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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nfres 6001 | 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 | csbres 6002 | Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.) (Revised by NM, 23-Aug-2018.) |
⊢ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) | ||
Theorem | res0 6003 | A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.) |
⊢ (𝐴 ↾ ∅) = ∅ | ||
Theorem | dfres3 6004 | Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴)) | ||
Theorem | opelres 6005 | Ordered pair elementhood in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.) (Revised by BJ, 18-Feb-2022.) Commute the consequent. (Revised by Peter Mazsa, 24-Sep-2022.) |
⊢ (𝐶 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝑅))) | ||
Theorem | brres 6006 | Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.) Commute the consequent. (Revised by Peter Mazsa, 24-Sep-2022.) |
⊢ (𝐶 ∈ 𝑉 → (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶))) | ||
Theorem | opelresi 6007 | Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.) |
⊢ 𝐶 ∈ V ⇒ ⊢ (〈𝐵, 𝐶〉 ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝑅)) | ||
Theorem | brresi 6008 | Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.) |
⊢ 𝐶 ∈ V ⇒ ⊢ (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶)) | ||
Theorem | opres 6009 | 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 6010 | 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 | opelidres 6011 | 〈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 6012 | The restriction of a restriction. (Contributed by NM, 27-Mar-2008.) |
⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶)) | ||
Theorem | resundi 6013 | 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 6014 | Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.) |
⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) | ||
Theorem | resindi 6015 | Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.) |
⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ 𝐶)) | ||
Theorem | resindir 6016 | Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.) |
⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) | ||
Theorem | inres 6017 | Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.) |
⊢ (𝐴 ∩ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ 𝐵) ↾ 𝐶) | ||
Theorem | resdifcom 6018 | Commutative law for restriction and difference. (Contributed by AV, 7-Jun-2021.) |
⊢ ((𝐴 ↾ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ↾ 𝐵) | ||
Theorem | resiun1 6019* | Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.) (Proof shortened by JJ, 25-Aug-2021.) |
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) | ||
Theorem | resiun2 6020* | Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.) |
⊢ (𝐶 ↾ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵) | ||
Theorem | resss 6021 | A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.) |
⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | ||
Theorem | rescom 6022 | Commutative law for restriction. (Contributed by NM, 27-Mar-1998.) |
⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ↾ 𝐵) | ||
Theorem | ssres 6023 | Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶)) | ||
Theorem | ssres2 6024 | 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 6025 | 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 6026 | 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 6027 | Absorption law for restriction, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵)) | ||
Theorem | resabs2 6028 | Absorption law for restriction. (Contributed by NM, 27-Mar-1998.) |
⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵)) | ||
Theorem | residm 6029 | Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.) |
⊢ ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵) | ||
Theorem | dmresss 6030 | The domain of a restriction is a subset of the original domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.) Proof shortened and axiom usage reduced. (Proof shortened by AV, 15-May-2025.) |
⊢ dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴 | ||
Theorem | dmres 6031 | 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 6032 | A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.) |
⊢ (𝐴 ⊆ dom 𝐵 ↔ dom (𝐵 ↾ 𝐴) = 𝐴) | ||
Theorem | dmresexg 6033 | The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.) |
⊢ (𝐵 ∈ 𝑉 → dom (𝐴 ↾ 𝐵) ∈ V) | ||
Theorem | resima 6034 | A restriction to an image. (Contributed by NM, 29-Sep-2004.) |
⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = (𝐴 “ 𝐵) | ||
Theorem | resima2 6035 | Image under a restricted class. (Contributed by FL, 31-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.) |
⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵)) | ||
Theorem | rnresss 6036 | The range of a restriction is a subset of the whole range. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴 | ||
Theorem | xpssres 6037 | Restriction of a constant function (or other Cartesian product). (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ (𝐶 ⊆ 𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵)) | ||
Theorem | elinxp 6038* | Membership in an intersection with a Cartesian product. (Contributed by Peter Mazsa, 9-Sep-2022.) |
⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝑅)) | ||
Theorem | elres 6039* | Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) |
⊢ (𝐴 ∈ (𝐵 ↾ 𝐶) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) | ||
Theorem | elsnres 6040* | Membership 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 6041 | Simplification law for restriction. (Contributed by NM, 16-Aug-1994.) |
⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → (𝐴 ↾ 𝐵) = 𝐴) | ||
Theorem | dmressnsn 6042 | The domain of a restriction to a singleton is a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.) |
⊢ (𝐴 ∈ dom 𝐹 → dom (𝐹 ↾ {𝐴}) = {𝐴}) | ||
Theorem | eldmressnsn 6043 | The element of the domain of a restriction to a singleton is the element of the singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.) |
⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ {𝐴})) | ||
Theorem | eldmeldmressn 6044 | An element of the domain (of a relation) is an element of the domain of the restriction (of the relation) to the singleton containing this element. (Contributed by Alexander van der Vekens, 22-Jul-2018.) |
⊢ (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ dom (𝐹 ↾ {𝑋})) | ||
Theorem | resdm 6045 | A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.) |
⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴) | ||
Theorem | resexg 6046 | 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 | resexd 6047 | The restriction of a set is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ V) | ||
Theorem | resex 6048 | The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ↾ 𝐵) ∈ V | ||
Theorem | resindm 6049 | 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 6050 | 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 | reldisjun 6051 | Split a relation into two disjoint parts based on its domain. (Contributed by Thierry Arnoux, 9-Oct-2023.) |
⊢ ((Rel 𝑅 ∧ dom 𝑅 = (𝐴 ∪ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑅 = ((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵))) | ||
Theorem | relresdm1 6052 | Restriction of a disjoint union to the domain of the first term. (Contributed by Thierry Arnoux, 9-Dec-2021.) |
⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ∪ 𝐵) ↾ dom 𝐴) = 𝐴) | ||
Theorem | resopab 6053* | Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | ||
Theorem | iss 6054 | 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 6055* | 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 6056* | Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.) |
⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | ||
Theorem | resmpt3 6057* | 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 6058 | 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 6059* | 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 6060* | Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.) |
⊢ (𝑅 ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} | ||
Theorem | mptss 6061* | Sufficient condition for inclusion among two functions in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶)) | ||
Theorem | elimampt 6062* | Membership in the image of a mapping. (Contributed by Thierry Arnoux, 3-Jan-2022.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ (𝜑 → 𝐷 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐷) ↔ ∃𝑥 ∈ 𝐷 𝐶 = 𝐵)) | ||
Theorem | elidinxp 6063* | Characterization of the elements of the intersection of the identity relation with a Cartesian product. (Contributed by Peter Mazsa, 9-Sep-2022.) |
⊢ (𝐶 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝐶 = 〈𝑥, 𝑥〉) | ||
Theorem | elidinxpid 6064* | Characterization of the elements of the intersection of the identity relation with a Cartesian square. (Contributed by Peter Mazsa, 9-Sep-2022.) |
⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ 𝐴 𝐵 = 〈𝑥, 𝑥〉) | ||
Theorem | elrid 6065* | Characterization of the elements of a restricted identity relation. (Contributed by BJ, 28-Aug-2022.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) |
⊢ (𝐴 ∈ ( I ↾ 𝑋) ↔ ∃𝑥 ∈ 𝑋 𝐴 = 〈𝑥, 𝑥〉) | ||
Theorem | idinxpres 6066 | The intersection of the identity relation with a cartesian product is the restriction of the identity relation to the intersection of the factors. (Contributed by FL, 2-Aug-2009.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) Generalize statement from cartesian square (now idinxpresid 6067) to cartesian product. (Revised by BJ, 23-Dec-2023.) |
⊢ ( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴 ∩ 𝐵)) | ||
Theorem | idinxpresid 6067 | The intersection of the identity relation with the cartesian square of a class is the restriction of the identity relation to that class. (Contributed by FL, 2-Aug-2009.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) (Proof shortened by BJ, 23-Dec-2023.) |
⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴) | ||
Theorem | idssxp 6068 | A diagonal set as a subset of a Cartesian square. (Contributed by Thierry Arnoux, 29-Dec-2019.) (Proof shortened by BJ, 9-Sep-2022.) |
⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) | ||
Theorem | opabresid 6069* | The restricted identity relation expressed as an ordered-pair class abstraction. (Contributed by FL, 25-Apr-2012.) |
⊢ ( I ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} | ||
Theorem | mptresid 6070* | The restricted identity relation expressed in maps-to notation. (Contributed by FL, 25-Apr-2012.) |
⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥) | ||
Theorem | dmresi 6071 | The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.) |
⊢ dom ( I ↾ 𝐴) = 𝐴 | ||
Theorem | restidsing 6072 | 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 | iresn0n0 6073 | The identity function restricted to a class 𝐴 is empty iff the class 𝐴 is empty. (Contributed by AV, 30-Jan-2024.) |
⊢ (𝐴 = ∅ ↔ ( I ↾ 𝐴) = ∅) | ||
Theorem | imaeq1 6074 | Equality theorem for image. (Contributed by NM, 14-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) | ||
Theorem | imaeq2 6075 | Equality theorem for image. (Contributed by NM, 14-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵)) | ||
Theorem | imaeq1i 6076 | Equality theorem for image. (Contributed by NM, 21-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 “ 𝐶) = (𝐵 “ 𝐶) | ||
Theorem | imaeq2i 6077 | Equality theorem for image. (Contributed by NM, 21-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 “ 𝐴) = (𝐶 “ 𝐵) | ||
Theorem | imaeq1d 6078 | Equality theorem for image. (Contributed by FL, 15-Dec-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) | ||
Theorem | imaeq2d 6079 | Equality theorem for image. (Contributed by FL, 15-Dec-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵)) | ||
Theorem | imaeq12d 6080 | Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐷)) | ||
Theorem | dfima2 6081* | 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.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} | ||
Theorem | dfima3 6082* | 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 6083* | 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 6084* | 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 6085* | 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 6086* | 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 6087 | 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 6088 | Deduction version of bound-variable hypothesis builder nfima 6087. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥(𝐴 “ 𝐵)) | ||
Theorem | imadmrn 6089 | 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 6090 | 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 | mptima 6091* | Image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) | ||
Theorem | mptimass 6092* | Image of a function in maps-to notation for a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝐶 ⊆ 𝐴) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ 𝐶 ↦ 𝐵)) | ||
Theorem | imai 6093 | Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.) |
⊢ ( I “ 𝐴) = 𝐴 | ||
Theorem | rnresi 6094 | The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.) |
⊢ ran ( I ↾ 𝐴) = 𝐴 | ||
Theorem | resiima 6095 | The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.) |
⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵) | ||
Theorem | ima0 6096 | Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.) |
⊢ (𝐴 “ ∅) = ∅ | ||
Theorem | 0ima 6097 | Image under the empty relation. (Contributed by FL, 11-Jan-2007.) |
⊢ (∅ “ 𝐴) = ∅ | ||
Theorem | csbima12 6098 | Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Revised by NM, 20-Aug-2018.) |
⊢ ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵) | ||
Theorem | imadisj 6099 | 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 | imadisjlnd 6100 | Deduction form of one negated side of imadisj 6099. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) ≠ ∅) ⇒ ⊢ (𝜑 → (𝐴 “ 𝐵) ≠ ∅) |
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