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Theorem List for Intuitionistic Logic Explorer - 5501-5600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremriotauni 5501 Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)
(∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = {𝑥𝐴𝜑})
 
Theoremnfriota1 5502* The abstraction variable in a restricted iota descriptor isn't free. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥(𝑥𝐴 𝜑)
 
Theoremnfriotadxy 5503* Deduction version of nfriota 5504. (Contributed by Jim Kingdon, 12-Jan-2019.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑𝑥𝐴)       (𝜑𝑥(𝑦𝐴 𝜓))
 
Theoremnfriota 5504* A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.)
𝑥𝜑    &   𝑥𝐴       𝑥(𝑦𝐴 𝜑)
 
Theoremcbvriota 5505* Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
 
Theoremcbvriotav 5506* Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
 
Theoremcsbriotag 5507* Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)
(𝐴𝑉𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
 
Theoremriotacl2 5508 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 5509* Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
(∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
 
Theoremriotasbc 5510 Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
(∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)
 
Theoremriotabidva 5511* Equivalent wff's yield equal restricted class abstractions (deduction rule). (rabbidva 2565 analog.) (Contributed by NM, 17-Jan-2012.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
 
Theoremriotabiia 5512 Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 2564 analog.) (Contributed by NM, 16-Jan-2012.)
(𝑥𝐴 → (𝜑𝜓))       (𝑥𝐴 𝜑) = (𝑥𝐴 𝜓)
 
Theoremriota1 5513* Property of restricted iota. Compare iota1 4908. (Contributed by Mario Carneiro, 15-Oct-2016.)
(∃!𝑥𝐴 𝜑 → ((𝑥𝐴𝜑) ↔ (𝑥𝐴 𝜑) = 𝑥))
 
Theoremriota1a 5514 Property of iota. (Contributed by NM, 23-Aug-2011.)
((𝑥𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
 
Theoremriota2df 5515* A deduction version of riota2f 5516. (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 5516* 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 5517* 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 5518* Properties of a restricted definite description operator. Todo (df-riota 5495 update): can some uses of riota2f 5516 be shortened with this? (Contributed by NM, 23-Nov-2013.)
𝑥𝜓    &   𝐵 = (𝑥𝐴 𝜑)    &   (𝑥 = 𝐵 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 → (𝐵𝐴𝜓))
 
Theoremriota5f 5519* 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 5520* 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 5521* 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 5522* 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 5523* 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 5524 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)})
 
Theoremeusvobj2 5525* 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 5526* 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 5527* 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 5528* 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 5529* 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) = ∅)
 
Theoremacexmidlema 5530* Lemma for acexmid 5538. (Contributed by Jim Kingdon, 6-Aug-2019.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   𝐶 = {𝐴, 𝐵}       ({∅} ∈ 𝐴𝜑)
 
Theoremacexmidlemb 5531* Lemma for acexmid 5538. (Contributed by Jim Kingdon, 6-Aug-2019.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   𝐶 = {𝐴, 𝐵}       (∅ ∈ 𝐵𝜑)
 
Theoremacexmidlemph 5532* Lemma for acexmid 5538. (Contributed by Jim Kingdon, 6-Aug-2019.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   𝐶 = {𝐴, 𝐵}       (𝜑𝐴 = 𝐵)
 
Theoremacexmidlemab 5533* Lemma for acexmid 5538. (Contributed by Jim Kingdon, 6-Aug-2019.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   𝐶 = {𝐴, 𝐵}       (((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅}) → ¬ 𝜑)
 
Theoremacexmidlemcase 5534* Lemma for acexmid 5538. Here we divide the proof into cases (based on the disjunction implicit in an unordered pair, not the sort of case elimination which relies on excluded middle).

The cases are (1) the choice function evaluated at 𝐴 equals {∅}, (2) the choice function evaluated at 𝐵 equals , and (3) the choice function evaluated at 𝐴 equals and the choice function evaluated at 𝐵 equals {∅}.

Because of the way we represent the choice function 𝑦, the choice function evaluated at 𝐴 is (𝑣𝐴𝑢𝑦(𝐴𝑢𝑣𝑢)) and the choice function evaluated at 𝐵 is (𝑣𝐵𝑢𝑦(𝐵𝑢𝑣𝑢)). Other than the difference in notation these work just as (𝑦𝐴) and (𝑦𝐵) would if 𝑦 were a function as defined by df-fun 4931.

Although it isn't exactly about the division into cases, it is also convenient for this lemma to also include the step that if the choice function evaluated at 𝐴 equals {∅}, then {∅} ∈ 𝐴 and likewise for 𝐵.

(Contributed by Jim Kingdon, 7-Aug-2019.)

𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   𝐶 = {𝐴, 𝐵}       (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ({∅} ∈ 𝐴 ∨ ∅ ∈ 𝐵 ∨ ((𝑣𝐴𝑢𝑦 (𝐴𝑢𝑣𝑢)) = ∅ ∧ (𝑣𝐵𝑢𝑦 (𝐵𝑢𝑣𝑢)) = {∅})))
 
Theoremacexmidlem1 5535* Lemma for acexmid 5538. List the cases identified in acexmidlemcase 5534 and hook them up to the lemmas which handle each case. (Contributed by Jim Kingdon, 7-Aug-2019.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   𝐶 = {𝐴, 𝐵}       (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → (𝜑 ∨ ¬ 𝜑))
 
Theoremacexmidlem2 5536* Lemma for acexmid 5538. This builds on acexmidlem1 5535 by noting that every element of 𝐶 is inhabited.

(Note that 𝑦 is not quite a function in the df-fun 4931 sense because it uses ordered pairs as described in opthreg 4307 rather than df-op 3411).

The set 𝐴 is also found in onsucelsucexmidlem 4281.

(Contributed by Jim Kingdon, 5-Aug-2019.)

𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}    &   𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}    &   𝐶 = {𝐴, 𝐵}       (∀𝑧𝐶𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → (𝜑 ∨ ¬ 𝜑))
 
Theoremacexmidlemv 5537* Lemma for acexmid 5538.

This is acexmid 5538 with additional distinct variable constraints, most notably between 𝜑 and 𝑥.

(Contributed by Jim Kingdon, 6-Aug-2019.)

𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)       (𝜑 ∨ ¬ 𝜑)
 
Theoremacexmid 5538* The axiom of choice implies excluded middle. Theorem 1.3 in [Bauer] p. 483.

The statement of the axiom of choice given here is ac2 in the Metamath Proof Explorer (version of 3-Aug-2019). In particular, note that the choice function 𝑦 provides a value when 𝑧 is inhabited (as opposed to non-empty as in some statements of the axiom of choice).

Essentially the same proof can also be found at "The axiom of choice implies instances of EM", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

Often referred to as Diaconescu's theorem, or Diaconescu-Goodman-Myhill theorem, after Radu Diaconescu who discovered it in 1975 in the framework of topos theory and N. D. Goodman and John Myhill in 1978 in the framework of set theory (although it already appeared as an exercise in Errett Bishop's book Foundations of Constructive Analysis from 1967).

(Contributed by Jim Kingdon, 4-Aug-2019.)

𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)       (𝜑 ∨ ¬ 𝜑)
 
2.6.10  Operations
 
Syntaxco 5539 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.
class (𝐴𝐹𝐵)
 
Syntaxcoprab 5540 Extend class notation to include class abstraction (class builder) of nested ordered pairs.
class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Syntaxcmpt2 5541 Extend the definition of a class to include maps-to notation for defining an operation via a rule.
class (𝑥𝐴, 𝑦𝐵𝐶)
 
Definitiondf-ov 5542 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 . 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 5568 and ovprc2 5569. On the other hand, we often find uses for this definition when 𝐹 is a proper class. 𝐹 is normally equal to a class of nested ordered pairs of the form defined by df-oprab 5543. (Contributed by NM, 28-Feb-1995.)
(𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
 
Definitiondf-oprab 5543* 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 5542 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 5663. (Contributed by NM, 12-Mar-1995.)
{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
 
Definitiondf-mpt2 5544* 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 𝐵(𝑥, 𝑦)." An extension of df-mpt 3847 for two arguments. (Contributed by NM, 17-Feb-2008.)
(𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
 
Theoremoveq 5545 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(𝐹 = 𝐺 → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))
 
Theoremoveq1 5546 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(𝐴 = 𝐵 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶))
 
Theoremoveq2 5547 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(𝐴 = 𝐵 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵))
 
Theoremoveq12 5548 Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveq1i 5549 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
𝐴 = 𝐵       (𝐴𝐹𝐶) = (𝐵𝐹𝐶)
 
Theoremoveq2i 5550 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
𝐴 = 𝐵       (𝐶𝐹𝐴) = (𝐶𝐹𝐵)
 
Theoremoveq12i 5551 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 5552 Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
𝐴 = 𝐵       (𝐶𝐴𝐷) = (𝐶𝐵𝐷)
 
Theoremoveq123i 5553 Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)
𝐴 = 𝐶    &   𝐵 = 𝐷    &   𝐹 = 𝐺       (𝐴𝐹𝐵) = (𝐶𝐺𝐷)
 
Theoremoveq1d 5554 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶))
 
Theoremoveq2d 5555 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵))
 
Theoremoveqd 5556 Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐷) = (𝐶𝐵𝐷))
 
Theoremoveq12d 5557 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 5558 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveqan12rd 5559 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜓𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveq123d 5560 Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐺𝐷))
 
Theoremnfovd 5561 Deduction version of bound-variable hypothesis builder nfov 5562. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐹)    &   (𝜑𝑥𝐵)       (𝜑𝑥(𝐴𝐹𝐵))
 
Theoremnfov 5562 Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.)
𝑥𝐴    &   𝑥𝐹    &   𝑥𝐵       𝑥(𝐴𝐹𝐵)
 
Theoremoprabidlem 5563* Slight elaboration of exdistrfor 1697. A lemma for oprabid 5564. (Contributed by Jim Kingdon, 15-Jan-2019.)
(∃𝑥𝑦(𝑥 = 𝑧𝜓) → ∃𝑥(𝑥 = 𝑧 ∧ ∃𝑦𝜓))
 
Theoremoprabid 5564 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Although this theorem would be useful with a distinct variable constraint between 𝑥, 𝑦, and 𝑧, we use ax-bndl 1415 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.)
(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
 
Theoremfnovex 5565 The result of an operation is a set. (Contributed by Jim Kingdon, 15-Jan-2019.)
((𝐹 Fn (𝐶 × 𝐷) ∧ 𝐴𝐶𝐵𝐷) → (𝐴𝐹𝐵) ∈ V)
 
Theoremovexg 5566 Evaluating a set operation at two sets gives a set. (Contributed by Jim Kingdon, 19-Aug-2021.)
((𝐴𝑉𝐹𝑊𝐵𝑋) → (𝐴𝐹𝐵) ∈ V)
 
Theoremovprc 5567 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 5568 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 5569 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) = ∅)
 
Theoremcsbov123g 5570 Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
(𝐴𝐷𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶))
 
Theoremcsbov12g 5571* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶))
 
Theoremcsbov1g 5572* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐶))
 
Theoremcsbov2g 5573* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐵𝐹𝐴 / 𝑥𝐶))
 
Theoremrspceov 5574* A frequently used special case of rspc2ev 2686 for operation values. (Contributed by NM, 21-Mar-2007.)
((𝐶𝐴𝐷𝐵𝑆 = (𝐶𝐹𝐷)) → ∃𝑥𝐴𝑦𝐵 𝑆 = (𝑥𝐹𝑦))
 
Theoremfnotovb 5575 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5242. (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 5576* A collection of ordered pairs with an extension of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑓(𝑉𝑊𝐸)𝑝𝜃))    &   ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ 𝜃} ∈ V)       ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜓)} ∈ V)
 
Theorem0neqopab 5577 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}
 
Theorembrabvv 5578* If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Jim Kingdon, 16-Jan-2019.)
(𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
 
Theoremdfoprab2 5579* 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 5580* An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)
Rel {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremnfoprab1 5581 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 5582 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 5583 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.)
𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremnfoprab 5584* Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.)
𝑤𝜑       𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremoprabbid 5585* 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 5586* 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 5587* 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 5588 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4039. (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 5589 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 4040. (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 5590 Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 4043. (Contributed by Mario Carneiro, 4-Jan-2017.)
({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ∀𝑥𝑦𝑧(𝜑𝜓))
 
Theoremmpt2eq123 5591* 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 5592* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
((𝐴 = 𝐶𝐵 = 𝐷) → (𝑥𝐴, 𝑦𝐵𝐸) = (𝑥𝐶, 𝑦𝐷𝐸))
 
Theoremmpt2eq123dva 5593* 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 5594* 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 5595 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 5596* 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 5597 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
((𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
 
Theoremnfmpt21 5598 Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
𝑥(𝑥𝐴, 𝑦𝐵𝐶)
 
Theoremnfmpt22 5599 Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
𝑦(𝑥𝐴, 𝑦𝐵𝐶)
 
Theoremnfmpt2 5600* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
𝑧𝐴    &   𝑧𝐵    &   𝑧𝐶       𝑧(𝑥𝐴, 𝑦𝐵𝐶)
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