C B)

Theoremreseq12i 4932 Equality inference for restrictions. (Contributed by set.mm contributors, 21-Oct-2014.)
A = B    &   C = D       (A C) = (B D)

Theoremreseq1d 4933 Equality deduction for restrictions. (Contributed by set.mm contributors, 21-Oct-2014.)
(φA = B)       (φ → (A C) = (B C))

Theoremreseq2d 4934 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
(φA = B)       (φ → (C A) = (C B))

Theoremreseq12d 4935 Equality deduction for restrictions. (Contributed by set.mm contributors, 21-Oct-2014.)
(φA = B)    &   (φC = D)       (φ → (A C) = (B D))

Theoremnfres 4936 Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)
xA    &   xB       x(A B)

Theoremimaeq1 4937 Equality theorem for image. (Contributed by set.mm contributors, 14-Aug-1994.)
(A = B → (AC) = (BC))

Theoremimaeq2 4938 Equality theorem for image. (Contributed by set.mm contributors, 14-Aug-1994.)
(A = B → (CA) = (CB))

Theoremimaeq1i 4939 Equality theorem for image. (Contributed by set.mm contributors, 21-Dec-2008.)
A = B       (AC) = (BC)

Theoremimaeq2i 4940 Equality theorem for image. (Contributed by set.mm contributors, 21-Dec-2008.)
A = B       (CA) = (CB)

Theoremimaeq1d 4941 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
(φA = B)       (φ → (AC) = (BC))

Theoremimaeq2d 4942 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
(φA = B)       (φ → (CA) = (CB))

Theoremimaeq12d 4943 Equality theorem for image. (Contributed by SF, 8-Jan-2018.)
(φA = B)    &   (φC = D)       (φ → (AC) = (BD))

Theoremelimapw1 4944* Membership in an image under a unit power class. (Contributed by set.mm contributors, 19-Feb-2015.)
(A (B1C) ↔ x C {x}, A B)

Theoremelimapw12 4945* Membership in an image under two unit power classes. (Contributed by set.mm contributors, 18-Mar-2015.)
(A (B11C) ↔ x C {{x}}, A B)

Theoremelimapw13 4946* Membership in an image under three unit power classes. (Contributed by set.mm contributors, 18-Mar-2015.)
(A (B111C) ↔ x C {{{x}}}, A B)

Theoremelima1c 4947* Membership in an image under cardinal one. (Contributed by set.mm contributors, 6-Feb-2015.)
(A (B “ 1c) ↔ x{x}, A B)

Theoremelimapw11c 4948* Membership in an image under the unit power class of cardinal one. (Contributed by set.mm contributors, 25-Feb-2015.)
(A (B11c) ↔ x{{x}}, A B)

Theorembrres 4949 Binary relation on a restriction. (Contributed by set.mm contributors, 12-Dec-2006.)
(A(C D)B ↔ (ACB A D))

Theoremopelres 4950 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by set.mm contributors, 13-Nov-1995.)
(A, B (C D) ↔ (A, B C A D))

Theoremdfima3 4951 Alternate definition of image. (Contributed by set.mm contributors, 19-Apr-2004.) (Revised by set.mm contributors, 27-Aug-2011.)
(AB) = ran (A B)

Theoremdfima4 4952* Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 14-Aug-1994.) (Revised by set.mm contributors, 27-Aug-2011.)
(AB) = {y x(x B x, y A)}

Theoremnfima 4953 Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
xA    &   xB       x(AB)

Theoremnfimad 4954 Deduction version of bound-variable hypothesis builder nfima 4953. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
(φxA)    &   (φxB)       (φx(AB))

Theoremcsbima12g 4955 Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
(A C[A / x](FB) = ([A / x]F[A / x]B))

Theoremrneq 4956 Equality theorem for range. (Contributed by set.mm contributors, 29-Dec-1996.)
(A = B → ran A = ran B)

Theoremrneqi 4957 Equality inference for range. (Contributed by set.mm contributors, 4-Mar-2004.)
A = B       ran A = ran B

Theoremrneqd 4958 Equality deduction for range. (Contributed by set.mm contributors, 4-Mar-2004.)
(φA = B)       (φ → ran A = ran B)

Theoremrnss 4959 Subset theorem for range. (Contributed by set.mm contributors, 22-Mar-1998.)
(A B → ran A ran B)

Theorembrelrn 4960 The second argument of a binary relation belongs to its range. (Contributed by set.mm contributors, 29-Jun-2008.)
(ACBB ran C)

Theoremopelrn 4961 Membership of second member of an ordered pair in a range. (Contributed by set.mm contributors, 8-Jan-2015.)
(A, B CB ran C)

Theoremdfrnf 4962* Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
xA    &   yA       ran A = {y x xAy}

Theoremnfrn 4963 Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)
xA       xran A

Theoremnfdm 4964 Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
xA       xdom A

Theoremdmiin 4965 Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
dom x A B x A dom B

Theoremcsbrng 4966 Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)
(A V[A / x]ran B = ran [A / x]B)

Theoremrnopab 4967* The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
ran {x, y φ} = {y xφ}

Theoremrnopab2 4968* The range of a function expressed as a class abstraction. (Contributed by set.mm contributors, 23-Mar-2006.)
ran {x, y (x A y = B)} = {y x A y = B}

Theoremrn0 4969 The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by set.mm contributors, 4-Jul-1994.)
ran =

Theoremrneq0 4970 A relation is empty iff its range is empty. (Contributed by set.mm contributors, 15-Sep-2004.) (Revised by Scott Fenton, 17-Apr-2021.)
(A = ↔ ran A = )

Theoremdmcoss 4971 Domain of a composition. Theorem 21 of [Suppes] p. 63. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 19-Mar-1998.) (Revised by set.mm contributors, 27-Aug-2011.)
dom (A B) dom B

Theoremrncoss 4972 Range of a composition. (Contributed by set.mm contributors, 19-Mar-1998.)
ran (A B) ran A

Theoremdmcosseq 4973 Domain of a composition. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 28-May-1998.) (Revised by set.mm contributors, 27-Aug-2011.)
(ran B dom A → dom (A B) = dom B)

Theoremdmcoeq 4974 Domain of a composition. (Contributed by set.mm contributors, 19-Mar-1998.)
(dom A = ran B → dom (A B) = dom B)

Theoremrncoeq 4975 Range of a composition. (Contributed by set.mm contributors, 19-Mar-1998.)
(dom A = ran B → ran (A B) = ran A)

Theoremcsbresg 4976 Distribute proper substitution through the restriction of a class. csbresg 4976 is derived from the virtual deduction proof csbresgVD in set.mm. (Contributed by Alan Sare, 10-Nov-2012.)
(A V[A / x](B C) = ([A / x]B [A / x]C))

Theoremres0 4977 A restriction to the empty set is empty. (Contributed by set.mm contributors, 12-Nov-1994.)
(A ) =

Theoremopres 4978 Ordered pair membership in a restriction when the first member belongs to the restricting class. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 30-Apr-2004.) (Revised by set.mm contributors, 27-Aug-2011.)
(A D → (A, B (C D) ↔ A, B C))

Theoremresieq 4979 A restricted identity relation is equivalent to equality in its domain. (Contributed by set.mm contributors, 30-Apr-2004.)
(B A → (B( I A)CB = C))

Theoremresres 4980 The restriction of a restriction. (Contributed by set.mm contributors, 27-Mar-2008.)
((A B) C) = (A (BC))

Theoremresundi 4981 Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by set.mm contributors, 30-Sep-2002.)
(A (BC)) = ((A B) ∪ (A C))

Theoremresundir 4982 Distributive law for restriction over union. (Contributed by set.mm contributors, 23-Sep-2004.)
((AB) C) = ((A C) ∪ (B C))

Theoremresindi 4983 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
(A (BC)) = ((A B) ∩ (A C))

Theoremresindir 4984 Class restriction distributes over intersection. (Contributed by set.mm contributors, 18-Dec-2008.)
((AB) C) = ((A C) ∩ (B C))

Theoreminres 4985 Move intersection into class restriction. (Contributed by set.mm contributors, 18-Dec-2008.)
(A ∩ (B C)) = ((AB) C)

Theoremdmres 4986 The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by set.mm contributors, 1-Aug-1994.)
dom (A B) = (B ∩ dom A)

Theoremssdmres 4987 A domain restricted to a subclass equals the subclass. (Contributed by set.mm contributors, 2-Mar-1997.) (Revised by set.mm contributors, 28-Aug-2004.)
(A dom B ↔ dom (B A) = A)

Theoremresss 4988 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by set.mm contributors, 2-Aug-1994.)
(A B) A

Theoremrescom 4989 Commutative law for restriction. (Contributed by set.mm contributors, 27-Mar-1998.)
((A B) C) = ((A C) B)

Theoremssres 4990 Subclass theorem for restriction. (Contributed by set.mm contributors, 16-Aug-1994.)
(A B → (A C) (B C))

Theoremssres2 4991 Subclass theorem for restriction. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 22-Mar-1998.) (Revised by set.mm contributors, 27-Aug-2011.)
(A B → (C A) (C B))

Theoremresabs1 4992 Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by set.mm contributors, 9-Aug-1994.)
(B C → ((A C) B) = (A B))

Theoremresabs2 4993 Absorption law for restriction. (Contributed by set.mm contributors, 27-Mar-1998.)
(B C → ((A B) C) = (A B))

Theoremresidm 4994 Idempotent law for restriction. (Contributed by set.mm contributors, 27-Mar-1998.)
((A B) B) = (A B)

Theoremelres 4995* Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)
(A (B C) ↔ x C y(A = x, y x, y B))

Theoremelsnres 4996* Memebership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.)
C V       (A (B {C}) ↔ y(A = C, y C, y B))

Theoremssreseq 4997 Simplification law for restriction. (Contributed by set.mm contributors, 16-Aug-1994.) (Revised by set.mm contributors, 15-Mar-2004.) (Revised by Scott Fenton, 18-Apr-2021.)
(dom A B → (A B) = A)

Theoremresdm 4998 A class restricted to its domain equals itself. (Contributed by set.mm contributors, 12-Dec-2006.) (Revised by Scott Fenton, 18-Apr-2021.)
(A dom A) = A

Theoremresopab 4999* Restriction of a class abstraction of ordered pairs. (Contributed by set.mm contributors, 5-Nov-2002.)
({x, y φ} A) = {x, y (x A φ)}

Theoremiss 5000 A subclass of the identity function is the identity function restricted to its domain. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 13-Dec-2003.) (Revised by set.mm contributors, 27-Aug-2011.)
(A I ↔ A = ( I dom A))

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