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Theorem List for Metamath Proof Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcaovcld 6601* Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)       (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸)
 
Theoremcaovcl 6602* Convert an operation closure law to class notation. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
((𝑥𝑆𝑦𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)       ((𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
 
Theoremcaovcomg 6603* Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))       ((𝜑 ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
 
Theoremcaovcomd 6604* Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)       (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
 
Theoremcaovcom 6605* Convert an operation commutative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 1-Jun-2013.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥𝐹𝑦) = (𝑦𝐹𝑥)       (𝐴𝐹𝐵) = (𝐵𝐹𝐴)
 
Theoremcaovassg 6606* Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.) (Revised by Mario Carneiro, 26-May-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))       ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
 
Theoremcaovassd 6607* Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
 
Theoremcaovass 6608* Convert an operation associative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))       ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))
 
Theoremcaovcang 6609* Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))       ((𝜑 ∧ (𝐴𝑇𝐵𝑆𝐶𝑆)) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))
 
Theoremcaovcand 6610* Convert an operation cancellation law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))    &   (𝜑𝐴𝑇)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))
 
Theoremcaovcanrd 6611* Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))    &   (𝜑𝐴𝑇)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))       (𝜑 → ((𝐵𝐹𝐴) = (𝐶𝐹𝐴) ↔ 𝐵 = 𝐶))
 
Theoremcaovcan 6612* Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.)
𝐶 ∈ V    &   ((𝑥𝑆𝑦𝑆) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) → 𝑦 = 𝑧))       ((𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) → 𝐵 = 𝐶))
 
Theoremcaovordig 6613* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))       ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
 
Theoremcaovordid 6614* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
 
Theoremcaovordg 6615* Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))       ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
 
Theoremcaovordd 6616* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
 
Theoremcaovord2d 6617* Operation ordering law with commuted arguments. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))       (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)))
 
Theoremcaovord3d 6618* Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   (𝜑𝐷𝑆)       (𝜑 → ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → (𝐴𝑅𝐶𝐷𝑅𝐵)))
 
Theoremcaovord 6619* Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑧𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))       (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
 
Theoremcaovord2 6620* Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑧𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   𝐶 ∈ V    &   (𝑥𝐹𝑦) = (𝑦𝐹𝑥)       (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)))
 
Theoremcaovord3 6621* Ordering law. (Contributed by NM, 29-Feb-1996.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑧𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))    &   𝐶 ∈ V    &   (𝑥𝐹𝑦) = (𝑦𝐹𝑥)    &   𝐷 ∈ V       (((𝐵𝑆𝐶𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐴𝑅𝐶𝐷𝑅𝐵))
 
Theoremcaovdig 6622* Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)
((𝜑 ∧ (𝑥𝐾𝑦𝑆𝑧𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)))       ((𝜑 ∧ (𝐴𝐾𝐵𝑆𝐶𝑆)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶)))
 
Theoremcaovdid 6623* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝐾𝑦𝑆𝑧𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)))    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶)))
 
Theoremcaovdir2d 6624* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))       (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)))
 
Theoremcaovdirg 6625* Convert an operation reverse distributive law to class notation. (Contributed by Mario Carneiro, 19-Oct-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)))       ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝐾)) → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))
 
Theoremcaovdird 6626* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)))    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝐾)       (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))
 
Theoremcaovdi 6627* Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 28-Jun-2013.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))       (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶))
 
Theoremcaov32d 6628* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))       (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵))
 
Theoremcaov12d 6629* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))       (𝜑 → (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶)))
 
Theoremcaov31d 6630* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))       (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴))
 
Theoremcaov13d 6631* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))       (𝜑 → (𝐴𝐹(𝐵𝐹𝐶)) = (𝐶𝐹(𝐵𝐹𝐴)))
 
Theoremcaov4d 6632* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))    &   (𝜑𝐷𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)       (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)))
 
Theoremcaov411d 6633* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))    &   (𝜑𝐷𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)       (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷)))
 
Theoremcaov42d 6634* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
(𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))    &   (𝜑𝐷𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)       (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐷𝐹𝐵)))
 
Theoremcaov32 6635* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   (𝑥𝐹𝑦) = (𝑦𝐹𝑥)    &   ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))       ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵)
 
Theoremcaov12 6636* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   (𝑥𝐹𝑦) = (𝑦𝐹𝑥)    &   ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))       (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶))
 
Theoremcaov31 6637* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   (𝑥𝐹𝑦) = (𝑦𝐹𝑥)    &   ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))       ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴)
 
Theoremcaov13 6638* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   (𝑥𝐹𝑦) = (𝑦𝐹𝑥)    &   ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))       (𝐴𝐹(𝐵𝐹𝐶)) = (𝐶𝐹(𝐵𝐹𝐴))
 
Theoremcaov4 6639* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   (𝑥𝐹𝑦) = (𝑦𝐹𝑥)    &   ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))    &   𝐷 ∈ V       ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷))
 
Theoremcaov411 6640* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   (𝑥𝐹𝑦) = (𝑦𝐹𝑥)    &   ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))    &   𝐷 ∈ V       ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷))
 
Theoremcaov42 6641* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   (𝑥𝐹𝑦) = (𝑦𝐹𝑥)    &   ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))    &   𝐷 ∈ V       ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐷𝐹𝐵))
 
Theoremcaovdir 6642* Reverse distributive law. (Contributed by NM, 26-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   (𝑥𝐺𝑦) = (𝑦𝐺𝑥)    &   (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))       ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))
 
Theoremcaovdilem 6643* Lemma used by real number construction. (Contributed by NM, 26-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   (𝑥𝐺𝑦) = (𝑦𝐺𝑥)    &   (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))    &   𝐷 ∈ V    &   𝐻 ∈ V    &   ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))       (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))
 
Theoremcaovlem2 6644* Lemma used in real number construction. (Contributed by NM, 26-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   (𝑥𝐺𝑦) = (𝑦𝐺𝑥)    &   (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))    &   𝐷 ∈ V    &   𝐻 ∈ V    &   ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))    &   𝑅 ∈ V    &   (𝑥𝐹𝑦) = (𝑦𝐹𝑥)    &   ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))       ((((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻)𝐹(((𝐴𝐺𝐷)𝐹(𝐵𝐺𝐶))𝐺𝑅)) = ((𝐴𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅)))𝐹(𝐵𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻))))
 
Theoremcaovmo 6645* Uniqueness of inverse element in commutative, associative operation with identity. Remark in proof of Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 4-Mar-1996.)
𝐵𝑆    &   dom 𝐹 = (𝑆 × 𝑆)    &    ¬ ∅ ∈ 𝑆    &   (𝑥𝐹𝑦) = (𝑦𝐹𝑥)    &   ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))    &   (𝑥𝑆 → (𝑥𝐹𝐵) = 𝑥)       ∃*𝑤(𝐴𝐹𝑤) = 𝐵
 
Theoremgrprinvlem 6646* Lemma for grprinvd 6647. (Contributed by NM, 9-Aug-2013.)
((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   (𝜑𝑂𝐵)    &   ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = 𝑥)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂)    &   ((𝜑𝜓) → 𝑋𝐵)    &   ((𝜑𝜓) → (𝑋 + 𝑋) = 𝑋)       ((𝜑𝜓) → 𝑋 = 𝑂)
 
Theoremgrprinvd 6647* Deduce right inverse from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   (𝜑𝑂𝐵)    &   ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = 𝑥)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂)    &   ((𝜑𝜓) → 𝑋𝐵)    &   ((𝜑𝜓) → 𝑁𝐵)    &   ((𝜑𝜓) → (𝑁 + 𝑋) = 𝑂)       ((𝜑𝜓) → (𝑋 + 𝑁) = 𝑂)
 
Theoremgrpridd 6648* Deduce right identity from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   (𝜑𝑂𝐵)    &   ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = 𝑥)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂)       ((𝜑𝑥𝐵) → (𝑥 + 𝑂) = 𝑥)
 
2.3.18  "Maps to" notation
 
Theoremmpt2ndm0 6649* The value of an operation given by a maps-to rule is the empty set if the arguments are not contained in the base sets of the rule. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)       (¬ (𝑉𝑋𝑊𝑌) → (𝑉𝐹𝑊) = ∅)
 
Theoremelmpt2cl 6650* If a two-parameter class is not empty, constrain the implicit pair. (Contributed by Stefan O'Rear, 7-Mar-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆𝐴𝑇𝐵))
 
Theoremelmpt2cl1 6651* If a two-parameter class is not empty, the first argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (𝑋 ∈ (𝑆𝐹𝑇) → 𝑆𝐴)
 
Theoremelmpt2cl2 6652* If a two-parameter class is not empty, the second argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (𝑋 ∈ (𝑆𝐹𝑇) → 𝑇𝐵)
 
Theoremelovmpt2 6653* Utility lemma for two-parameter classes.

EDITORIAL: can simplify isghm 17375, islmhm 18752. (Contributed by Stefan O'Rear, 21-Jan-2015.)

𝐷 = (𝑎𝐴, 𝑏𝐵𝐶)    &   𝐶 ∈ V    &   ((𝑎 = 𝑋𝑏 = 𝑌) → 𝐶 = 𝐸)       (𝐹 ∈ (𝑋𝐷𝑌) ↔ (𝑋𝐴𝑌𝐵𝐹𝐸))
 
Theoremelovmpt2rab 6654* Implications for the value of an operation, defined by the maps-to notation with a class abstraction as a result, having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑀𝜑})    &   ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑀 ∈ V)       (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑀))
 
Theoremelovmpt2rab1 6655* Implications for the value of an operation, defined by the maps-to notation with a class abstraction as a result, having an element. Here, the base set of the class abstraction depends on the first operand. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑥 / 𝑚𝑀𝜑})    &   ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 / 𝑚𝑀 ∈ V)       (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑋 / 𝑚𝑀))
 
Theorem2mpt20 6656* If the operation value of the operation value of two nested maps-to notation is not empty, all involved arguments belong to the corresponding base classes of the maps-to notations. (Contributed by AV, 21-May-2021.)
𝑂 = (𝑥𝐴, 𝑦𝐵𝐸)    &   ((𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) = (𝑠𝐶, 𝑡𝐷𝐹))       (¬ ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
 
Theoremrelmptopab 6657* Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 7-Aug-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
𝐹 = (𝑥𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ 𝜑})       Rel (𝐹𝐵)
 
Theoremf1ocnvd 6658* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
𝐹 = (𝑥𝐴𝐶)    &   ((𝜑𝑥𝐴) → 𝐶𝑊)    &   ((𝜑𝑦𝐵) → 𝐷𝑋)    &   (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))       (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))
 
Theoremf1od 6659* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)
𝐹 = (𝑥𝐴𝐶)    &   ((𝜑𝑥𝐴) → 𝐶𝑊)    &   ((𝜑𝑦𝐵) → 𝐷𝑋)    &   (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))       (𝜑𝐹:𝐴1-1-onto𝐵)
 
Theoremf1ocnv2d 6660* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
𝐹 = (𝑥𝐴𝐶)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑦𝐵) → 𝐷𝐴)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))       (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))
 
Theoremf1o2d 6661* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)
𝐹 = (𝑥𝐴𝐶)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑦𝐵) → 𝐷𝐴)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))       (𝜑𝐹:𝐴1-1-onto𝐵)
 
Theoremf1opw2 6662* A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 6663 avoids the Axiom of Replacement. (Contributed by Mario Carneiro, 26-Jun-2015.)
(𝜑𝐹:𝐴1-1-onto𝐵)    &   (𝜑 → (𝐹𝑎) ∈ V)    &   (𝜑 → (𝐹𝑏) ∈ V)       (𝜑 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏)):𝒫 𝐴1-1-onto→𝒫 𝐵)
 
Theoremf1opw 6663* A one-to-one mapping induces a one-to-one mapping on power sets. (Contributed by Stefan O'Rear, 18-Nov-2014.) (Revised by Mario Carneiro, 26-Jun-2015.)
(𝐹:𝐴1-1-onto𝐵 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹𝑏)):𝒫 𝐴1-1-onto→𝒫 𝐵)
 
Theoremelovmpt3imp 6664* If the value of a function which is the result of an operation defined by the maps-to notation is not empty, the operands must be sets. Remark: a function which is the result of an operation can be regared as operation with 3 operands - therefore the abbreviation "mpt3" is used in the label. (Contributed by AV, 16-May-2019.)
𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧𝑀𝐵))       (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
 
Theoremovmpt3rab1 6665* The value of an operation defined by the maps-to notation with a function into a class abstraction as a result. The domain of the function and the base set of the class abstraction may depend on the operands, using implicit substitution. (Contributed by AV, 16-Jul-2018.) (Revised by AV, 16-May-2019.)
𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧𝑀 ↦ {𝑎𝑁𝜑}))    &   ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑀 = 𝐾)    &   ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑁 = 𝐿)    &   ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑𝜓))    &   𝑥𝜓    &   𝑦𝜓       ((𝑋𝑉𝑌𝑊𝐾𝑈) → (𝑋𝑂𝑌) = (𝑧𝐾 ↦ {𝑎𝐿𝜓}))
 
Theoremovmpt3rabdm 6666* If the value of a function which is the result of an operation defined by the maps-to notation is not empty, the operands and the argument of the function must be sets. (Contributed by AV, 16-May-2019.)
𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧𝑀 ↦ {𝑎𝑁𝜑}))    &   ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑀 = 𝐾)    &   ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑁 = 𝐿)       (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → dom (𝑋𝑂𝑌) = 𝐾)
 
Theoremelovmpt3rab1 6667* Implications for the value of an operation defined by the maps-to notation with a function into a class abstraction as a result having an element. The domain of the function and the base set of the class abstraction may depend on the operands, using implicit substitution. (Contributed by AV, 16-Jul-2018.) (Revised by AV, 16-May-2019.)
𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧𝑀 ↦ {𝑎𝑁𝜑}))    &   ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑀 = 𝐾)    &   ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑁 = 𝐿)       ((𝐾𝑈𝐿𝑇) → (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑍𝐾𝐴𝐿))))
 
Theoremelovmpt3rab 6668* Implications for the value of an operation defined by the maps-to notation with a class abstration as a result having an element. (Contributed by AV, 17-Jul-2018.) (Revised by AV, 16-May-2019.)
𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧𝑀 ↦ {𝑎𝑁𝜑}))       ((𝑀𝑈𝑁𝑇) → (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑍𝑀𝐴𝑁))))
 
2.3.19  Function operation
 
Syntaxcof 6669 Extend class notation to include mapping of an operation to a function operation.
class 𝑓 𝑅
 
Syntaxcofr 6670 Extend class notation to include mapping of a binary relation to a function relation.
class 𝑟 𝑅
 
Definitiondf-of 6671* Define the function operation map. The definition is designed so that if 𝑅 is a binary operation, then 𝑓 𝑅 is the analogous operation on functions which corresponds to applying 𝑅 pointwise to the values of the functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
 
Definitiondf-ofr 6672* Define the function relation map. The definition is designed so that if 𝑅 is a binary relation, then 𝑟 𝑅 is the analogous relation on functions which is true when each element of the left function relates to the corresponding element of the right function. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝑟 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓𝑥)𝑅(𝑔𝑥)}
 
Theoremofeq 6673 Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
(𝑅 = 𝑆 → ∘𝑓 𝑅 = ∘𝑓 𝑆)
 
Theoremofreq 6674 Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝑅 = 𝑆 → ∘𝑟 𝑅 = ∘𝑟 𝑆)
 
Theoremofexg 6675 A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)
(𝐴𝑉 → ( ∘𝑓 𝑅𝐴) ∈ V)
 
Theoremnfof 6676* Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
𝑥𝑅       𝑥𝑓 𝑅
 
Theoremnfofr 6677* Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝑥𝑅       𝑥𝑟 𝑅
 
Theoremoffval 6678* Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝑆    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)    &   ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝑆 ↦ (𝐶𝑅𝐷)))
 
Theoremofrfval 6679* Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝑆    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)    &   ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)       (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥𝑆 𝐶𝑅𝐷))
 
Theoremofval 6680 Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝑆    &   ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)    &   ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)       ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))
 
Theoremofrval 6681 Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝑆    &   ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)    &   ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)       ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝐶𝑅𝐷)
 
Theoremoffn 6682 The function operation produces a function. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝑆       (𝜑 → (𝐹𝑓 𝑅𝐺) Fn 𝑆)
 
Theoremoffval2f 6683* The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 23-Jun-2017.)
𝑥𝜑    &   𝑥𝐴    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   ((𝜑𝑥𝐴) → 𝐶𝑋)    &   (𝜑𝐹 = (𝑥𝐴𝐵))    &   (𝜑𝐺 = (𝑥𝐴𝐶))       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
 
Theoremofmresval 6684 Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
(𝜑𝐹𝐴)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐹( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹𝑓 𝑅𝐺))
 
Theoremfnfvof 6685 Function value of a pointwise composition. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Jun-2015.)
(((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐴𝑉𝑋𝐴)) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
 
Theoremoff 6686* The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐵𝑇)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝐶       (𝜑 → (𝐹𝑓 𝑅𝐺):𝐶𝑈)
 
Theoremofres 6687 Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝐶       (𝜑 → (𝐹𝑓 𝑅𝐺) = ((𝐹𝐶) ∘𝑓 𝑅(𝐺𝐶)))
 
Theoremoffval2 6688* The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   ((𝜑𝑥𝐴) → 𝐶𝑋)    &   (𝜑𝐹 = (𝑥𝐴𝐵))    &   (𝜑𝐺 = (𝑥𝐴𝐶))       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
 
Theoremofrfval2 6689* The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   ((𝜑𝑥𝐴) → 𝐶𝑋)    &   (𝜑𝐹 = (𝑥𝐴𝐵))    &   (𝜑𝐺 = (𝑥𝐴𝐶))       (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥𝐴 𝐵𝑅𝐶))
 
Theoremofmpteq 6690* Value of a pointwise operation on two functions defined using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014.)
((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → ((𝑥𝐴𝐵) ∘𝑓 𝑅(𝑥𝐴𝐶)) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
 
Theoremofco 6691 The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐻:𝐷𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐷𝑋)    &   (𝐴𝐵) = 𝐶       (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)))
 
Theoremoffveq 6692* Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐴)    &   (𝜑𝐻 Fn 𝐴)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)    &   ((𝜑𝑥𝐴) → (𝐺𝑥) = 𝐶)    &   ((𝜑𝑥𝐴) → (𝐵𝑅𝐶) = (𝐻𝑥))       (𝜑 → (𝐹𝑓 𝑅𝐺) = 𝐻)
 
Theoremoffveqb 6693* Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
(𝜑𝐴𝑉)    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐴)    &   (𝜑𝐻 Fn 𝐴)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)    &   ((𝜑𝑥𝐴) → (𝐺𝑥) = 𝐶)       (𝜑 → (𝐻 = (𝐹𝑓 𝑅𝐺) ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐵𝑅𝐶)))
 
Theoremofc1 6694 Left operation by a constant. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐹 Fn 𝐴)    &   ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)       ((𝜑𝑋𝐴) → (((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹)‘𝑋) = (𝐵𝑅𝐶))
 
Theoremofc2 6695 Right operation by a constant. (Contributed by NM, 7-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐹 Fn 𝐴)    &   ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)       ((𝜑𝑋𝐴) → ((𝐹𝑓 𝑅(𝐴 × {𝐵}))‘𝑋) = (𝐶𝑅𝐵))
 
Theoremofc12 6696 Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)       (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)}))
 
Theoremcaofref 6697* Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   ((𝜑𝑥𝑆) → 𝑥𝑅𝑥)       (𝜑𝐹𝑟 𝑅𝐹)
 
Theoremcaofinvl 6698* Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐵𝑊)    &   (𝜑𝑁:𝑆𝑆)    &   (𝜑𝐺 = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))))    &   ((𝜑𝑥𝑆) → ((𝑁𝑥)𝑅𝑥) = 𝐵)       (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝐴 × {𝐵}))
 
Theoremcaofid0l 6699* Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐵𝑊)    &   ((𝜑𝑥𝑆) → (𝐵𝑅𝑥) = 𝑥)       (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹) = 𝐹)
 
Theoremcaofid0r 6700* Transfer a right identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐵𝑊)    &   ((𝜑𝑥𝑆) → (𝑥𝑅𝐵) = 𝑥)       (𝜑 → (𝐹𝑓 𝑅(𝐴 × {𝐵})) = 𝐹)
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
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