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Theorem List for Metamath Proof Explorer - 6401-6500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremriotacl2 6401 Membership law for "the unique element in 𝐴 such that 𝜑."

(Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

(∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
 
Theoremriotacl 6402* Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
(∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
 
Theoremriotasbc 6403 Substitution law for descriptions. Compare iotasbc 37539. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
(∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)
 
Theoremriotabidva 6404* Equivalent wff's yield equal restricted class abstractions (deduction rule). (rabbidva 3067 analog.) (Contributed by NM, 17-Jan-2012.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
 
Theoremriotabiia 6405 Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 3065 analog.) (Contributed by NM, 16-Jan-2012.)
(𝑥𝐴 → (𝜑𝜓))       (𝑥𝐴 𝜑) = (𝑥𝐴 𝜓)
 
Theoremriota1 6406* Property of restricted iota. Compare iota1 5667. (Contributed by Mario Carneiro, 15-Oct-2016.)
(∃!𝑥𝐴 𝜑 → ((𝑥𝐴𝜑) ↔ (𝑥𝐴 𝜑) = 𝑥))
 
Theoremriota1a 6407 Property of iota. (Contributed by NM, 23-Aug-2011.)
((𝑥𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
 
Theoremriota2df 6408* A deduction version of riota2f 6409. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝜑    &   (𝜑𝑥𝐵)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑𝐵𝐴)    &   ((𝜑𝑥 = 𝐵) → (𝜓𝜒))       ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝜒 ↔ (𝑥𝐴 𝜓) = 𝐵))
 
Theoremriota2f 6409* This theorem shows a condition that allows us to represent a descriptor with a class expression 𝐵. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐵    &   𝑥𝜓    &   (𝑥 = 𝐵 → (𝜑𝜓))       ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
 
Theoremriota2 6410* This theorem shows a condition that allows us to represent a descriptor with a class expression 𝐵. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 10-Dec-2016.)
(𝑥 = 𝐵 → (𝜑𝜓))       ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
 
Theoremriotaprop 6411* Properties of a restricted definite description operator. TODO (df-riota 6388 update): can some uses of riota2f 6409 be shortened with this? (Contributed by NM, 23-Nov-2013.)
𝑥𝜓    &   𝐵 = (𝑥𝐴 𝜑)    &   (𝑥 = 𝐵 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 → (𝐵𝐴𝜓))
 
Theoremriota5f 6412* A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝑥𝐵)    &   (𝜑𝐵𝐴)    &   ((𝜑𝑥𝐴) → (𝜓𝑥 = 𝐵))       (𝜑 → (𝑥𝐴 𝜓) = 𝐵)
 
Theoremriota5 6413* A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.)
(𝜑𝐵𝐴)    &   ((𝜑𝑥𝐴) → (𝜓𝑥 = 𝐵))       (𝜑 → (𝑥𝐴 𝜓) = 𝐵)
 
Theoremriotass2 6414* Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)
(((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜓))
 
Theoremriotass 6415* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜑))
 
Theoremmoriotass 6416* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.)
((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜑))
 
Theoremsnriota 6417 A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)
(∃!𝑥𝐴 𝜑 → {𝑥𝐴𝜑} = {(𝑥𝐴 𝜑)})
 
Theoremriotaxfrd 6418* Change the variable 𝑥 in the expression for "the unique 𝑥 such that 𝜓 " to another variable 𝑦 contained in expression 𝐵. Use reuhypd 4720 to eliminate the last hypothesis. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑦𝐶    &   ((𝜑𝑦𝐴) → 𝐵𝐴)    &   ((𝜑 ∧ (𝑦𝐴 𝜒) ∈ 𝐴) → 𝐶𝐴)    &   (𝑥 = 𝐵 → (𝜓𝜒))    &   (𝑦 = (𝑦𝐴 𝜒) → 𝐵 = 𝐶)    &   ((𝜑𝑥𝐴) → ∃!𝑦𝐴 𝑥 = 𝐵)       ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝑥𝐴 𝜓) = 𝐶)
 
Theoremeusvobj2 6419* Specify the same property in two ways when class 𝐵(𝑦) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
𝐵 ∈ V       (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑥 = 𝐵))
 
Theoremeusvobj1 6420* Specify the same object in two ways when class 𝐵(𝑦) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
𝐵 ∈ V       (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (℩𝑥𝑦𝐴 𝑥 = 𝐵) = (℩𝑥𝑦𝐴 𝑥 = 𝐵))
 
Theoremf1ofveu 6421* There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.)
((𝐹:𝐴1-1-onto𝐵𝐶𝐵) → ∃!𝑥𝐴 (𝐹𝑥) = 𝐶)
 
Theoremf1ocnvfv3 6422* Value of the converse of a one-to-one onto function. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
((𝐹:𝐴1-1-onto𝐵𝐶𝐵) → (𝐹𝐶) = (𝑥𝐴 (𝐹𝑥) = 𝐶))
 
Theoremriotaund 6423* Restricted iota equals the empty set when not meaningful. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 13-Sep-2018.)
(¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = ∅)
 
Theoremriotassuni 6424* The restricted iota class is limited in size by the base set. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝑥𝐴 𝜑) ⊆ (𝒫 𝐴 𝐴)
 
Theoremriotaclb 6425* Bidirectional closure of restricted iota when domain is not empty. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.) (Revised by NM, 13-Sep-2018.)
(¬ ∅ ∈ 𝐴 → (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴))
 
2.3.17  Operations
 
Syntaxco 6426 Extend class notation to include the value of an operation 𝐹 (such as +) for two arguments 𝐴 and 𝐵. Note that the syntax is simply three class symbols in a row surrounded by parentheses. Since operation values are the only possible class expressions consisting of three class expressions in a row surrounded by parentheses, the syntax is unambiguous. (For an example of how syntax could become ambiguous if we are not careful, see the comment in cneg 10018.)
class (𝐴𝐹𝐵)
 
Syntaxcoprab 6427 Extend class notation to include class abstraction (class builder) of nested ordered pairs.
class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Syntaxcmpt2 6428 Extend the definition of a class to include maps-to notation for defining an operation via a rule.
class (𝑥𝐴, 𝑦𝐵𝐶)
 
Definitiondf-ov 6429 Define the value of an operation. Definition of operation value in [Enderton] p. 79. Note that the syntax is simply three class expressions in a row bracketed by parentheses. There are no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation 𝐹 and its arguments 𝐴 and 𝐵- will be useful for proving meaningful theorems. For example, if class 𝐹 is the operation + and arguments 𝐴 and 𝐵 are 3 and 2, the expression (3 + 2) can be proved to equal 5 (see 3p2e5 10915). This definition is well-defined, although not very meaningful, when classes 𝐴 and/or 𝐵 are proper classes (i.e. are not sets); see ovprc1 6459 and ovprc2 6460. On the other hand, we often find uses for this definition when 𝐹 is a proper class, such as +𝑜 in oav 7354. 𝐹 is normally equal to a class of nested ordered pairs of the form defined by df-oprab 6430. (Contributed by NM, 28-Feb-1995.)
(𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
 
Definitiondf-oprab 6430* Define the class abstraction (class builder) of a collection of nested ordered pairs (for use in defining operations). This is a special case of Definition 4.16 of [TakeutiZaring] p. 14. Normally 𝑥, 𝑦, and 𝑧 are distinct, although the definition doesn't strictly require it. See df-ov 6429 for the value of an operation. The brace notation is called "class abstraction" by Quine; it is also called a "class builder" in the literature. The value of the most common operation class builder is given by ovmpt2 6571. (Contributed by NM, 12-Mar-1995.)
{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
 
Definitiondf-mpt2 6431* Define maps-to notation for defining an operation via a rule. Read as "the operation defined by the map from 𝑥, 𝑦 (in 𝐴 × 𝐵) to 𝐶(𝑥, 𝑦)." An extension of df-mpt 4543 for two arguments. (Contributed by NM, 17-Feb-2008.)
(𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
 
Theoremoveq 6432 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(𝐹 = 𝐺 → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))
 
Theoremoveq1 6433 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(𝐴 = 𝐵 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶))
 
Theoremoveq2 6434 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(𝐴 = 𝐵 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵))
 
Theoremoveq12 6435 Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveq1i 6436 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
𝐴 = 𝐵       (𝐴𝐹𝐶) = (𝐵𝐹𝐶)
 
Theoremoveq2i 6437 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
𝐴 = 𝐵       (𝐶𝐹𝐴) = (𝐶𝐹𝐵)
 
Theoremoveq12i 6438 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 6439 Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
𝐴 = 𝐵       (𝐶𝐴𝐷) = (𝐶𝐵𝐷)
 
Theoremoveq123i 6440 Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)
𝐴 = 𝐶    &   𝐵 = 𝐷    &   𝐹 = 𝐺       (𝐴𝐹𝐵) = (𝐶𝐺𝐷)
 
Theoremoveq1d 6441 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶))
 
Theoremoveq2d 6442 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵))
 
Theoremoveqd 6443 Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐷) = (𝐶𝐵𝐷))
 
Theoremoveq12d 6444 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 6445 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveqan12rd 6446 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜓𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveq123d 6447 Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐺𝐷))
 
Theoremovrspc2v 6448* 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 6449* Restricted specialization of operands, using implicit substitution. (Contributed by Mario Carneiro, 6-Dec-2014.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))       ((𝜑 ∧ (𝑋𝐴𝑌𝐵)) → (𝑋𝐹𝑌) = (𝑋𝐺𝑌))
 
Theoremoveqdr 6450 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 6451 Deduction version of bound-variable hypothesis builder nfov 6452. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐹)    &   (𝜑𝑥𝐵)       (𝜑𝑥(𝐴𝐹𝐵))
 
Theoremnfov 6452 Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.)
𝑥𝐴    &   𝑥𝐹    &   𝑥𝐵       𝑥(𝐴𝐹𝐵)
 
Theoremoprabid 6453 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 6454 The result of an operation is a set. (Contributed by NM, 13-Mar-1995.)
(𝐴𝐹𝐵) ∈ V
 
Theoremovexi 6455 The result of an operation is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴 = (𝐵𝐹𝐶)       𝐴 ∈ V
 
Theoremovssunirn 6456 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 6457 Operation value of the empty set. (Contributed by AV, 15-May-2021.)
(𝐴𝐵) = ∅
 
Theoremovprc 6458 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 6459 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 6460 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 6461 Reverse closure for an operation value. (Contributed by Mario Carneiro, 5-May-2015.)
Rel dom 𝐹       (𝐶 ∈ (𝐴𝐹𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
Theoremcsbov123 6462 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 6463* Move class substitution in and out of an operation. (Contributed by NM, 23-Aug-2018.)
𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐵𝐴 / 𝑥𝐹𝐶)
 
Theoremcsbov12g 6464* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶))
 
Theoremcsbov1g 6465* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐶))
 
Theoremcsbov2g 6466* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐵𝐹𝐴 / 𝑥𝐶))
 
Theoremrspceov 6467* A frequently used special case of rspc2ev 3199 for operation values. (Contributed by NM, 21-Mar-2007.)
((𝐶𝐴𝐷𝐵𝑆 = (𝐶𝐹𝐷)) → ∃𝑥𝐴𝑦𝐵 𝑆 = (𝑥𝐹𝑦))
 
Theoremelovimad 6468 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)))
 
Theoremfnotovb 6469 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6031. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))
 
Theoremopabbrex 6470 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)
 
Theoremfvmptopab1 6471* 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.)
((𝜑𝑧 = 𝑍) → (𝜒𝜓))    &   (𝜑𝑍𝑉)    &   (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V)    &   𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)})       (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
 
Theoremfvmptopab2 6472* 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 conditions 𝜓 and 𝜏. (Contributed by AV, 31-Jan-2021.)
((𝜑𝑧 = 𝑍) → (𝜒𝜓))    &   (𝜑𝑍𝑉)    &   (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V)    &   𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒𝜃)})    &   ((𝜑𝑧 = 𝑍) → (𝜃𝜏))       (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓𝜏)})
 
Theorem0neqopab 6473 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 6474 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 6475 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 6476* 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 6477* An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)
Rel {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremoprabv 6478* 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 6479 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 6480 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 6481 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.)
𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremnfoprab 6482* Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.)
𝑤𝜑       𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremoprabbid 6483* 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 6484* 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 6485* 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 6486 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4820. (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 6487 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 4821. (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 6488 Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 4824. (Contributed by Mario Carneiro, 4-Jan-2017.)
({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ∀𝑥𝑦𝑧(𝜑𝜓))
 
Theoremmpt2eq123 6489* 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 6490* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
((𝐴 = 𝐶𝐵 = 𝐷) → (𝑥𝐴, 𝑦𝐵𝐸) = (𝑥𝐶, 𝑦𝐷𝐸))
 
Theoremmpt2eq123dva 6491* 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 6492* 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 6493 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 6494* 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 6495 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
((𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
 
Theoremmpt2eq3dv 6496* 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 6497 Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
𝑥(𝑥𝐴, 𝑦𝐵𝐶)
 
Theoremnfmpt22 6498 Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
𝑦(𝑥𝐴, 𝑦𝐵𝐶)
 
Theoremnfmpt2 6499* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
𝑧𝐴    &   𝑧𝐵    &   𝑧𝐶       𝑧(𝑥𝐴, 𝑦𝐵𝐶)
 
Theoremmpt20 6500 A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.)
(𝑥 ∈ ∅, 𝑦𝐵𝐶) = ∅
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