Type | Label | Description |
Statement |
|
Theorem | reseq12d 4901 |
Equality deduction for restrictions. (Contributed by NM,
21-Oct-2014.)
|
⊢ (𝜑 → 𝐴 = 𝐵)
& ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷)) |
|
Theorem | nfres 4902 |
Bound-variable hypothesis builder for restriction. (Contributed by NM,
15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵) |
|
Theorem | csbresg 4903 |
Distribute proper substitution through the restriction of a class.
(Contributed by Alan Sare, 10-Nov-2012.)
|
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶)) |
|
Theorem | res0 4904 |
A restriction to the empty set is empty. (Contributed by NM,
12-Nov-1994.)
|
⊢ (𝐴 ↾ ∅) =
∅ |
|
Theorem | opelres 4905 |
Ordered pair membership in a restriction. Exercise 13 of
[TakeutiZaring] p. 25.
(Contributed by NM, 13-Nov-1995.)
|
⊢ 𝐵 ∈ V ⇒ ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)) |
|
Theorem | brres 4906 |
Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
|
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴(𝐶 ↾ 𝐷)𝐵 ↔ (𝐴𝐶𝐵 ∧ 𝐴 ∈ 𝐷)) |
|
Theorem | opelresg 4907 |
Ordered pair membership in a restriction. Exercise 13 of
[TakeutiZaring] p. 25.
(Contributed by NM, 14-Oct-2005.)
|
⊢ (𝐵 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))) |
|
Theorem | brresg 4908 |
Binary relation on a restriction. (Contributed by Mario Carneiro,
4-Nov-2015.)
|
⊢ (𝐵 ∈ 𝑉 → (𝐴(𝐶 ↾ 𝐷)𝐵 ↔ (𝐴𝐶𝐵 ∧ 𝐴 ∈ 𝐷))) |
|
Theorem | opres 4909 |
Ordered pair membership in a restriction when the first member belongs
to the restricting class. (Contributed by NM, 30-Apr-2004.) (Proof
shortened by Andrew Salmon, 27-Aug-2011.)
|
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∈ 𝐷 → (⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶)) |
|
Theorem | resieq 4910 |
A restricted identity relation is equivalent to equality in its domain.
(Contributed by NM, 30-Apr-2004.)
|
⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵( I ↾ 𝐴)𝐶 ↔ 𝐵 = 𝐶)) |
|
Theorem | opelresi 4911 |
⟨𝐴,
𝐴⟩ belongs to a
restriction of the identity class iff 𝐴
belongs to the restricting class. (Contributed by FL, 27-Oct-2008.)
(Revised by NM, 30-Mar-2016.)
|
⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ 𝐴 ∈ 𝐵)) |
|
Theorem | resres 4912 |
The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
|
⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶)) |
|
Theorem | resundi 4913 |
Distributive law for restriction over union. Theorem 31 of [Suppes]
p. 65. (Contributed by NM, 30-Sep-2002.)
|
⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) |
|
Theorem | resundir 4914 |
Distributive law for restriction over union. (Contributed by NM,
23-Sep-2004.)
|
⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) |
|
Theorem | resindi 4915 |
Class restriction distributes over intersection. (Contributed by FL,
6-Oct-2008.)
|
⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ 𝐶)) |
|
Theorem | resindir 4916 |
Class restriction distributes over intersection. (Contributed by NM,
18-Dec-2008.)
|
⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) |
|
Theorem | inres 4917 |
Move intersection into class restriction. (Contributed by NM,
18-Dec-2008.)
|
⊢ (𝐴 ∩ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ 𝐵) ↾ 𝐶) |
|
Theorem | resdifcom 4918 |
Commutative law for restriction and difference. (Contributed by AV,
7-Jun-2021.)
|
⊢ ((𝐴 ↾ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ↾ 𝐵) |
|
Theorem | resiun1 4919* |
Distribution of restriction over indexed union. (Contributed by Mario
Carneiro, 29-May-2015.)
|
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪
𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) |
|
Theorem | resiun2 4920* |
Distribution of restriction over indexed union. (Contributed by Mario
Carneiro, 29-May-2015.)
|
⊢ (𝐶 ↾ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪
𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵) |
|
Theorem | dmres 4921 |
The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25.
(Contributed by NM, 1-Aug-1994.)
|
⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) |
|
Theorem | ssdmres 4922 |
A domain restricted to a subclass equals the subclass. (Contributed by
NM, 2-Mar-1997.)
|
⊢ (𝐴 ⊆ dom 𝐵 ↔ dom (𝐵 ↾ 𝐴) = 𝐴) |
|
Theorem | dmresexg 4923 |
The domain of a restriction to a set exists. (Contributed by NM,
7-Apr-1995.)
|
⊢ (𝐵 ∈ 𝑉 → dom (𝐴 ↾ 𝐵) ∈ V) |
|
Theorem | resss 4924 |
A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25.
(Contributed by NM, 2-Aug-1994.)
|
⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 |
|
Theorem | rescom 4925 |
Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
|
⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ↾ 𝐵) |
|
Theorem | ssres 4926 |
Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶)) |
|
Theorem | ssres2 4927 |
Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵)) |
|
Theorem | relres 4928 |
A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25.
(Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon,
27-Aug-2011.)
|
⊢ Rel (𝐴 ↾ 𝐵) |
|
Theorem | resabs1 4929 |
Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25.
(Contributed by NM, 9-Aug-1994.)
|
⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵)) |
|
Theorem | resabs1d 4930 |
Absorption law for restriction, deduction form. (Contributed by Glauco
Siliprandi, 11-Dec-2019.)
|
⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵)) |
|
Theorem | resabs2 4931 |
Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
|
⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵)) |
|
Theorem | residm 4932 |
Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
|
⊢ ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵) |
|
Theorem | resima 4933 |
A restriction to an image. (Contributed by NM, 29-Sep-2004.)
|
⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = (𝐴 “ 𝐵) |
|
Theorem | resima2 4934 |
Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
|
⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵)) |
|
Theorem | xpssres 4935 |
Restriction of a constant function (or other cross product). (Contributed
by Stefan O'Rear, 24-Jan-2015.)
|
⊢ (𝐶 ⊆ 𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵)) |
|
Theorem | elres 4936* |
Membership in a restriction. (Contributed by Scott Fenton,
17-Mar-2011.)
|
⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵)) |
|
Theorem | elsnres 4937* |
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)) |
|
Theorem | relssres 4938 |
Simplification law for restriction. (Contributed by NM,
16-Aug-1994.)
|
⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → (𝐴 ↾ 𝐵) = 𝐴) |
|
Theorem | resdm 4939 |
A relation restricted to its domain equals itself. (Contributed by NM,
12-Dec-2006.)
|
⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴) |
|
Theorem | resexg 4940 |
The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.)
|
⊢ (𝐴 ∈ 𝑉 → (𝐴 ↾ 𝐵) ∈ V) |
|
Theorem | resex 4941 |
The restriction of a set is a set. (Contributed by Jeff Madsen,
19-Jun-2011.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ↾ 𝐵) ∈ V |
|
Theorem | resindm 4942 |
When restricting a relation, intersecting with the domain of the relation
has no effect. (Contributed by FL, 6-Oct-2008.)
|
⊢ (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵)) |
|
Theorem | resdmdfsn 4943 |
Restricting a relation to its domain without a set is the same as
restricting the relation to the universe without this set. (Contributed
by AV, 2-Dec-2018.)
|
⊢ (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))) |
|
Theorem | resopab 4944* |
Restriction of a class abstraction of ordered pairs. (Contributed by
NM, 5-Nov-2002.)
|
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
|
Theorem | resiexg 4945 |
The existence of a restricted identity function, proved without using
the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.)
|
⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V) |
|
Theorem | iss 4946 |
A subclass of the identity function is the identity function restricted
to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by
Andrew Salmon, 27-Aug-2011.)
|
⊢ (𝐴 ⊆ I ↔ 𝐴 = ( I ↾ dom 𝐴)) |
|
Theorem | resopab2 4947* |
Restriction of a class abstraction of ordered pairs. (Contributed by
NM, 24-Aug-2007.)
|
⊢ (𝐴 ⊆ 𝐵 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
|
Theorem | resmpt 4948* |
Restriction of the mapping operation. (Contributed by Mario Carneiro,
15-Jul-2013.)
|
⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) |
|
Theorem | resmpt3 4949* |
Unconditional restriction of the mapping operation. (Contributed by
Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro,
22-Mar-2015.)
|
⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) |
|
Theorem | resmptf 4950 |
Restriction of the mapping operation. (Contributed by Thierry Arnoux,
28-Mar-2017.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) |
|
Theorem | resmptd 4951* |
Restriction of the mapping operation, deduction form. (Contributed by
Glauco Siliprandi, 11-Dec-2019.)
|
⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) |
|
Theorem | dfres2 4952* |
Alternate definition of the restriction operation. (Contributed by
Mario Carneiro, 5-Nov-2013.)
|
⊢ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} |
|
Theorem | opabresid 4953* |
The restricted identity expressed with the class builder. (Contributed
by FL, 25-Apr-2012.)
|
⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐴) |
|
Theorem | mptresid 4954* |
The restricted identity expressed with the maps-to notation.
(Contributed by FL, 25-Apr-2012.)
|
⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ( I ↾ 𝐴) |
|
Theorem | dmresi 4955 |
The domain of a restricted identity function. (Contributed by NM,
27-Aug-2004.)
|
⊢ dom ( I ↾ 𝐴) = 𝐴 |
|
Theorem | restidsing 4956 |
Restriction of the identity to a singleton. (Contributed by FL,
2-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.) (Proof shortened by
Peter Mazsa, 6-Oct-2022.)
|
⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴}) |
|
Theorem | resid 4957 |
Any relation restricted to the universe is itself. (Contributed by NM,
16-Mar-2004.)
|
⊢ (Rel 𝐴 → (𝐴 ↾ V) = 𝐴) |
|
Theorem | imaeq1 4958 |
Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
|
⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) |
|
Theorem | imaeq2 4959 |
Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
|
⊢ (𝐴 = 𝐵 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵)) |
|
Theorem | imaeq1i 4960 |
Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 “ 𝐶) = (𝐵 “ 𝐶) |
|
Theorem | imaeq2i 4961 |
Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 “ 𝐴) = (𝐶 “ 𝐵) |
|
Theorem | imaeq1d 4962 |
Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) |
|
Theorem | imaeq2d 4963 |
Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵)) |
|
Theorem | imaeq12d 4964 |
Equality theorem for image. (Contributed by Mario Carneiro,
4-Dec-2016.)
|
⊢ (𝜑 → 𝐴 = 𝐵)
& ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐷)) |
|
Theorem | dfima2 4965* |
Alternate definition of image. Compare definition (d) of [Enderton]
p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew
Salmon, 27-Aug-2011.)
|
⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} |
|
Theorem | dfima3 4966* |
Alternate definition of image. Compare definition (d) of [Enderton]
p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew
Salmon, 27-Aug-2011.)
|
⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} |
|
Theorem | elimag 4967* |
Membership in an image. Theorem 34 of [Suppes]
p. 65. (Contributed by
NM, 20-Jan-2007.)
|
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴)) |
|
Theorem | elima 4968* |
Membership in an image. Theorem 34 of [Suppes]
p. 65. (Contributed by
NM, 19-Apr-2004.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴) |
|
Theorem | elima2 4969* |
Membership in an image. Theorem 34 of [Suppes]
p. 65. (Contributed by
NM, 11-Aug-2004.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝐴)) |
|
Theorem | elima3 4970* |
Membership in an image. Theorem 34 of [Suppes]
p. 65. (Contributed by
NM, 14-Aug-1994.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵)) |
|
Theorem | nfima 4971 |
Bound-variable hypothesis builder for image. (Contributed by NM,
30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 “ 𝐵) |
|
Theorem | nfimad 4972 |
Deduction version of bound-variable hypothesis builder nfima 4971.
(Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro,
15-Oct-2016.)
|
⊢ (𝜑 → Ⅎ𝑥𝐴)
& ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥(𝐴 “ 𝐵)) |
|
Theorem | imadmrn 4973 |
The image of the domain of a class is the range of the class.
(Contributed by NM, 14-Aug-1994.)
|
⊢ (𝐴 “ dom 𝐴) = ran 𝐴 |
|
Theorem | imassrn 4974 |
The image of a class is a subset of its range. Theorem 3.16(xi) of
[Monk1] p. 39. (Contributed by NM,
31-Mar-1995.)
|
⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴 |
|
Theorem | imaexg 4975 |
The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed
by NM, 24-Jul-1995.)
|
⊢ (𝐴 ∈ 𝑉 → (𝐴 “ 𝐵) ∈ V) |
|
Theorem | imaex 4976 |
The image of a set is a set. Theorem 3.17 of [Monk1] p. 39.
(Contributed by JJ, 24-Sep-2021.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 “ 𝐵) ∈ V |
|
Theorem | imai 4977 |
Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38.
(Contributed by NM, 30-Apr-1998.)
|
⊢ ( I “ 𝐴) = 𝐴 |
|
Theorem | rnresi 4978 |
The range of the restricted identity function. (Contributed by NM,
27-Aug-2004.)
|
⊢ ran ( I ↾ 𝐴) = 𝐴 |
|
Theorem | resiima 4979 |
The image of a restriction of the identity function. (Contributed by FL,
31-Dec-2006.)
|
⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵) |
|
Theorem | ima0 4980 |
Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed
by NM, 20-May-1998.)
|
⊢ (𝐴 “ ∅) =
∅ |
|
Theorem | 0ima 4981 |
Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
|
⊢ (∅ “ 𝐴) = ∅ |
|
Theorem | csbima12g 4982 |
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⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵)) |
|
Theorem | imadisj 4983 |
A class whose image under another is empty is disjoint with the other's
domain. (Contributed by FL, 24-Jan-2007.)
|
⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅) |
|
Theorem | cnvimass 4984 |
A preimage under any class is included in the domain of the class.
(Contributed by FL, 29-Jan-2007.)
|
⊢ (◡𝐴 “ 𝐵) ⊆ dom 𝐴 |
|
Theorem | cnvimarndm 4985 |
The preimage of the range of a class is the domain of the class.
(Contributed by Jeff Hankins, 15-Jul-2009.)
|
⊢ (◡𝐴 “ ran 𝐴) = dom 𝐴 |
|
Theorem | imasng 4986* |
The image of a singleton. (Contributed by NM, 8-May-2005.)
|
⊢ (𝐴 ∈ 𝐵 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦}) |
|
Theorem | elreimasng 4987 |
Elementhood in the image of a singleton. (Contributed by Jim Kingdon,
10-Dec-2018.)
|
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑉) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵)) |
|
Theorem | elimasn 4988 |
Membership in an image of a singleton. (Contributed by NM,
15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|
⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈
V ⇒ ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴) |
|
Theorem | elimasng 4989 |
Membership in an image of a singleton. (Contributed by Raph Levien,
21-Oct-2006.)
|
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)) |
|
Theorem | args 4990* |
Two ways to express the class of unique-valued arguments of 𝐹,
which is the same as the domain of 𝐹 whenever 𝐹 is a function.
The left-hand side of the equality is from Definition 10.2 of [Quine]
p. 65. Quine uses the notation "arg 𝐹 " for this class
(for which
we have no separate notation). (Contributed by NM, 8-May-2005.)
|
⊢ {𝑥 ∣ ∃𝑦(𝐹 “ {𝑥}) = {𝑦}} = {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} |
|
Theorem | eliniseg 4991 |
Membership in an initial segment. The idiom (◡𝐴 “ {𝐵}),
meaning {𝑥 ∣ 𝑥𝐴𝐵}, is used to specify an initial
segment in
(for example) Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by
NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|
⊢ 𝐶 ∈ V ⇒ ⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵)) |
|
Theorem | epini 4992 |
Any set is equal to its preimage under the converse epsilon relation.
(Contributed by Mario Carneiro, 9-Mar-2013.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (◡ E “ {𝐴}) = 𝐴 |
|
Theorem | iniseg 4993* |
An idiom that signifies an initial segment of an ordering, used, for
example, in Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by
NM, 28-Apr-2004.)
|
⊢ (𝐵 ∈ 𝑉 → (◡𝐴 “ {𝐵}) = {𝑥 ∣ 𝑥𝐴𝐵}) |
|
Theorem | dfse2 4994* |
Alternate definition of set-like relation. (Contributed by Mario
Carneiro, 23-Jun-2015.)
|
⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V) |
|
Theorem | exse2 4995 |
Any set relation is set-like. (Contributed by Mario Carneiro,
22-Jun-2015.)
|
⊢ (𝑅 ∈ 𝑉 → 𝑅 Se 𝐴) |
|
Theorem | imass1 4996 |
Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶)) |
|
Theorem | imass2 4997 |
Subset theorem for image. Exercise 22(a) of [Enderton] p. 53.
(Contributed by NM, 22-Mar-1998.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵)) |
|
Theorem | ndmima 4998 |
The image of a singleton outside the domain is empty. (Contributed by NM,
22-May-1998.)
|
⊢ (¬ 𝐴 ∈ dom 𝐵 → (𝐵 “ {𝐴}) = ∅) |
|
Theorem | relcnv 4999 |
A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed
by NM, 29-Oct-1996.)
|
⊢ Rel ◡𝐴 |
|
Theorem | relbrcnvg 5000 |
When 𝑅 is a relation, the sethood
assumptions on brcnv 4803 can be
omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
|
⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)) |