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Theorem List for Intuitionistic Logic Explorer - 5701-5800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoveq2i 5701 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
𝐴 = 𝐵       (𝐶𝐹𝐴) = (𝐶𝐹𝐵)
 
Theoremoveq12i 5702 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 5703 Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
𝐴 = 𝐵       (𝐶𝐴𝐷) = (𝐶𝐵𝐷)
 
Theoremoveq123i 5704 Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)
𝐴 = 𝐶    &   𝐵 = 𝐷    &   𝐹 = 𝐺       (𝐴𝐹𝐵) = (𝐶𝐺𝐷)
 
Theoremoveq1d 5705 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶))
 
Theoremoveq2d 5706 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵))
 
Theoremoveqd 5707 Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐷) = (𝐶𝐵𝐷))
 
Theoremoveq12d 5708 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 5709 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveqan12rd 5710 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜓𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveq123d 5711 Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐺𝐷))
 
Theoremfvoveq1d 5712 Equality deduction for nested function and operation value. (Contributed by AV, 23-Jul-2022.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹‘(𝐴𝑂𝐶)) = (𝐹‘(𝐵𝑂𝐶)))
 
Theoremfvoveq1 5713 Equality theorem for nested function and operation value. Closed form of fvoveq1d 5712. (Contributed by AV, 23-Jul-2022.)
(𝐴 = 𝐵 → (𝐹‘(𝐴𝑂𝐶)) = (𝐹‘(𝐵𝑂𝐶)))
 
Theoremovanraleqv 5714* Equality theorem for a conjunction with an operation values within a restricted universal quantification. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)
(𝐵 = 𝑋 → (𝜑𝜓))       (𝐵 = 𝑋 → (∀𝑥𝑉 (𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ ∀𝑥𝑉 (𝜓 ∧ (𝐴 · 𝑋) = 𝐶)))
 
Theoremimbrov2fvoveq 5715 Equality theorem for nested function and operation value in an implication for a binary relation. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 17-Aug-2022.)
(𝑋 = 𝑌 → (𝜑𝜓))       (𝑋 = 𝑌 → ((𝜑 → (𝐹‘((𝐺𝑋) · 𝑂))𝑅𝐴) ↔ (𝜓 → (𝐹‘((𝐺𝑌) · 𝑂))𝑅𝐴)))
 
Theoremnfovd 5716 Deduction version of bound-variable hypothesis builder nfov 5717. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐹)    &   (𝜑𝑥𝐵)       (𝜑𝑥(𝐴𝐹𝐵))
 
Theoremnfov 5717 Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.)
𝑥𝐴    &   𝑥𝐹    &   𝑥𝐵       𝑥(𝐴𝐹𝐵)
 
Theoremoprabidlem 5718* Slight elaboration of exdistrfor 1735. A lemma for oprabid 5719. (Contributed by Jim Kingdon, 15-Jan-2019.)
(∃𝑥𝑦(𝑥 = 𝑧𝜓) → ∃𝑥(𝑥 = 𝑧 ∧ ∃𝑦𝜓))
 
Theoremoprabid 5719 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 1451 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.)
(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
 
Theoremfnovex 5720 The result of an operation is a set. (Contributed by Jim Kingdon, 15-Jan-2019.)
((𝐹 Fn (𝐶 × 𝐷) ∧ 𝐴𝐶𝐵𝐷) → (𝐴𝐹𝐵) ∈ V)
 
Theoremovexg 5721 Evaluating a set operation at two sets gives a set. (Contributed by Jim Kingdon, 19-Aug-2021.)
((𝐴𝑉𝐹𝑊𝐵𝑋) → (𝐴𝐹𝐵) ∈ V)
 
Theoremovprc 5722 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 5723 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 5724 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 5725 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 5726* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶))
 
Theoremcsbov1g 5727* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐶))
 
Theoremcsbov2g 5728* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐵𝐹𝐴 / 𝑥𝐶))
 
Theoremrspceov 5729* A frequently used special case of rspc2ev 2750 for operation values. (Contributed by NM, 21-Mar-2007.)
((𝐶𝐴𝐷𝐵𝑆 = (𝐶𝐹𝐷)) → ∃𝑥𝐴𝑦𝐵 𝑆 = (𝑥𝐹𝑦))
 
Theoremfnotovb 5730 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5381. (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 5731* 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 5732 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 5733* 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 5734* 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 5735* An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)
Rel {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremnfoprab1 5736 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 5737 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 5738 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.)
𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremnfoprab 5739* Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.)
𝑤𝜑       𝑤{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremoprabbid 5740* Equivalent wff's yield equal operation class abstractions (deduction form). (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 5741* Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.)
(𝜑 → (𝜓𝜒))       (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒})
 
Theoremoprabbii 5742* 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 5743 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4126. (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 5744 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 4127. (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 5745 Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 4130. (Contributed by Mario Carneiro, 4-Jan-2017.)
({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ∀𝑥𝑦𝑧(𝜑𝜓))
 
Theoremmpt2eq123 5746* 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 5747* An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
((𝐴 = 𝐶𝐵 = 𝐷) → (𝑥𝐴, 𝑦𝐵𝐸) = (𝑥𝐶, 𝑦𝐷𝐸))
 
Theoremmpt2eq123dva 5748* 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 5749* 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 5750 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 5751* 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 5752 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
((𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
 
Theoremnfmpt21 5753 Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
𝑥(𝑥𝐴, 𝑦𝐵𝐶)
 
Theoremnfmpt22 5754 Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
𝑦(𝑥𝐴, 𝑦𝐵𝐶)
 
Theoremnfmpt2 5755* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
𝑧𝐴    &   𝑧𝐵    &   𝑧𝐶       𝑧(𝑥𝐴, 𝑦𝐵𝐶)
 
Theoremmpt20 5756 A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.)
(𝑥 ∈ ∅, 𝑦𝐵𝐶) = ∅
 
Theoremoprab4 5757* 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 5758* 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 5759* 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 5760* 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 5761* 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 5762* 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 5763* 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 5764* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 5765 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 5765* 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 5766* Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 3955, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)
(𝑥 = 𝑧𝐶 = 𝐸)    &   (𝑦 = 𝑤𝐸 = 𝐷)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐵𝐷)
 
Theoremdmoprab 5767* 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 5768* The domain of an operation class abstraction. (Contributed by NM, 24-Aug-1995.)
dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
 
Theoremrnoprab 5769* 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 5770* 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 5771* The domain of an operation class abstraction is a relation. (Contributed by NM, 17-Mar-1995.)
Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremoprabss 5772* Structure of an operation class abstraction. (Contributed by NM, 28-Nov-2006.)
{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ ((V × V) × V)
 
Theoremeloprabga 5773* The law of concretion for operation class abstraction. Compare elopab 4109. (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 5774* The law of concretion for operation class abstraction. Compare elopab 4109. (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 5775* 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 5776* 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 5777* 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 5778* 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)
 
Theoremresoprab 5779* Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.)
({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
 
Theoremresoprab2 5780* 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 5781* Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.)
((𝐶𝐴𝐷𝐵) → ((𝑥𝐴, 𝑦𝐵𝐸) ↾ (𝐶 × 𝐷)) = (𝑥𝐶, 𝑦𝐷𝐸))
 
Theoremfunoprabg 5782* "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 28-Aug-2007.)
(∀𝑥𝑦∃*𝑧𝜑 → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
 
Theoremfunoprab 5783* "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 17-Mar-1995.)
∃*𝑧𝜑       Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Theoremfnoprabg 5784* Functionality and domain of an operation class abstraction. (Contributed by NM, 28-Aug-2007.)
(∀𝑥𝑦(𝜑 → ∃!𝑧𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑})
 
Theoremmpt2fun 5785* The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       Fun 𝐹
 
Theoremfnoprab 5786* Functionality and domain of an operation class abstraction. (Contributed by NM, 15-May-1995.)
(𝜑 → ∃!𝑧𝜓)       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
Theoremffnov 5787* An operation maps to a class to which all values belong. (Contributed by NM, 7-Feb-2004.)
(𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶))
 
Theoremfovcl 5788 Closure law for an operation. (Contributed by NM, 19-Apr-2007.)
𝐹:(𝑅 × 𝑆)⟶𝐶       ((𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
 
Theoremeqfnov 5789* Equality of two operations is determined by their values. (Contributed by NM, 1-Sep-2005.)
((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))))
 
Theoremeqfnov2 5790* Two operators with the same domain are equal iff their values at each point in the domain are equal. (Contributed by Jeff Madsen, 7-Jun-2010.)
((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐴 × 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))
 
Theoremfnovim 5791* Representation of a function in terms of its values. (Contributed by Jim Kingdon, 16-Jan-2019.)
(𝐹 Fn (𝐴 × 𝐵) → 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
 
Theoremmpt22eqb 5792* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 5790. (Contributed by Mario Carneiro, 4-Jan-2017.)
(∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ((𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷) ↔ ∀𝑥𝐴𝑦𝐵 𝐶 = 𝐷))
 
Theoremrnmpt2 5793* The range of an operation given by the maps-to notation. (Contributed by FL, 20-Jun-2011.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
 
Theoremreldmmpt2 5794* The domain of an operation defined by maps-to notation is a relation. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       Rel dom 𝐹
 
Theoremelrnmpt2g 5795* Membership in the range of an operation class abstraction. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (𝐷𝑉 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶))
 
Theoremelrnmpt2 5796* Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   𝐶 ∈ V       (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶)
 
Theoremralrnmpt2 5797* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   (𝑧 = 𝐶 → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜓))
 
Theoremrexrnmpt2 5798* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   (𝑧 = 𝐶 → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓))
 
Theoremovid 5799* The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
((𝑥𝑅𝑦𝑆) → ∃!𝑧𝜑)    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}       ((𝑥𝑅𝑦𝑆) → ((𝑥𝐹𝑦) = 𝑧𝜑))
 
Theoremovidig 5800* The value of an operation class abstraction. Compare ovidi 5801. The condition (𝑥𝑅𝑦𝑆) is been removed. (Contributed by Mario Carneiro, 29-Dec-2014.)
∃*𝑧𝜑    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}       (𝜑 → (𝑥𝐹𝑦) = 𝑧)
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