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Theorem List for Metamath Proof Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcaov411d 6901* 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 6902* 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 6903* 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 6904* 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 6905* 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 6906* 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 6907* 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 6908* 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 6909* 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 6910* 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 6911* 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 6912* 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 6913* 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 6914* Lemma for grprinvd 6915. (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 6915* 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 6916* 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)
 
2.3.19  "Maps to" notation
 
Theoremmpt2ndm0 6917* 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 6918* 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 6919* 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 6920* 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 6921* Utility lemma for two-parameter classes.

EDITORIAL: can simplify isghm 17707, islmhm 19075. (Contributed by Stefan O'Rear, 21-Jan-2015.)

𝐷 = (𝑎𝐴, 𝑏𝐵𝐶)    &   𝐶 ∈ V    &   ((𝑎 = 𝑋𝑏 = 𝑌) → 𝐶 = 𝐸)       (𝐹 ∈ (𝑋𝐷𝑌) ↔ (𝑋𝐴𝑌𝐵𝐹𝐸))
 
Theoremelovmpt2rab 6922* 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 6923* 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 6924* 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 6925* 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 6926* 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 6927* 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 6928* 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 6929* 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 6930* A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 6931 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 6931* 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 6932* 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 6933* 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 6934* 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 6935* 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 6936* 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))))
 
2.3.20  Function operation
 
Syntaxcof 6937 Extend class notation to include mapping of an operation to a function operation.
class 𝑓 𝑅
 
Syntaxcofr 6938 Extend class notation to include mapping of a binary relation to a function relation.
class 𝑟 𝑅
 
Definitiondf-of 6939* 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 6940* 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 6941 Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
(𝑅 = 𝑆 → ∘𝑓 𝑅 = ∘𝑓 𝑆)
 
Theoremofreq 6942 Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝑅 = 𝑆 → ∘𝑟 𝑅 = ∘𝑟 𝑆)
 
Theoremofexg 6943 A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)
(𝐴𝑉 → ( ∘𝑓 𝑅𝐴) ∈ V)
 
Theoremnfof 6944* Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
𝑥𝑅       𝑥𝑓 𝑅
 
Theoremnfofr 6945* Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝑥𝑅       𝑥𝑟 𝑅
 
Theoremoffval 6946* 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 6947* 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 6948 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 6949 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 6950 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 6951* 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 6952 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 6953 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 6954* 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 6955 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 6956* 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 6957* 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 6958* 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 6959 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 6960* 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 6961* 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 6962 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 6963 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 6964 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 6965* Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   ((𝜑𝑥𝑆) → 𝑥𝑅𝑥)       (𝜑𝐹𝑟 𝑅𝐹)
 
Theoremcaofinvl 6966* 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 6967* 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 6968* 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)
 
Theoremcaofid1 6969* Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)    &   ((𝜑𝑥𝑆) → (𝑥𝑅𝐵) = 𝐶)       (𝜑 → (𝐹𝑓 𝑅(𝐴 × {𝐵})) = (𝐴 × {𝐶}))
 
Theoremcaofid2 6970* Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)    &   ((𝜑𝑥𝑆) → (𝐵𝑅𝑥) = 𝐶)       (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹) = (𝐴 × {𝐶}))
 
Theoremcaofcom 6971* 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 6972* 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))
 
Theoremcaofass 6973* Transfer an associative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑆)    &   (𝜑𝐻:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))       (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘𝑓 𝑇𝐻) = (𝐹𝑓 𝑂(𝐺𝑓 𝑃𝐻)))
 
Theoremcaoftrn 6974* 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))
 
Theoremcaofdi 6975* Transfer a distributive law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐾)    &   (𝜑𝐺:𝐴𝑆)    &   (𝜑𝐻:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝐾𝑦𝑆𝑧𝑆)) → (𝑥𝑇(𝑦𝑅𝑧)) = ((𝑥𝑇𝑦)𝑂(𝑥𝑇𝑧)))       (𝜑 → (𝐹𝑓 𝑇(𝐺𝑓 𝑅𝐻)) = ((𝐹𝑓 𝑇𝐺) ∘𝑓 𝑂(𝐹𝑓 𝑇𝐻)))
 
Theoremcaofdir 6976* Transfer a reverse distributive law to the function operation. (Contributed by NM, 19-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐾)    &   (𝜑𝐺:𝐴𝑆)    &   (𝜑𝐻:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧)))       (𝜑 → ((𝐺𝑓 𝑅𝐻) ∘𝑓 𝑇𝐹) = ((𝐺𝑓 𝑇𝐹) ∘𝑓 𝑂(𝐻𝑓 𝑇𝐹)))
 
Theoremcaonncan 6977* Transfer nncan 10348-shaped laws to vectors of numbers. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(𝜑𝐼𝑉)    &   (𝜑𝐴:𝐼𝑆)    &   (𝜑𝐵:𝐼𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)       (𝜑 → (𝐴𝑓 𝑀(𝐴𝑓 𝑀𝐵)) = 𝐵)
 
2.3.21  Proper subset relation
 
Syntaxcrpss 6978 Extend class notation to include the reified proper subset relation.
class []
 
Definitiondf-rpss 6979* Define a relation which corresponds to proper subsethood df-pss 3623 on sets. This allows us to use proper subsethood with general concepts that require relations, such as strict ordering, see sorpss 6984. (Contributed by Stefan O'Rear, 2-Nov-2014.)
[] = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
 
Theoremrelrpss 6980 The proper subset relation is a relation. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Rel []
 
Theorembrrpssg 6981 The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.)
(𝐵𝑉 → (𝐴 [] 𝐵𝐴𝐵))
 
Theorembrrpss 6982 The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.)
𝐵 ∈ V       (𝐴 [] 𝐵𝐴𝐵)
 
Theoremporpss 6983 Every class is partially ordered by proper subsets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
[] Po 𝐴
 
Theoremsorpss 6984* Express strict ordering under proper subsets, i.e. the notion of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
( [] Or 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥))
 
Theoremsorpssi 6985 Property of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
(( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐶𝐵))
 
Theoremsorpssun 6986 A chain of sets is closed under binary union. (Contributed by Mario Carneiro, 16-May-2015.)
(( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶) ∈ 𝐴)
 
Theoremsorpssin 6987 A chain of sets is closed under binary intersection. (Contributed by Mario Carneiro, 16-May-2015.)
(( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶) ∈ 𝐴)
 
Theoremsorpssuni 6988* In a chain of sets, a maximal element is the union of the chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)
( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣 𝑌𝑌))
 
Theoremsorpssint 6989* In a chain of sets, a minimal element is the intersection of the chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)
( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑣𝑢 𝑌𝑌))
 
Theoremsorpsscmpl 6990* The componentwise complement of a chain of sets is also a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
( [] Or 𝑌 → [] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌})
 
2.4  ZF Set Theory - add the Axiom of Union
 
2.4.1  Introduce the Axiom of Union
 
Axiomax-un 6991* Axiom of Union. An axiom of Zermelo-Fraenkel set theory. It states that a set 𝑦 exists that includes the union of a given set 𝑥 i.e. the collection of all members of the members of 𝑥. The variant axun2 6993 states that the union itself exists. A version with the standard abbreviation for union is uniex2 6994. A version using class notation is uniex 6995.

The union of a class df-uni 4469 should not be confused with the union of two classes df-un 3612. Their relationship is shown in unipr 4481. (Contributed by NM, 23-Dec-1993.)

𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
 
Theoremzfun 6992* Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)
 
Theoremaxun2 6993* A variant of the Axiom of Union ax-un 6991. For any set 𝑥, there exists a set 𝑦 whose members are exactly the members of the members of 𝑥 i.e. the union of 𝑥. Axiom Union of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
 
Theoremuniex2 6994* The Axiom of Union using the standard abbreviation for union. Given any set 𝑥, its union 𝑦 exists. (Contributed by NM, 4-Jun-2006.)
𝑦 𝑦 = 𝑥
 
Theoremuniex 6995 The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 3238), then the union of 𝐴 is also a set. Same as Axiom 3 of [TakeutiZaring] p. 16. (Contributed by NM, 11-Aug-1993.)
𝐴 ∈ V        𝐴 ∈ V
 
Theoremvuniex 6996 The union of a setvar is a set. (Contributed by BJ, 3-May-2021.)
𝑥 ∈ V
 
Theoremuniexg 6997 The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent 𝐴𝑉 instead of 𝐴 ∈ V to make the theorem more general and thus shorten some proofs; obviously the universal class constant V is one possible substitution for class variable 𝑉. (Contributed by NM, 25-Nov-1994.)
(𝐴𝑉 𝐴 ∈ V)
 
Theoremunex 6998 The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 1-Jul-1994.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵) ∈ V
 
Theoremtpex 6999 An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.)
{𝐴, 𝐵, 𝐶} ∈ V
 
Theoremunexb 7000 Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
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