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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dfiun3 5301 | Alternate definition of indexed union when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐵 ∈ V ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) | ||
Theorem | dfiin3 5302 | Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐵 ∈ V ⇒ ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵) | ||
Theorem | riinint 5303* | Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) = ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆))) | ||
Theorem | relrn0 5304 | A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.) |
⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅)) | ||
Theorem | dmrnssfld 5305 | The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.) |
⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 | ||
Theorem | dmcoss 5306 | Domain of a composition. Theorem 21 of [Suppes] p. 63. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 | ||
Theorem | rncoss 5307 | Range of a composition. (Contributed by NM, 19-Mar-1998.) |
⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴 | ||
Theorem | dmcosseq 5308 | Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) = dom 𝐵) | ||
Theorem | dmcoeq 5309 | Domain of a composition. (Contributed by NM, 19-Mar-1998.) |
⊢ (dom 𝐴 = ran 𝐵 → dom (𝐴 ∘ 𝐵) = dom 𝐵) | ||
Theorem | rncoeq 5310 | Range of a composition. (Contributed by NM, 19-Mar-1998.) |
⊢ (dom 𝐴 = ran 𝐵 → ran (𝐴 ∘ 𝐵) = ran 𝐴) | ||
Theorem | reseq1 5311 | Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)) | ||
Theorem | reseq2 5312 | Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) | ||
Theorem | reseq1i 5313 | Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶) | ||
Theorem | reseq2i 5314 | Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵) | ||
Theorem | reseq12i 5315 | Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷) | ||
Theorem | reseq1d 5316 | Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)) | ||
Theorem | reseq2d 5317 | Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) | ||
Theorem | reseq12d 5318 | Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷)) | ||
Theorem | nfres 5319 | 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 5320 | 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 5321 | A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.) |
⊢ (𝐴 ↾ ∅) = ∅ | ||
Theorem | opelres 5322 | 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 5323 | Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴(𝐶 ↾ 𝐷)𝐵 ↔ (𝐴𝐶𝐵 ∧ 𝐴 ∈ 𝐷)) | ||
Theorem | opelresg 5324 | 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 5325 | Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴(𝐶 ↾ 𝐷)𝐵 ↔ (𝐴𝐶𝐵 ∧ 𝐴 ∈ 𝐷))) | ||
Theorem | opres 5326 | 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 5327 | 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 5328 | 〈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 5329 | The restriction of a restriction. (Contributed by NM, 27-Mar-2008.) |
⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶)) | ||
Theorem | resundi 5330 | 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 5331 | Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.) |
⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) | ||
Theorem | resindi 5332 | Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.) |
⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ 𝐶)) | ||
Theorem | resindir 5333 | Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.) |
⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) | ||
Theorem | inres 5334 | Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.) |
⊢ (𝐴 ∩ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ 𝐵) ↾ 𝐶) | ||
Theorem | resdifcom 5335 | Commutative law for restriction and difference. (Contributed by AV, 7-Jun-2021.) |
⊢ ((𝐴 ↾ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ↾ 𝐵) | ||
Theorem | resiun1 5336* | 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 | resiun1OLD 5337* | Obsolete proof of resiun1 5336 as of 25-Aug-2021. (Contributed by Mario Carneiro, 29-May-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) | ||
Theorem | resiun2 5338* | Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.) |
⊢ (𝐶 ↾ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 ↾ 𝐵) | ||
Theorem | dmres 5339 | 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 5340 | A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.) |
⊢ (𝐴 ⊆ dom 𝐵 ↔ dom (𝐵 ↾ 𝐴) = 𝐴) | ||
Theorem | dmresexg 5341 | The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.) |
⊢ (𝐵 ∈ 𝑉 → dom (𝐴 ↾ 𝐵) ∈ V) | ||
Theorem | resss 5342 | A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.) |
⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | ||
Theorem | rescom 5343 | Commutative law for restriction. (Contributed by NM, 27-Mar-1998.) |
⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ↾ 𝐵) | ||
Theorem | ssres 5344 | Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶)) | ||
Theorem | ssres2 5345 | 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 5346 | 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 5347 | 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 5348 | Absorption law for restriction, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵)) | ||
Theorem | resabs2 5349 | Absorption law for restriction. (Contributed by NM, 27-Mar-1998.) |
⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵)) | ||
Theorem | residm 5350 | Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.) |
⊢ ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵) | ||
Theorem | resima 5351 | A restriction to an image. (Contributed by NM, 29-Sep-2004.) |
⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = (𝐴 “ 𝐵) | ||
Theorem | resima2 5352 | 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 | resima2OLD 5353 | Obsolete proof of resima2 5352 as of 25-Aug-2021. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵)) | ||
Theorem | xpssres 5354 | 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 | elres 5355* | 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 5356* | 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 5357 | Simplification law for restriction. (Contributed by NM, 16-Aug-1994.) |
⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → (𝐴 ↾ 𝐵) = 𝐴) | ||
Theorem | dmressnsn 5358 | 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 5359 | 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 5360 | 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 5361 | A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.) |
⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴) | ||
Theorem | resexg 5362 | 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 5363 | The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ↾ 𝐵) ∈ V | ||
Theorem | resindm 5364 | Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.) |
⊢ (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵)) | ||
Theorem | resdmdfsn 5365 | Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself. (Contributed by AV, 2-Dec-2018.) |
⊢ (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))) | ||
Theorem | resopab 5366* | 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 5367 | 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 5368* | 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 5369* | Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.) |
⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | ||
Theorem | resmpt3 5370* | 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 | resmptd 5371* | 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 5372* | Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.) |
⊢ (𝑅 ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} | ||
Theorem | mptss 5373* | Sufficient condition for inclusion in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶)) | ||
Theorem | opabresid 5374* | The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.) |
⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐴) | ||
Theorem | mptresid 5375* | The restricted identity expressed with the "maps to" notation. (Contributed by FL, 25-Apr-2012.) |
⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ( I ↾ 𝐴) | ||
Theorem | dmresi 5376 | The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.) |
⊢ dom ( I ↾ 𝐴) = 𝐴 | ||
Theorem | restidsing 5377 | Restriction of the identity to a singleton. (Contributed by FL, 2-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.) |
⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴}) | ||
Theorem | restidsingOLD 5378 | Obsolete proof of restidsing 5377 as of 25-Aug-2021. (Contributed by FL, 2-Aug-2009.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴}) | ||
Theorem | resid 5379 | Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.) |
⊢ (Rel 𝐴 → (𝐴 ↾ V) = 𝐴) | ||
Theorem | imaeq1 5380 | Equality theorem for image. (Contributed by NM, 14-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) | ||
Theorem | imaeq2 5381 | Equality theorem for image. (Contributed by NM, 14-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵)) | ||
Theorem | imaeq1i 5382 | Equality theorem for image. (Contributed by NM, 21-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 “ 𝐶) = (𝐵 “ 𝐶) | ||
Theorem | imaeq2i 5383 | Equality theorem for image. (Contributed by NM, 21-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 “ 𝐴) = (𝐶 “ 𝐵) | ||
Theorem | imaeq1d 5384 | Equality theorem for image. (Contributed by FL, 15-Dec-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) | ||
Theorem | imaeq2d 5385 | Equality theorem for image. (Contributed by FL, 15-Dec-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵)) | ||
Theorem | imaeq12d 5386 | Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐷)) | ||
Theorem | dfima2 5387* | 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 5388* | 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 5389* | 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 5390* | 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 5391* | 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 5392* | 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 5393 | 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 5394 | Deduction version of bound-variable hypothesis builder nfima 5393. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥(𝐴 “ 𝐵)) | ||
Theorem | imadmrn 5395 | 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 5396 | 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 | imai 5397 | Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.) |
⊢ ( I “ 𝐴) = 𝐴 | ||
Theorem | rnresi 5398 | The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.) |
⊢ ran ( I ↾ 𝐴) = 𝐴 | ||
Theorem | resiima 5399 | The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.) |
⊢ (𝐵 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵) | ||
Theorem | ima0 5400 | Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.) |
⊢ (𝐴 “ ∅) = ∅ |
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