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Theorem List for Metamath Proof Explorer - 5301-5400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdfiun3 5301 Alternate definition of indexed union when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐵 ∈ V        𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)

Theoremdfiin3 5302 Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐵 ∈ V        𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)

Theoremriinint 5303* Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝑋𝑉 ∧ ∀𝑘𝐼 𝑆𝑋) → (𝑋 𝑘𝐼 𝑆) = ({𝑋} ∪ ran (𝑘𝐼𝑆)))

Theoremrelrn0 5304 A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)
(Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))

Theoremdmrnssfld 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

Theoremdmcoss 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 (𝐴𝐵) ⊆ dom 𝐵

Theoremrncoss 5307 Range of a composition. (Contributed by NM, 19-Mar-1998.)
ran (𝐴𝐵) ⊆ ran 𝐴

Theoremdmcosseq 5308 Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(ran 𝐵 ⊆ dom 𝐴 → dom (𝐴𝐵) = dom 𝐵)

Theoremdmcoeq 5309 Domain of a composition. (Contributed by NM, 19-Mar-1998.)
(dom 𝐴 = ran 𝐵 → dom (𝐴𝐵) = dom 𝐵)

Theoremrncoeq 5310 Range of a composition. (Contributed by NM, 19-Mar-1998.)
(dom 𝐴 = ran 𝐵 → ran (𝐴𝐵) = ran 𝐴)

Theoremreseq1 5311 Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremreseq2 5312 Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremreseq1i 5313 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)

Theoremreseq2i 5314 Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)

Theoremreseq12i 5315 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)

Theoremreseq1d 5316 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Theoremreseq2d 5317 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Theoremreseq12d 5318 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremnfres 5319 Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)

Theoremcsbres 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(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)

Theoremres0 5321 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
(𝐴 ↾ ∅) = ∅

Theoremopelres 5322 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)
𝐵 ∈ V       (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷))

Theorembrres 5323 Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
𝐵 ∈ V       (𝐴(𝐶𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐴𝐷))

Theoremopelresg 5324 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.)
(𝐵𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷)))

Theorembrresg 5325 Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)
(𝐵𝑉 → (𝐴(𝐶𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐴𝐷)))

Theoremopres 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       (𝐴𝐷 → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))

Theoremresieq 5327 A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)
((𝐵𝐴𝐶𝐴) → (𝐵( I ↾ 𝐴)𝐶𝐵 = 𝐶))

Theoremopelresi 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.)
(𝐴𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ 𝐴𝐵))

Theoremresres 5329 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
((𝐴𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵𝐶))

Theoremresundi 5330 Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
(𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Theoremresundir 5331 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremresindi 5332 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
(𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Theoremresindir 5333 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Theoreminres 5334 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
(𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ↾ 𝐶)

Theoremresdifcom 5335 Commutative law for restriction and difference. (Contributed by AV, 7-Jun-2021.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ↾ 𝐵)

Theoremresiun1 5336* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.) (Proof shortened by JJ, 25-Aug-2021.)
( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)

Theoremresiun1OLD 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.)
( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)

Theoremresiun2 5338* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
(𝐶 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶𝐵)

Theoremdmres 5339 The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)
dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)

Theoremssdmres 5340 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
(𝐴 ⊆ dom 𝐵 ↔ dom (𝐵𝐴) = 𝐴)

Theoremdmresexg 5341 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
(𝐵𝑉 → dom (𝐴𝐵) ∈ V)

Theoremresss 5342 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
(𝐴𝐵) ⊆ 𝐴

Theoremrescom 5343 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ↾ 𝐵)

Theoremssres 5344 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremssres2 5345 Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

Theoremrelres 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 (𝐴𝐵)

Theoremresabs1 5347 Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)
(𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))

Theoremresabs1d 5348 Absorption law for restriction, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐵𝐶)       (𝜑 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))

Theoremresabs2 5349 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
(𝐵𝐶 → ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐵))

Theoremresidm 5350 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
((𝐴𝐵) ↾ 𝐵) = (𝐴𝐵)

Theoremresima 5351 A restriction to an image. (Contributed by NM, 29-Sep-2004.)
((𝐴𝐵) “ 𝐵) = (𝐴𝐵)

Theoremresima2 5352 Image under a restricted class. (Contributed by FL, 31-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.)
(𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))

Theoremresima2OLD 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.)
(𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))

Theoremxpssres 5354 Restriction of a constant function (or other Cartesian product). (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐶𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))

Theoremelres 5355* Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)
(𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))

Theoremelsnres 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))

Theoremrelssres 5357 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
((Rel 𝐴 ∧ dom 𝐴𝐵) → (𝐴𝐵) = 𝐴)

Theoremdmressnsn 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})

Theoremeldmressnsn 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 𝐹𝐴 ∈ dom (𝐹 ↾ {𝐴}))

Theoremeldmeldmressn 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 𝐹𝑋 ∈ dom (𝐹 ↾ {𝑋}))

Theoremresdm 5361 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
(Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)

Theoremresexg 5362 The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)

Theoremresex 5363 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
𝐴 ∈ V       (𝐴𝐵) ∈ V

Theoremresindm 5364 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
(Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))

Theoremresdmdfsn 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})))

Theoremresopab 5366* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}

Theoremiss 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))

Theoremresopab2 5368* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)
(𝐴𝐵 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})

Theoremresmpt 5369* Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)
(𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))

Theoremresmpt3 5370* Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)
((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴𝐵) ↦ 𝐶)

Theoremresmptd 5371* Restriction of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐵𝐴)       (𝜑 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))

Theoremdfres2 5372* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
(𝑅𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}

Theoremmptss 5373* Sufficient condition for inclusion in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))

Theoremopabresid 5374* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)
{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ( I ↾ 𝐴)

Theoremmptresid 5375* The restricted identity expressed with the "maps to" notation. (Contributed by FL, 25-Apr-2012.)
(𝑥𝐴𝑥) = ( I ↾ 𝐴)

Theoremdmresi 5376 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
dom ( I ↾ 𝐴) = 𝐴

Theoremrestidsing 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})

TheoremrestidsingOLD 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})

Theoremresid 5379 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)
(Rel 𝐴 → (𝐴 ↾ V) = 𝐴)

Theoremimaeq1 5380 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremimaeq2 5381 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremimaeq1i 5382 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)

Theoremimaeq2i 5383 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)

Theoremimaeq1d 5384 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Theoremimaeq2d 5385 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Theoremimaeq12d 5386 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremdfima2 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.)
(𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}

Theoremdfima3 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.)
(𝐴𝐵) = {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}

Theoremelimag 5389* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)
(𝐴𝑉 → (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴))

Theoremelima 5390* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)
𝐴 ∈ V       (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴)

Theoremelima2 5391* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)
𝐴 ∈ V       (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐶𝑥𝐵𝐴))

Theoremelima3 5392* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)
𝐴 ∈ V       (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))

Theoremnfima 5393 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 5394 Deduction version of bound-variable hypothesis builder nfima 5393. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥(𝐴𝐵))

Theoremimadmrn 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

Theoremimassrn 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.)
(𝐴𝐵) ⊆ ran 𝐴

Theoremimai 5397 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
( I “ 𝐴) = 𝐴

Theoremrnresi 5398 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)
ran ( I ↾ 𝐴) = 𝐴

Theoremresiima 5399 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
(𝐵𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵)

Theoremima0 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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