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Theorem List for Metamath Proof Explorer - 6701-6800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoveq2i 6701 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
𝐴 = 𝐵       (𝐶𝐹𝐴) = (𝐶𝐹𝐵)
 
Theoremoveq12i 6702 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐹𝐶) = (𝐵𝐹𝐷)
 
Theoremoveqi 6703 Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
𝐴 = 𝐵       (𝐶𝐴𝐷) = (𝐶𝐵𝐷)
 
Theoremoveq123i 6704 Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)
𝐴 = 𝐶    &   𝐵 = 𝐷    &   𝐹 = 𝐺       (𝐴𝐹𝐵) = (𝐶𝐺𝐷)
 
Theoremoveq1d 6705 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶))
 
Theoremoveq2d 6706 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵))
 
Theoremoveqd 6707 Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐷) = (𝐶𝐵𝐷))
 
Theoremoveq12d 6708 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveqan12d 6709 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveqan12rd 6710 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜓𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveq123d 6711 Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐺𝐷))
 
Theoremovrspc2v 6712* If an operation value is element of a class for all operands of two classes, then the operation value is an element of the class for specific operands of the two classes. (Contributed by Mario Carneiro, 6-Dec-2014.)
(((𝑋𝐴𝑌𝐵) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶) → (𝑋𝐹𝑌) ∈ 𝐶)
 
Theoremoveqrspc2v 6713* Restricted specialization of operands, using implicit substitution. (Contributed by Mario Carneiro, 6-Dec-2014.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))       ((𝜑 ∧ (𝑋𝐴𝑌𝐵)) → (𝑋𝐹𝑌) = (𝑋𝐺𝑌))
 
Theoremoveqdr 6714 Equality of two operations for any two operands. Useful in proofs using *propd theorems. (Contributed by Mario Carneiro, 29-Jun-2015.)
(𝜑𝐹 = 𝐺)       ((𝜑𝜓) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
 
Theoremnfovd 6715 Deduction version of bound-variable hypothesis builder nfov 6716. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐹)    &   (𝜑𝑥𝐵)       (𝜑𝑥(𝐴𝐹𝐵))
 
Theoremnfov 6716 Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.)
𝑥𝐴    &   𝑥𝐹    &   𝑥𝐵       𝑥(𝐴𝐹𝐵)
 
Theoremoprabid 6717 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 20-Mar-2013.)
(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
 
Theoremovex 6718 The result of an operation is a set. (Contributed by NM, 13-Mar-1995.)
(𝐴𝐹𝐵) ∈ V
 
Theoremovexi 6719 The result of an operation is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴 = (𝐵𝐹𝐶)       𝐴 ∈ V
 
Theoremovexd 6720 The result of an operation is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑 → (𝐴𝐹𝐵) ∈ V)
 
Theoremovssunirn 6721 The result of an operation value is always a subset of the union of the range. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝑋𝐹𝑌) ⊆ ran 𝐹
 
Theorem0ov 6722 Operation value of the empty set. (Contributed by AV, 15-May-2021.)
(𝐴𝐵) = ∅
 
Theoremovprc 6723 The value of an operation when the one of the arguments is a proper class. Note: this theorem is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by Mario Carneiro, 26-Apr-2015.)
Rel dom 𝐹       (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)
 
Theoremovprc1 6724 The value of an operation when the first argument is a proper class. (Contributed by NM, 16-Jun-2004.)
Rel dom 𝐹       𝐴 ∈ V → (𝐴𝐹𝐵) = ∅)
 
Theoremovprc2 6725 The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
Rel dom 𝐹       𝐵 ∈ V → (𝐴𝐹𝐵) = ∅)
 
Theoremovrcl 6726 Reverse closure for an operation value. (Contributed by Mario Carneiro, 5-May-2015.)
Rel dom 𝐹       (𝐶 ∈ (𝐴𝐹𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
Theoremcsbov123 6727 Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Revised by NM, 23-Aug-2018.)
𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)
 
Theoremcsbov 6728* Move class substitution in and out of an operation. (Contributed by NM, 23-Aug-2018.)
𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶)
 
Theoremcsbov12g 6729* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶))
 
Theoremcsbov1g 6730* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐶))
 
Theoremcsbov2g 6731* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐵𝐹𝐴 / 𝑥𝐶))
 
Theoremrspceov 6732* A frequently used special case of rspc2ev 3355 for operation values. (Contributed by NM, 21-Mar-2007.)
((𝐶𝐴𝐷𝐵𝑆 = (𝐶𝐹𝐷)) → ∃𝑥𝐴𝑦𝐵 𝑆 = (𝑥𝐹𝑦))
 
Theoremelovimad 6733 Elementhood of the image set of an operation value. (Contributed by Thierry Arnoux, 13-Mar-2017.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)    &   (𝜑 → Fun 𝐹)    &   (𝜑 → (𝐶 × 𝐷) ⊆ dom 𝐹)       (𝜑 → (𝐴𝐹𝐵) ∈ (𝐹 “ (𝐶 × 𝐷)))
 
Theoremfnbrovb 6734 Value of a binary operation expressed as a binary relation. See also fnbrfvb 6274 for functions on Cartesian products. (Contributed by BJ, 15-Feb-2022.)
((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴𝑉𝐵𝑊)) → ((𝐴𝐹𝐵) = 𝐶 ↔ ⟨𝐴, 𝐵𝐹𝐶))
 
Theoremfnotovb 6735 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6275. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof shortened by BJ, 15-Feb-2022.)
((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))
 
TheoremfnotovbOLD 6736 Old proof of fnotovb 6735 obsolete as of 15-Feb-2022. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))
 
Theoremopabbrex 6737 A collection of ordered pairs with an extension of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by BJ/AV, 20-Jun-2019.) (Proof shortened by OpenAI, 25-Mar-2020.)
((∀𝑥𝑦(𝑥𝑅𝑦𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
 
Theoremopabresex2d 6738* Restrictions of a collection of ordered pairs of related elements are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.)
((𝜑𝑥(𝑊𝐺)𝑦) → 𝜓)    &   (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ 𝑉)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} ∈ V)
 
Theoremfvmptopab 6739* The function value of a mapping 𝑀 to a restricted binary relation expressed as an ordered-pair class abstraction: The restricted binary relation is a binary relation given as value of a function 𝐹 restricted by the condition 𝜓. (Contributed by AV, 31-Jan-2021.) (Revised by AV, 29-Oct-2021.)
((𝜑𝑧 = 𝑍) → (𝜒𝜓))    &   (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V)    &   𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)})       (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
 
Theorem0neqopab 6740 The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
Theorembrabv 6741 If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
(𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
 
Theorembrfvopab 6742 The classes involved in a binary relation of a function value which is an ordered-pair class abstraction are sets. (Contributed by AV, 7-Jan-2021.)
(𝑋 ∈ V → (𝐹𝑋) = {⟨𝑦, 𝑧⟩ ∣ 𝜑})       (𝐴(𝐹𝑋)𝐵 → (𝑋 ∈ V ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V))
 
Theoremdfoprab2 6743* Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995.)
{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
 
Theoremreloprab 6744* An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)
Rel {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremoprabv 6745* If a pair and a class are in a relationship given by a class abstraction of a collection of nested ordered pairs, the involved classes are sets. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
(⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V))
 
Theoremnfoprab1 6746 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)
𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremnfoprab2 6747 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 30-Jul-2012.)
𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremnfoprab3 6748 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.)
𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremnfoprab 6749* Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.)
𝑤𝜑       𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremoprabbid 6750* Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.)
𝑥𝜑    &   𝑦𝜑    &   𝑧𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒})
 
Theoremoprabbidv 6751* Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.)
(𝜑 → (𝜓𝜒))       (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒})
 
Theoremoprabbii 6752* Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
(𝜑𝜓)       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
 
Theoremssoprab2 6753 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 5030. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
(∀𝑥𝑦𝑧(𝜑𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓})
 
Theoremssoprab2b 6754 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 5031. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ∀𝑥𝑦𝑧(𝜑𝜓))
 
Theoremeqoprab2b 6755 Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 5034. (Contributed by Mario Carneiro, 4-Jan-2017.)
({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ∀𝑥𝑦𝑧(𝜑𝜓))
 
Theoremmpt2eq123 6756* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.)
((𝐴 = 𝐷 ∧ ∀𝑥𝐴 (𝐵 = 𝐸 ∧ ∀𝑦𝐵 𝐶 = 𝐹)) → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
 
Theoremmpt2eq12 6757* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
((𝐴 = 𝐶𝐵 = 𝐷) → (𝑥𝐴, 𝑦𝐵𝐸) = (𝑥𝐶, 𝑦𝐷𝐸))
 
Theoremmpt2eq123dva 6758* An equality deduction for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑𝐴 = 𝐷)    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐸)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶 = 𝐹)       (𝜑 → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
 
Theoremmpt2eq123dv 6759* An equality deduction for the maps to notation. (Contributed by NM, 12-Sep-2011.)
(𝜑𝐴 = 𝐷)    &   (𝜑𝐵 = 𝐸)    &   (𝜑𝐶 = 𝐹)       (𝜑 → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
 
Theoremmpt2eq123i 6760 An equality inference for the maps to notation. (Contributed by NM, 15-Jul-2013.)
𝐴 = 𝐷    &   𝐵 = 𝐸    &   𝐶 = 𝐹       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹)
 
Theoremmpt2eq3dva 6761* Slightly more general equality inference for the maps to notation. (Contributed by NM, 17-Oct-2013.)
((𝜑𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)       (𝜑 → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷))
 
Theoremmpt2eq3ia 6762 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
((𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
 
Theoremmpt2eq3dv 6763* An equality deduction for the maps to notation restricted to the value of the operation. (Contributed by SO, 16-Jul-2018.)
(𝜑𝐶 = 𝐷)       (𝜑 → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷))
 
Theoremnfmpt21 6764 Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
𝑥(𝑥𝐴, 𝑦𝐵𝐶)
 
Theoremnfmpt22 6765 Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
𝑦(𝑥𝐴, 𝑦𝐵𝐶)
 
Theoremnfmpt2 6766* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
𝑧𝐴    &   𝑧𝐵    &   𝑧𝐶       𝑧(𝑥𝐴, 𝑦𝐵𝐶)
 
Theoremmpt20 6767 A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.)
(𝑥 ∈ ∅, 𝑦𝐵𝐶) = ∅
 
Theoremoprab4 6768* Two ways to state the domain of an operation. (Contributed by FL, 24-Jan-2010.)
{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
 
Theoremcbvoprab1 6769* Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)
𝑤𝜑    &   𝑥𝜓    &   (𝑥 = 𝑤 → (𝜑𝜓))       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
 
Theoremcbvoprab2 6770* Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑤𝜑    &   𝑦𝜓    &   (𝑦 = 𝑤 → (𝜑𝜓))       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜓}
 
Theoremcbvoprab12 6771* Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
𝑤𝜑    &   𝑣𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑤𝑦 = 𝑣) → (𝜑𝜓))       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}
 
Theoremcbvoprab12v 6772* Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.)
((𝑥 = 𝑤𝑦 = 𝑣) → (𝜑𝜓))       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}
 
Theoremcbvoprab3 6773* Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.)
𝑤𝜑    &   𝑧𝜓    &   (𝑧 = 𝑤 → (𝜑𝜓))       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓}
 
Theoremcbvoprab3v 6774* Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.) (Revised by David Abernethy, 19-Jun-2012.)
(𝑧 = 𝑤 → (𝜑𝜓))       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓}
 
Theoremcbvmpt2x 6775* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6776 allows 𝐵 to be a function of 𝑥. (Contributed by NM, 29-Dec-2014.)
𝑧𝐵    &   𝑥𝐷    &   𝑧𝐶    &   𝑤𝐶    &   𝑥𝐸    &   𝑦𝐸    &   (𝑥 = 𝑧𝐵 = 𝐷)    &   ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐸)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐷𝐸)
 
Theoremcbvmpt2 6776* Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.)
𝑧𝐶    &   𝑤𝐶    &   𝑥𝐷    &   𝑦𝐷    &   ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐷)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐵𝐷)
 
Theoremcbvmpt2v 6777* Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4782, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)
(𝑥 = 𝑧𝐶 = 𝐸)    &   (𝑦 = 𝑤𝐸 = 𝐷)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐵𝐷)
 
Theoremelimdelov 6778 Eliminate a hypothesis which is a predicate expressing membership in the result of an operator (deduction version). (Contributed by Paul Chapman, 25-Mar-2008.)
(𝜑𝐶 ∈ (𝐴𝐹𝐵))    &   𝑍 ∈ (𝑋𝐹𝑌)       if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌))
 
Theoremovif 6779 Move a conditional outside of an operation. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(if(𝜑, 𝐴, 𝐵)𝐹𝐶) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐶))
 
Theoremovif2 6780 Move a conditional outside of an operation. (Contributed by Thierry Arnoux, 1-Oct-2018.)
(𝐴𝐹if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐹𝐶))
 
Theoremovif12 6781 Move a conditional outside of an operation. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷))
 
Theoremifov 6782 Move a conditional outside of an operation. (Contributed by AV, 11-Nov-2019.)
(𝐴if(𝜑, 𝐹, 𝐺)𝐵) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐺𝐵))
 
Theoremdmoprab 6783* The domain of an operation class abstraction. (Contributed by NM, 17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)
dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝜑}
 
Theoremdmoprabss 6784* The domain of an operation class abstraction. (Contributed by NM, 24-Aug-1995.)
dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
 
Theoremrnoprab 6785* The range of an operation class abstraction. (Contributed by NM, 30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.)
ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦𝜑}
 
Theoremrnoprab2 6786* The range of a restricted operation class abstraction. (Contributed by Scott Fenton, 21-Mar-2012.)
ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑}
 
Theoremreldmoprab 6787* The domain of an operation class abstraction is a relation. (Contributed by NM, 17-Mar-1995.)
Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremoprabss 6788* Structure of an operation class abstraction. (Contributed by NM, 28-Nov-2006.)
{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ ((V × V) × V)
 
Theoremeloprabga 6789* The law of concretion for operation class abstraction. Compare elopab 5012. (Contributed by NM, 14-Sep-1999.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓))
 
Theoremeloprabg 6790* The law of concretion for operation class abstraction. Compare elopab 5012. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜃))
 
Theoremssoprab2i 6791* Inference of operation class abstraction subclass from implication. (Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy, 19-Jun-2012.)
(𝜑𝜓)       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
 
Theoremmpt2v 6792* Operation with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
(𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝐶}
 
Theoremmpt2mptx 6793* Express a two-argument function as a one-argument function, or vice-versa. In this version 𝐵(𝑥) is not assumed to be constant w.r.t 𝑥. (Contributed by Mario Carneiro, 29-Dec-2014.)
(𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)       (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
 
Theoremmpt2mpt 6794* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 17-Dec-2013.) (Revised by Mario Carneiro, 29-Dec-2014.)
(𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)       (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
 
Theoremmpt2difsnif 6795 A mapping with two arguments with the first argument from a difference set with a singleton and a conditional as result. (Contributed by AV, 13-Feb-2019.)
(𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗𝐵𝐷)
 
Theoremmpt2snif 6796 A mapping with two arguments with the first argument from a singleton and a conditional as result. (Contributed by AV, 14-Feb-2019.)
(𝑖 ∈ {𝑋}, 𝑗𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗𝐵𝐶)
 
Theoremfconstmpt2 6797* Representation of a constant operation using the mapping operation. (Contributed by SO, 11-Jul-2018.)
((𝐴 × 𝐵) × {𝐶}) = (𝑥𝐴, 𝑦𝐵𝐶)
 
Theoremresoprab 6798* Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.)
({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
 
Theoremresoprab2 6799* Restriction of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐶𝐴𝐷𝐵) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ↾ (𝐶 × 𝐷)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)})
 
Theoremresmpt2 6800* Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.)
((𝐶𝐴𝐷𝐵) → ((𝑥𝐴, 𝑦𝐵𝐸) ↾ (𝐶 × 𝐷)) = (𝑥𝐶, 𝑦𝐷𝐸))
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