P. 63 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6201-6300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	caov42d 6201*	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 6202*	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 6203*	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 6204*	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 6205*	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	caovdilemd 6206*	Lemma used by real number construction. (Contributed by Jim Kingdon, 16-Sep-2019.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐹(𝑦𝐺𝑧)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) ∈ 𝑆)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))))
Theorem	caovlem2d 6207*	Rearrangement of expression involving multiplication (𝐺) and addition (𝐹). (Contributed by Jim Kingdon, 3-Jan-2020.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐹(𝑦𝐺𝑧)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) ∈ 𝑆)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    ⇒   ⊢ (𝜑 → ((((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻)𝐹(((𝐴𝐺𝐷)𝐹(𝐵𝐺𝐶))𝐺𝑅)) = ((𝐴𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅)))𝐹(𝐵𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻)))))
Theorem	caovimo 6208*	Uniqueness of inverse element in commutative, associative operation with identity. The identity element is 𝐵. (Contributed by Jim Kingdon, 18-Sep-2019.)
⊢ 𝐵 ∈ 𝑆    &   ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))    &   ⊢ (𝑥 ∈ 𝑆 → (𝑥𝐹𝐵) = 𝑥)    ⇒   ⊢ (𝐴 ∈ 𝑆 → ∃*𝑤(𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵))
2.6.12  Maps-to notation
Theorem	elmpocl 6209*	If a two-parameter class is inhabited, 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 6210*	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 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 𝑆 ∈ 𝐴)
Theorem	elmpocl2 6211*	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 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 𝑇 ∈ 𝐵)
Theorem	elovmpod 6212*	Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.) Variant of elovmpo 6213 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 6213*	Utility lemma for two-parameter classes. (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 6214*	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 6215*	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.) (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	relmptopab 6216*	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 6217*	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 6218*	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 6219*	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 6220*	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 6221*	A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 6222 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 6222*	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	suppssfv 6223*	Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.)
⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ 𝐴) “ (V ∖ {𝑌})) ⊆ 𝐿)    &   ⊢ (𝜑 → (𝐹‘𝑌) = 𝑍)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ (𝐹‘𝐴)) “ (V ∖ {𝑍})) ⊆ 𝐿)
Theorem	suppssov1 6224*	Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ 𝐴) “ (V ∖ {𝑌})) ⊆ 𝐿)    &   ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑅)    ⇒   ⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) ⊆ 𝐿)
2.6.13  Function operation
Syntax	cof 6225	Extend class notation to include mapping of an operation to a function operation.
class ∘𝑓 𝑅
Syntax	cofr 6226	Extend class notation to include mapping of a binary relation to a function relation.
class ∘𝑟 𝑅
Definition	df-of 6227*	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 6228*	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)(𝑓‘𝑥)𝑅(𝑔‘𝑥)}
Theorem	ofeqd 6229	Equality theorem for function operation, deduction form. (Contributed by SN, 11-Nov-2024.)
⊢ (𝜑 → 𝑅 = 𝑆)    ⇒   ⊢ (𝜑 → ∘𝑓 𝑅 = ∘𝑓 𝑆)
Theorem	ofeq 6230	Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
⊢ (𝑅 = 𝑆 → ∘𝑓 𝑅 = ∘𝑓 𝑆)
Theorem	ofreq 6231	Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (𝑅 = 𝑆 → ∘𝑟 𝑅 = ∘𝑟 𝑆)
Theorem	ofexg 6232	A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)
⊢ (𝐴 ∈ 𝑉 → ( ∘𝑓 𝑅 ↾ 𝐴) ∈ V)
Theorem	nfof 6233	Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
⊢ Ⅎ𝑥𝑅    ⇒   ⊢ Ⅎ𝑥 ∘𝑓 𝑅
Theorem	nfofr 6234	Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ Ⅎ𝑥𝑅    ⇒   ⊢ Ⅎ𝑥 ∘𝑟 𝑅
Theorem	offval 6235*	Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝐴 ∩ 𝐵) = 𝑆    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷)    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ (𝐶𝑅𝐷)))
Theorem	ofrfval 6236*	Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝐴 ∩ 𝐵) = 𝑆    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷)    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷))
Theorem	ofvalg 6237	Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Jim Kingdon, 22-Nov-2023.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝐴 ∩ 𝐵) = 𝑆    &   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷)    &   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐶𝑅𝐷) ∈ 𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))
Theorem	ofrval 6238	Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝐴 ∩ 𝐵) = 𝑆    &   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷)    ⇒   ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝐶𝑅𝐷)
Theorem	ofmresval 6239	Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
⊢ (𝜑 → 𝐹 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹 ∘𝑓 𝑅𝐺))
Theorem	off 6240*	The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝑇)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝐴 ∩ 𝐵) = 𝐶    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈)
Theorem	offeq 6241*	Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Jim Kingdon, 26-Nov-2023.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝑇)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝐴 ∩ 𝐵) = 𝐶    &   ⊢ (𝜑 → 𝐻:𝐶⟶𝑈)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐸)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐷𝑅𝐸) = (𝐻‘𝑥))    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = 𝐻)
Theorem	ofres 6242	Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝐴 ∩ 𝐵) = 𝐶    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = ((𝐹 ↾ 𝐶) ∘𝑓 𝑅(𝐺 ↾ 𝐶)))
Theorem	offval2 6243*	The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶))    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
Theorem	ofrfval2 6244*	The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶))    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀𝑥 ∈ 𝐴 𝐵𝑅𝐶))
Theorem	suppssof1 6245*	Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
⊢ (𝜑 → (◡𝐴 “ (V ∖ {𝑌})) ⊆ 𝐿)    &   ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)    &   ⊢ (𝜑 → 𝐴:𝐷⟶𝑉)    &   ⊢ (𝜑 → 𝐵:𝐷⟶𝑅)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (◡(𝐴 ∘𝑓 𝑂𝐵) “ (V ∖ {𝑍})) ⊆ 𝐿)
Theorem	ofco 6246	The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → 𝐻:𝐷⟶𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ (𝐴 ∩ 𝐵) = 𝐶    ⇒   ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘ 𝐻) ∘𝑓 𝑅(𝐺 ∘ 𝐻)))
Theorem	offveqb 6247*	Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐴)    &   ⊢ (𝜑 → 𝐻 Fn 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = 𝐶)    ⇒   ⊢ (𝜑 → (𝐻 = (𝐹 ∘𝑓 𝑅𝐺) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶)))
Theorem	offveq 6248*	Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐴)    &   ⊢ (𝜑 → 𝐻 Fn 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝐶) = (𝐻‘𝑥))    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = 𝐻)
Theorem	ofc1g 6249	Left operation by a constant. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐵𝑅𝐶) ∈ 𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹)‘𝑋) = (𝐵𝑅𝐶))
Theorem	ofc2g 6250	Right operation by a constant. (Contributed by NM, 7-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐶𝑅𝐵) ∈ 𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘𝑓 𝑅(𝐴 × {𝐵}))‘𝑋) = (𝐶𝑅𝐵))
Theorem	ofc12 6251	Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)}))
Theorem	caofref 6252*	Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥𝑅𝑥)    ⇒   ⊢ (𝜑 → 𝐹 ∘𝑟 𝑅𝐹)
Theorem	caofinvl 6253*	Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑁:𝑆⟶𝑆)    &   ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑁‘𝑥)𝑅𝑥) = 𝐵)    ⇒   ⊢ (𝜑 → (𝐺 ∘𝑓 𝑅𝐹) = (𝐴 × {𝐵}))
Theorem	caofid0l 6254*	Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐵𝑅𝑥) = 𝑥)    ⇒   ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹) = 𝐹)
Theorem	caofid0r 6255*	Transfer a right identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝑅𝐵) = 𝑥)    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅(𝐴 × {𝐵})) = 𝐹)
Theorem	caofid1 6256*	Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝑅𝐵) = 𝐶)    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅(𝐴 × {𝐵})) = (𝐴 × {𝐶}))
Theorem	caofid2 6257*	Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐵𝑅𝑥) = 𝐶)    ⇒   ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹) = (𝐴 × {𝐶}))
Theorem	caofcom 6258*	Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥))    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝐺 ∘𝑓 𝑅𝐹))
Theorem	caofrss 6259*	Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦 → 𝑥𝑇𝑦))    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐺 → 𝐹 ∘𝑟 𝑇𝐺))
Theorem	caoftrn 6260*	Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) → 𝑥𝑈𝑧))    ⇒   ⊢ (𝜑 → ((𝐹 ∘𝑟 𝑅𝐺 ∧ 𝐺 ∘𝑟 𝑇𝐻) → 𝐹 ∘𝑟 𝑈𝐻))
Theorem	caofdig 6261*	Transfer a distributive law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐾)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦) ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑇𝑦) ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑇(𝑦𝑅𝑧)) = ((𝑥𝑇𝑦)𝑂(𝑥𝑇𝑧)))    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑓 𝑇(𝐺 ∘𝑓 𝑅𝐻)) = ((𝐹 ∘𝑓 𝑇𝐺) ∘𝑓 𝑂(𝐹 ∘𝑓 𝑇𝐻)))
2.6.14  Functions (continued)
Theorem	resfunexgALT 6262	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 5867 but requires ax-pow 4259 and ax-un 4525. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V)
Theorem	cofunexg 6263	Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∘ 𝐵) ∈ V)
Theorem	cofunex2g 6264	Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)
⊢ ((𝐴 ∈ 𝑉 ∧ Fun ◡𝐵) → (𝐴 ∘ 𝐵) ∈ V)
Theorem	fnexALT 6265	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 5408. This version of fnex 5868 uses ax-pow 4259 and ax-un 4525, whereas fnex 5868 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V)
Theorem	funexw 6266	Weak version of funex 5869 that holds without ax-coll 4199. If the domain and codomain of a function exist, so does the function. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → 𝐹 ∈ V)
Theorem	mptexw 6267*	Weak version of mptex 5872 that holds without ax-coll 4199. If the domain and codomain of a function given by maps-to notation are sets, the function is a set. (Contributed by Rohan Ridenour, 13-Aug-2023.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V
Theorem	funrnex 6268	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 5869. (Contributed by NM, 11-Nov-1995.)
⊢ (dom 𝐹 ∈ 𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V))
Theorem	focdmex 6269	If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
⊢ (𝐴 ∈ 𝐶 → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V))
Theorem	f1dmex 6270	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:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
Theorem	abrexex 6271*	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 5871, funex 5869, fnex 5868, resfunexg 5867, and funimaexg 5408. See also abrexex2 6278. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V
Theorem	abrexexg 6272*	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.)
⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
Theorem	iunexg 6273*	The existence of an indexed union. 𝑥 is normally a free-variable parameter in 𝐵. (Contributed by NM, 23-Mar-2006.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)
Theorem	abrexex2g 6274*	Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ 𝑊) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V)
Theorem	opabex3d 6275*	Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)
⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝜓} ∈ V)    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} ∈ V)
Theorem	opabex3 6276*	Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∣ 𝜑} ∈ V)    ⇒   ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V
Theorem	iunex 6277*	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    ⇒   ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V
Theorem	abrexex2 6278*	Existence of an existentially restricted class abstraction. 𝜑 is normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 6271. (Contributed by NM, 12-Sep-2004.)
⊢ 𝐴 ∈ V    &   ⊢ {𝑦 ∣ 𝜑} ∈ V    ⇒   ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V
Theorem	abexssex 6279*	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    &   ⊢ {𝑦 ∣ 𝜑} ∈ V    ⇒   ⊢ {𝑦 ∣ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V
Theorem	abexex 6280*	A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
⊢ 𝐴 ∈ V    &   ⊢ (𝜑 → 𝑥 ∈ 𝐴)    &   ⊢ {𝑦 ∣ 𝜑} ∈ V    ⇒   ⊢ {𝑦 ∣ ∃𝑥𝜑} ∈ V
Theorem	oprabexd 6281*	Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑧𝜓)    &   ⊢ (𝜑 → 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)})    ⇒   ⊢ (𝜑 → 𝐹 ∈ V)
Theorem	oprabex 6282*	Existence of an operation class abstraction. (Contributed by NM, 19-Oct-2004.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃*𝑧𝜑)    &   ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)}    ⇒   ⊢ 𝐹 ∈ V
Theorem	oprabex3 6283*	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)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}    ⇒   ⊢ 𝐹 ∈ V
Theorem	oprabrexex2 6284*	Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
⊢ 𝐴 ∈ V    &   ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ∈ V    ⇒   ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤 ∈ 𝐴 𝜑} ∈ V
Theorem	ab2rexex 6285*	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 6271. (Contributed by NM, 20-Sep-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V
Theorem	ab2rexex2 6286*	Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 6278. (Contributed by NM, 20-Sep-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ {𝑧 ∣ 𝜑} ∈ V    ⇒   ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑} ∈ V
Theorem	xpexgALT 6287	The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4835 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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V)
Theorem	offval3 6288*	General value of (𝐹 ∘𝑓 𝑅𝐺) with no assumptions on functionality of 𝐹 and 𝐺. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
Theorem	offres 6289	Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘𝑓 𝑅𝐺) ↾ 𝐷) = ((𝐹 ↾ 𝐷) ∘𝑓 𝑅(𝐺 ↾ 𝐷)))
Theorem	ofmres 6290*	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 6291, allowing it to be used as a function or structure argument. By ofmresval 6239, 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)) = (𝑓 ∈ 𝐴, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘𝑓 𝑅𝑔))
Theorem	ofmresex 6291	Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) ∈ V)
Theorem	uchoice 6292*	Principle of unique choice. This is also called non-choice. The name choice results in its similarity to something like acfun 7405 (with the key difference being the change of ∃ to ∃!) but unique choice in fact follows from the axiom of collection and our other axioms. This is somewhat similar to Corollary 3.9.2 of [HoTT], p. (varies) but is better described by the paragraph at the end of Section 3.9 which starts "A similar issue arises in set-theoretic mathematics". (Contributed by Jim Kingdon, 13-Sep-2025.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑))
2.6.15  First and second members of an ordered pair
Syntax	c1st 6293	Extend the definition of a class to include the first member an ordered pair function.
class 1st
Syntax	c2nd 6294	Extend the definition of a class to include the second member an ordered pair function.
class 2nd
Definition	df-1st 6295	Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 6301 proves that it does this. For example, (1st ‘⟨ 3 , 4 ⟩) = 3 . Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 5213 and op1stb 4570). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
Definition	df-2nd 6296	Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 6302 proves that it does this. For example, (2nd ‘⟨ 3 , 4 ⟩) = 4 . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 5216 and op2ndb 5215). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
Theorem	1stvalg 6297	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 → (1st ‘𝐴) = ∪ dom {𝐴})
Theorem	2ndvalg 6298	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 → (2nd ‘𝐴) = ∪ ran {𝐴})
Theorem	1st0 6299	The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
⊢ (1st ‘∅) = ∅
Theorem	2nd0 6300	The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
⊢ (2nd ‘∅) = ∅