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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | oprssov 7601 | The value of a member of the domain of a subclass of an operation. (Contributed by NM, 23-Aug-2007.) |
⊢ (((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵)) | ||
Theorem | fovcdm 7602 | An operation's value belongs to its codomain. (Contributed by NM, 27-Aug-2006.) |
⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) | ||
Theorem | fovcdmda 7603 | An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.) |
⊢ (𝜑 → 𝐹:(𝑅 × 𝑆)⟶𝐶) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶) | ||
Theorem | fovcdmd 7604 | An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.) |
⊢ (𝜑 → 𝐹:(𝑅 × 𝑆)⟶𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝑅) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐶) | ||
Theorem | fnrnov 7605* | The range of an operation expressed as a collection of the operation's values. (Contributed by NM, 29-Oct-2006.) |
⊢ (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)}) | ||
Theorem | foov 7606* | An onto mapping of an operation expressed in terms of operation values. (Contributed by NM, 29-Oct-2006.) |
⊢ (𝐹:(𝐴 × 𝐵)–onto→𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦))) | ||
Theorem | fnovrn 7607 | An operation's value belongs to its range. (Contributed by NM, 10-Feb-2007.) |
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶𝐹𝐷) ∈ ran 𝐹) | ||
Theorem | ovelrn 7608* | A member of an operation's range is a value of the operation. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 30-Jan-2014.) |
⊢ (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦))) | ||
Theorem | funimassov 7609* | Membership relation for the values of a function whose image is a subclass. (Contributed by Mario Carneiro, 23-Dec-2013.) |
⊢ ((Fun 𝐹 ∧ (𝐴 × 𝐵) ⊆ dom 𝐹) → ((𝐹 “ (𝐴 × 𝐵)) ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶)) | ||
Theorem | ovelimab 7610* | Operation value in an image. (Contributed by Mario Carneiro, 23-Dec-2013.) (Revised by Mario Carneiro, 29-Jan-2014.) |
⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝐷 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐷 = (𝑥𝐹𝑦))) | ||
Theorem | ovima0 7611 | An operation value is a member of the image plus null. (Contributed by Thierry Arnoux, 25-Jun-2019.) |
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})) | ||
Theorem | ovconst2 7612 | The value of a constant operation. (Contributed by NM, 5-Nov-2006.) |
⊢ 𝐶 ∈ V ⇒ ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (𝑅((𝐴 × 𝐵) × {𝐶})𝑆) = 𝐶) | ||
Theorem | oprssdm 7613* | Domain of closure of an operation. (Contributed by NM, 24-Aug-1995.) |
⊢ ¬ ∅ ∈ 𝑆 & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆) ⇒ ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹 | ||
Theorem | nssdmovg 7614 | The value of an operation outside its domain. (Contributed by Alexander van der Vekens, 7-Sep-2017.) |
⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅) | ||
Theorem | ndmovg 7615 | The value of an operation outside its domain. (Contributed by NM, 28-Mar-2008.) |
⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅) | ||
Theorem | ndmov 7616 | The value of an operation outside its domain. (Contributed by NM, 24-Aug-1995.) |
⊢ dom 𝐹 = (𝑆 × 𝑆) ⇒ ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅) | ||
Theorem | ndmovcl 7617 | The closure of an operation outside its domain, when the domain includes the empty set. This technical lemma can make the operation more convenient to work in some cases. It is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by NM, 24-Sep-2004.) |
⊢ dom 𝐹 = (𝑆 × 𝑆) & ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆) & ⊢ ∅ ∈ 𝑆 ⇒ ⊢ (𝐴𝐹𝐵) ∈ 𝑆 | ||
Theorem | ndmovrcl 7618 | Reverse closure law, when an operation's domain doesn't contain the empty set. (Contributed by NM, 3-Feb-1996.) |
⊢ dom 𝐹 = (𝑆 × 𝑆) & ⊢ ¬ ∅ ∈ 𝑆 ⇒ ⊢ ((𝐴𝐹𝐵) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) | ||
Theorem | ndmovcom 7619 | Any operation is commutative outside its domain. (Contributed by NM, 24-Aug-1995.) |
⊢ dom 𝐹 = (𝑆 × 𝑆) ⇒ ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) | ||
Theorem | ndmovass 7620 | Any operation is associative outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.) |
⊢ dom 𝐹 = (𝑆 × 𝑆) & ⊢ ¬ ∅ ∈ 𝑆 ⇒ ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) | ||
Theorem | ndmovdistr 7621 | Any operation is distributive outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.) |
⊢ dom 𝐹 = (𝑆 × 𝑆) & ⊢ ¬ ∅ ∈ 𝑆 & ⊢ dom 𝐺 = (𝑆 × 𝑆) ⇒ ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶))) | ||
Theorem | ndmovord 7622 | Elimination of redundant antecedents in an ordering law. (Contributed by NM, 7-Mar-1996.) |
⊢ dom 𝐹 = (𝑆 × 𝑆) & ⊢ 𝑅 ⊆ (𝑆 × 𝑆) & ⊢ ¬ ∅ ∈ 𝑆 & ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) ⇒ ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) | ||
Theorem | ndmovordi 7623 | Elimination of redundant antecedent in an ordering law. (Contributed by NM, 25-Jun-1998.) |
⊢ dom 𝐹 = (𝑆 × 𝑆) & ⊢ 𝑅 ⊆ (𝑆 × 𝑆) & ⊢ ¬ ∅ ∈ 𝑆 & ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) ⇒ ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐴𝑅𝐵) | ||
Theorem | caovclg 7624* | Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸) | ||
Theorem | caovcld 7625* | Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸) & ⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸) | ||
Theorem | caovcl 7626* | Convert an operation closure law to class notation. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.) |
⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆) ⇒ ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆) | ||
Theorem | caovcomg 7627* | Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) | ||
Theorem | caovcomd 7628* | Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) | ||
Theorem | caovcom 7629* | 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) | ||
Theorem | caovassg 7630* | Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.) (Revised by Mario Carneiro, 26-May-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) | ||
Theorem | caovassd 7631* | Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) | ||
Theorem | caovass 7632* | 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)) | ||
Theorem | caovcang 7633* | Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧)) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | caovcand 7634* | Convert an operation cancellation law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧)) & ⊢ (𝜑 → 𝐴 ∈ 𝑇) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | caovcanrd 7635* | Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧)) & ⊢ (𝜑 → 𝐴 ∈ 𝑇) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) ⇒ ⊢ (𝜑 → ((𝐵𝐹𝐴) = (𝐶𝐹𝐴) ↔ 𝐵 = 𝐶)) | ||
Theorem | caovcan 7636* | Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) |
⊢ 𝐶 ∈ V & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) → 𝑦 = 𝑧)) ⇒ ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) → 𝐵 = 𝐶)) | ||
Theorem | caovordig 7637* | Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) | ||
Theorem | caovordid 7638* | Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) | ||
Theorem | caovordg 7639* | Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) | ||
Theorem | caovordd 7640* | Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) | ||
Theorem | caovord2d 7641* | Operation ordering law with commuted arguments. (Contributed by Mario Carneiro, 30-Dec-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))) | ||
Theorem | caovord3d 7642* | Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) & ⊢ (𝜑 → 𝐷 ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → (𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵))) | ||
Theorem | caovord 7643* | Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) ⇒ ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) | ||
Theorem | caovord2 7644* | Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) & ⊢ 𝐶 ∈ V & ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) ⇒ ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))) | ||
Theorem | caovord3 7645* | Ordering law. (Contributed by NM, 29-Feb-1996.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) & ⊢ 𝐶 ∈ V & ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) & ⊢ 𝐷 ∈ V ⇒ ⊢ (((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵)) | ||
Theorem | caovdig 7646* | Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧))) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶))) | ||
Theorem | caovdid 7647* | Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧))) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶))) | ||
Theorem | caovdir2d 7648* | Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥)) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))) | ||
Theorem | caovdirg 7649* | Convert an operation reverse distributive law to class notation. (Contributed by Mario Carneiro, 19-Oct-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧))) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐾)) → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶))) | ||
Theorem | caovdird 7650* | Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧))) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝐾) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶))) | ||
Theorem | caovdi 7651* | 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)) | ||
Theorem | caov32d 7652* | Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵)) | ||
Theorem | caov12d 7653* | Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) ⇒ ⊢ (𝜑 → (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶))) | ||
Theorem | caov31d 7654* | Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴)) | ||
Theorem | caov13d 7655* | Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) ⇒ ⊢ (𝜑 → (𝐴𝐹(𝐵𝐹𝐶)) = (𝐶𝐹(𝐵𝐹𝐴))) | ||
Theorem | caov4d 7656* | Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) & ⊢ (𝜑 → 𝐷 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷))) | ||
Theorem | caov411d 7657* | Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) & ⊢ (𝜑 → 𝐷 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷))) | ||
Theorem | caov42d 7658* | Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) & ⊢ (𝜑 → 𝐷 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐷𝐹𝐵))) | ||
Theorem | caov32 7659* | 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) | ||
Theorem | caov12 7660* | 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)) | ||
Theorem | caov31 7661* | 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) | ||
Theorem | caov13 7662* | 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)) | ||
Theorem | caov4 7663* | 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)) | ||
Theorem | caov411 7664* | 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)) | ||
Theorem | caov42 7665* | 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)) | ||
Theorem | caovdir 7666* | 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)) | ||
Theorem | caovdilem 7667* | 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))) | ||
Theorem | caovlem2 7668* | 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)))) | ||
Theorem | caovmo 7669* | 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.) |
⊢ 𝐵 ∈ 𝑆 & ⊢ dom 𝐹 = (𝑆 × 𝑆) & ⊢ ¬ ∅ ∈ 𝑆 & ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) & ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) & ⊢ (𝑥 ∈ 𝑆 → (𝑥𝐹𝐵) = 𝑥) ⇒ ⊢ ∃*𝑤(𝐴𝐹𝑤) = 𝐵 | ||
Theorem | imaeqexov 7670* | Substitute an operation value into an existential quantifier over an image. (Contributed by Scott Fenton, 20-Jan-2025.) |
⊢ (𝑥 = (𝑦𝐹𝑧) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓)) | ||
Theorem | imaeqalov 7671* | Substitute an operation value into a universal quantifier over an image. (Contributed by Scott Fenton, 20-Jan-2025.) |
⊢ (𝑥 = (𝑦𝐹𝑧) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓)) | ||
Theorem | mpondm0 7672* | 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 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ⇒ ⊢ (¬ (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉𝐹𝑊) = ∅) | ||
Theorem | elmpocl 7673* | If a two-parameter class is not empty, constrain the implicit pair. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵)) | ||
Theorem | elmpocl1 7674* | 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 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 𝑆 ∈ 𝐴) | ||
Theorem | elmpocl2 7675* | 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 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 𝑇 ∈ 𝐵) | ||
Theorem | elovmpod 7676* | Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.) Variant of elovmpo 7677 in deduction form. (Revised by AV, 20-Apr-2025.) |
⊢ 𝑂 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐸 ∈ (𝑋𝑂𝑌) ↔ 𝐸 ∈ 𝐷)) | ||
Theorem | elovmpo 7677* |
Utility lemma for two-parameter classes.
EDITORIAL: can simplify islmhm 21043. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
⊢ 𝐷 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) & ⊢ 𝐶 ∈ V & ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → 𝐶 = 𝐸) ⇒ ⊢ (𝐹 ∈ (𝑋𝐷𝑌) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹 ∈ 𝐸)) | ||
Theorem | elovmporab 7678* | 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 ↦ {𝑧 ∈ 𝑀 ∣ 𝜑}) & ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑀 ∈ V) ⇒ ⊢ (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ 𝑀)) | ||
Theorem | elovmporab1w 7679* | 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. Version of elovmporab1 7680 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by Alexander van der Vekens, 15-Jul-2018.) Avoid ax-13 2374. (Revised by GG, 26-Jan-2024.) |
⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) & ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ⦋𝑋 / 𝑚⦌𝑀 ∈ V) ⇒ ⊢ (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀)) | ||
Theorem | elovmporab1 7680* | 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. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker elovmporab1w 7679 when possible. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (New usage is discouraged.) |
⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) & ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ⦋𝑋 / 𝑚⦌𝑀 ∈ V) ⇒ ⊢ (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀)) | ||
Theorem | 2mpo0 7681* | 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 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) & ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑂𝑌) = (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐹)) ⇒ ⊢ (¬ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅) | ||
Theorem | relmptopab 7682* | 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 = (𝑥 ∈ 𝐴 ↦ {〈𝑦, 𝑧〉 ∣ 𝜑}) ⇒ ⊢ Rel (𝐹‘𝐵) | ||
Theorem | f1ocnvd 7683* | Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑋) & ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) ⇒ ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))) | ||
Theorem | f1od 7684* | Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑋) & ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) ⇒ ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | ||
Theorem | f1ocnv2d 7685* | Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)) ⇒ ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))) | ||
Theorem | f1o2d 7686* | Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)) ⇒ ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | ||
Theorem | f1opw2 7687* | A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 7688 avoids the Axiom of Replacement. (Contributed by Mario Carneiro, 26-Jun-2015.) |
⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) & ⊢ (𝜑 → (◡𝐹 “ 𝑎) ∈ V) & ⊢ (𝜑 → (𝐹 “ 𝑏) ∈ V) ⇒ ⊢ (𝜑 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)):𝒫 𝐴–1-1-onto→𝒫 𝐵) | ||
Theorem | f1opw 7688* | 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:𝐴–1-1-onto→𝐵 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)):𝒫 𝐴–1-1-onto→𝒫 𝐵) | ||
Theorem | elovmpt3imp 7689* | 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 ↦ (𝑧 ∈ 𝑀 ↦ 𝐵)) ⇒ ⊢ (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → (𝑋 ∈ V ∧ 𝑌 ∈ V)) | ||
Theorem | ovmpt3rab1 7690* | 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 ↦ (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑})) & ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑀 = 𝐾) & ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑁 = 𝐿) & ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓)) & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜓 ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → (𝑋𝑂𝑌) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓})) | ||
Theorem | ovmpt3rabdm 7691* | 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 ↦ (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑})) & ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑀 = 𝐾) & ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑁 = 𝐿) ⇒ ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ 𝐿 ∈ 𝑇) → dom (𝑋𝑂𝑌) = 𝐾) | ||
Theorem | elovmpt3rab1 7692* | 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 ↦ (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑})) & ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑀 = 𝐾) & ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑁 = 𝐿) ⇒ ⊢ ((𝐾 ∈ 𝑈 ∧ 𝐿 ∈ 𝑇) → (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑍 ∈ 𝐾 ∧ 𝐴 ∈ 𝐿)))) | ||
Theorem | elovmpt3rab 7693* | 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 AV, 17-Jul-2018.) (Revised by AV, 16-May-2019.) |
⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑})) ⇒ ⊢ ((𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑇) → (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑍 ∈ 𝑀 ∧ 𝐴 ∈ 𝑁)))) | ||
Syntax | cof 7694 | Extend class notation to include mapping of an operation to a function operation. |
class ∘f 𝑅 | ||
Syntax | cofr 7695 | Extend class notation to include mapping of a binary relation to a function relation. |
class ∘r 𝑅 | ||
Definition | df-of 7696* | 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) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) | ||
Definition | df-ofr 7697* | 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)(𝑓‘𝑥)𝑅(𝑔‘𝑥)} | ||
Theorem | ofeqd 7698 | Equality theorem for function operation, deduction form. (Contributed by SN, 11-Nov-2024.) |
⊢ (𝜑 → 𝑅 = 𝑆) ⇒ ⊢ (𝜑 → ∘f 𝑅 = ∘f 𝑆) | ||
Theorem | ofeq 7699 | Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.) |
⊢ (𝑅 = 𝑆 → ∘f 𝑅 = ∘f 𝑆) | ||
Theorem | ofreq 7700 | Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
⊢ (𝑅 = 𝑆 → ∘r 𝑅 = ∘r 𝑆) |
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