HomeHome Intuitionistic Logic Explorer
Theorem List (p. 60 of 130)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 5901-6000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcaovdird 5901* 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 5902* 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 5903* 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 5904* 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 5905* 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 5906* 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 5907* 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 5908* 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 5909* 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 5910* 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 5911* 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 5912* 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 5913* 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))
 
Theoremcaovdilemd 5914* Lemma used by real number construction. (Contributed by Jim Kingdon, 16-Sep-2019.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐹(𝑦𝐺𝑧)))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐺𝑦) ∈ 𝑆)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑𝐷𝑆)    &   (𝜑𝐻𝑆)       (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))))
 
Theoremcaovlem2d 5915* Rearrangement of expression involving multiplication (𝐺) and addition (𝐹). (Contributed by Jim Kingdon, 3-Jan-2020.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐹(𝑦𝐺𝑧)))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐺𝑦) ∈ 𝑆)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑𝐷𝑆)    &   (𝜑𝐻𝑆)    &   (𝜑𝑅𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)       (𝜑 → ((((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻)𝐹(((𝐴𝐺𝐷)𝐹(𝐵𝐺𝐶))𝐺𝑅)) = ((𝐴𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅)))𝐹(𝐵𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻)))))
 
Theoremcaovimo 5916* Uniqueness of inverse element in commutative, associative operation with identity. The identity element is 𝐵. (Contributed by Jim Kingdon, 18-Sep-2019.)
𝐵𝑆    &   ((𝑥𝑆𝑦𝑆) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))    &   (𝑥𝑆 → (𝑥𝐹𝐵) = 𝑥)       (𝐴𝑆 → ∃*𝑤(𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵))
 
Theoremgrprinvlem 5917* Lemma for grprinvd 5918. (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 5918* 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 5919* 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)
 
2.6.11  Maps-to notation
 
Theoremelmpocl 5920* If a two-parameter class is inhabited, constrain the implicit pair. (Contributed by Stefan O'Rear, 7-Mar-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆𝐴𝑇𝐵))
 
Theoremelmpocl1 5921* If a two-parameter class is inhabited, 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)
 
Theoremelmpocl2 5922* If a two-parameter class is inhabited, 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)
 
Theoremelovmpo 5923* Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.)
𝐷 = (𝑎𝐴, 𝑏𝐵𝐶)    &   𝐶 ∈ V    &   ((𝑎 = 𝑋𝑏 = 𝑌) → 𝐶 = 𝐸)       (𝐹 ∈ (𝑋𝐷𝑌) ↔ (𝑋𝐴𝑌𝐵𝐹𝐸))
 
Theoremf1ocnvd 5924* 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 5925* 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 5926* 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 5927* 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 5928* A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 5929 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 5929* 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)
 
Theoremsuppssfv 5930* Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.)
(𝜑 → ((𝑥𝐷𝐴) “ (V ∖ {𝑌})) ⊆ 𝐿)    &   (𝜑 → (𝐹𝑌) = 𝑍)    &   ((𝜑𝑥𝐷) → 𝐴𝑉)       (𝜑 → ((𝑥𝐷 ↦ (𝐹𝐴)) “ (V ∖ {𝑍})) ⊆ 𝐿)
 
Theoremsuppssov1 5931* Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
(𝜑 → ((𝑥𝐷𝐴) “ (V ∖ {𝑌})) ⊆ 𝐿)    &   ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)    &   ((𝜑𝑥𝐷) → 𝐴𝑉)    &   ((𝜑𝑥𝐷) → 𝐵𝑅)       (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) ⊆ 𝐿)
 
2.6.12  Function operation
 
Syntaxcof 5932 Extend class notation to include mapping of an operation to a function operation.
class 𝑓 𝑅
 
Syntaxcofr 5933 Extend class notation to include mapping of a binary relation to a function relation.
class 𝑟 𝑅
 
Definitiondf-of 5934* 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 5935* Define the function relation map. The definition is designed so that if 𝑅 is a binary relation, then 𝑓 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 5936 Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
(𝑅 = 𝑆 → ∘𝑓 𝑅 = ∘𝑓 𝑆)
 
Theoremofreq 5937 Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝑅 = 𝑆 → ∘𝑟 𝑅 = ∘𝑟 𝑆)
 
Theoremofexg 5938 A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)
(𝐴𝑉 → ( ∘𝑓 𝑅𝐴) ∈ V)
 
Theoremnfof 5939 Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
𝑥𝑅       𝑥𝑓 𝑅
 
Theoremnfofr 5940 Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝑥𝑅       𝑥𝑟 𝑅
 
Theoremoffval 5941* 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 5942* 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))
 
Theoremofvalg 5943 Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Jim Kingdon, 22-Nov-2023.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝑆    &   ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)    &   ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)    &   ((𝜑𝑋𝑆) → (𝐶𝑅𝐷) ∈ 𝑈)       ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))
 
Theoremofrval 5944 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)
 
Theoremofmresval 5945 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))
 
Theoremoff 5946* 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)
 
Theoremoffeq 5947* Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Jim Kingdon, 26-Nov-2023.)
((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐵𝑇)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝐶    &   (𝜑𝐻:𝐶𝑈)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐷)    &   ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐸)    &   ((𝜑𝑥𝐶) → (𝐷𝑅𝐸) = (𝐻𝑥))       (𝜑 → (𝐹𝑓 𝑅𝐺) = 𝐻)
 
Theoremofres 5948 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 5949* 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 5950* 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))
 
Theoremsuppssof1 5951* Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
(𝜑 → (𝐴 “ (V ∖ {𝑌})) ⊆ 𝐿)    &   ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)    &   (𝜑𝐴:𝐷𝑉)    &   (𝜑𝐵:𝐷𝑅)    &   (𝜑𝐷𝑊)       (𝜑 → ((𝐴𝑓 𝑂𝐵) “ (V ∖ {𝑍})) ⊆ 𝐿)
 
Theoremofco 5952 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)))
 
Theoremoffveqb 5953* 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)))
 
Theoremofc12 5954 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 5955* Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   ((𝜑𝑥𝑆) → 𝑥𝑅𝑥)       (𝜑𝐹𝑟 𝑅𝐹)
 
Theoremcaofinvl 5956* 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}))
 
Theoremcaofcom 5957* Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥))       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝐺𝑓 𝑅𝐹))
 
Theoremcaofrss 5958* Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦𝑥𝑇𝑦))       (𝜑 → (𝐹𝑟 𝑅𝐺𝐹𝑟 𝑇𝐺))
 
Theoremcaoftrn 5959* Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑆)    &   (𝜑𝐻:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧))       (𝜑 → ((𝐹𝑟 𝑅𝐺𝐺𝑟 𝑇𝐻) → 𝐹𝑟 𝑈𝐻))
 
2.6.13  Functions (continued)
 
TheoremresfunexgALT 5960 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. This version has a shorter proof than resfunexg 5593 but requires ax-pow 4056 and ax-un 4313. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
 
Theoremcofunexg 5961 Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
 
Theoremcofunex2g 5962 Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)
((𝐴𝑉 ∧ Fun 𝐵) → (𝐴𝐵) ∈ V)
 
TheoremfnexALT 5963 If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of funimaexg 5163. This version of fnex 5594 uses ax-pow 4056 and ax-un 4313, whereas fnex 5594 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)
 
Theoremfunrnex 5964 If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 5595. (Contributed by NM, 11-Nov-1995.)
(dom 𝐹𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V))
 
Theoremfornex 5965 If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
(𝐴𝐶 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
 
Theoremf1dmex 5966 If the codomain of a one-to-one function exists, so does its domain. This can be thought of as a form of the Axiom of Replacement. (Contributed by NM, 4-Sep-2004.)
((𝐹:𝐴1-1𝐵𝐵𝐶) → 𝐴 ∈ V)
 
Theoremabrexex 5967* Existence of a class abstraction of existentially restricted sets. 𝑥 is normally a free-variable parameter in the class expression substituted for 𝐵, which can be thought of as 𝐵(𝑥). This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path mptexg 5597, funex 5595, fnex 5594, resfunexg 5593, and funimaexg 5163. See also abrexex2 5974. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
𝐴 ∈ V       {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V
 
Theoremabrexexg 5968* Existence of a class abstraction of existentially restricted sets. 𝑥 is normally a free-variable parameter in 𝐵. The antecedent assures us that 𝐴 is a set. (Contributed by NM, 3-Nov-2003.)
(𝐴𝑉 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
 
Theoremiunexg 5969* The existence of an indexed union. 𝑥 is normally a free-variable parameter in 𝐵. (Contributed by NM, 23-Mar-2006.)
((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵𝑊) → 𝑥𝐴 𝐵 ∈ V)
 
Theoremabrexex2g 5970* Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐴𝑉 ∧ ∀𝑥𝐴 {𝑦𝜑} ∈ 𝑊) → {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V)
 
Theoremopabex3d 5971* Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)
(𝜑𝐴 ∈ V)    &   ((𝜑𝑥𝐴) → {𝑦𝜓} ∈ V)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜓)} ∈ V)
 
Theoremopabex3 5972* Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐴 ∈ V    &   (𝑥𝐴 → {𝑦𝜑} ∈ V)       {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
 
Theoremiunex 5973* The existence of an indexed union. 𝑥 is normally a free-variable parameter in the class expression substituted for 𝐵, which can be read informally as 𝐵(𝑥). (Contributed by NM, 13-Oct-2003.)
𝐴 ∈ V    &   𝐵 ∈ V        𝑥𝐴 𝐵 ∈ V
 
Theoremabrexex2 5974* Existence of an existentially restricted class abstraction. 𝜑 is normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 5967. (Contributed by NM, 12-Sep-2004.)
𝐴 ∈ V    &   {𝑦𝜑} ∈ V       {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
 
Theoremabexssex 5975* Existence of a class abstraction with an existentially quantified expression. Both 𝑥 and 𝑦 can be free in 𝜑. (Contributed by NM, 29-Jul-2006.)
𝐴 ∈ V    &   {𝑦𝜑} ∈ V       {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)} ∈ V
 
Theoremabexex 5976* A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
𝐴 ∈ V    &   (𝜑𝑥𝐴)    &   {𝑦𝜑} ∈ V       {𝑦 ∣ ∃𝑥𝜑} ∈ V
 
Theoremoprabexd 5977* Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → ∃*𝑧𝜓)    &   (𝜑𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)})       (𝜑𝐹 ∈ V)
 
Theoremoprabex 5978* Existence of an operation class abstraction. (Contributed by NM, 19-Oct-2004.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥𝐴𝑦𝐵) → ∃*𝑧𝜑)    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)}       𝐹 ∈ V
 
Theoremoprabex3 5979* Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)
𝐻 ∈ V    &   𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}       𝐹 ∈ V
 
Theoremoprabrexex2 5980* Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
𝐴 ∈ V    &   {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ∈ V       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤𝐴 𝜑} ∈ V
 
Theoremab2rexex 5981* Existence of a class abstraction of existentially restricted sets. Variables 𝑥 and 𝑦 are normally free-variable parameters in the class expression substituted for 𝐶, which can be thought of as 𝐶(𝑥, 𝑦). See comments for abrexex 5967. (Contributed by NM, 20-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ∈ V
 
Theoremab2rexex2 5982* Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 5974. (Contributed by NM, 20-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   {𝑧𝜑} ∈ V       {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑} ∈ V
 
TheoremxpexgALT 5983 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4611 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
 
Theoremoffval3 5984* General value of (𝐹𝑓 𝑅𝐺) with no assumptions on functionality of 𝐹 and 𝐺. (Contributed by Stefan O'Rear, 24-Jan-2015.)
((𝐹𝑉𝐺𝑊) → (𝐹𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
 
Theoremoffres 5985 Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)
((𝐹𝑉𝐺𝑊) → ((𝐹𝑓 𝑅𝐺) ↾ 𝐷) = ((𝐹𝐷) ∘𝑓 𝑅(𝐺𝐷)))
 
Theoremofmres 5986* Equivalent expressions for a restriction of the function operation map. Unlike 𝑓 𝑅 which is a proper class, ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) can be a set by ofmresex 5987, allowing it to be used as a function or structure argument. By ofmresval 5945, the restricted operation map values are the same as the original values, allowing theorems for 𝑓 𝑅 to be reused. (Contributed by NM, 20-Oct-2014.)
( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) = (𝑓𝐴, 𝑔𝐵 ↦ (𝑓𝑓 𝑅𝑔))
 
Theoremofmresex 5987 Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) ∈ V)
 
2.6.14  First and second members of an ordered pair
 
Syntaxc1st 5988 Extend the definition of a class to include the first member an ordered pair function.
class 1st
 
Syntaxc2nd 5989 Extend the definition of a class to include the second member an ordered pair function.
class 2nd
 
Definitiondf-1st 5990 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 5996 proves that it does this. For example, (1st ‘⟨ 3 , 4 ) = 3 . Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 4976 and op1stb 4357). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
1st = (𝑥 ∈ V ↦ dom {𝑥})
 
Definitiondf-2nd 5991 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 5997 proves that it does this. For example, (2nd ‘⟨ 3 , 4 ) = 4 . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 4979 and op2ndb 4978). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
2nd = (𝑥 ∈ V ↦ ran {𝑥})
 
Theorem1stvalg 5992 The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
(𝐴 ∈ V → (1st𝐴) = dom {𝐴})
 
Theorem2ndvalg 5993 The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
(𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
 
Theorem1st0 5994 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
(1st ‘∅) = ∅
 
Theorem2nd0 5995 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
(2nd ‘∅) = ∅
 
Theoremop1st 5996 Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
 
Theoremop2nd 5997 Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵
 
Theoremop1std 5998 Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐶 = ⟨𝐴, 𝐵⟩ → (1st𝐶) = 𝐴)
 
Theoremop2ndd 5999 Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐶 = ⟨𝐴, 𝐵⟩ → (2nd𝐶) = 𝐵)
 
Theoremop1stg 6000 Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
((𝐴𝑉𝐵𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-12930
  Copyright terms: Public domain < Previous  Next >