P. 72 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 7101-7200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nelrnfvne 7101	A function value cannot be any element not contained in the range of the function. (Contributed by AV, 28-Jan-2020.)
⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹 ∧ 𝑌 ∉ ran 𝐹) → (𝐹‘𝑋) ≠ 𝑌)
Theorem	fveqdmss 7102*	If the empty set is not contained in the range of a function, and the function values of another class (not necessarily a function) are equal to the function values of the function for all elements of the domain of the function, then the domain of the function is contained in the domain of the class. (Contributed by AV, 28-Jan-2020.)
⊢ 𝐷 = dom 𝐵    ⇒   ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → 𝐷 ⊆ dom 𝐴)
Theorem	fveqressseq 7103*	If the empty set is not contained in the range of a function, and the function values of another class (not necessarily a function) are equal to the function values of the function for all elements of the domain of the function, then the class restricted to the domain of the function is the function itself. (Contributed by AV, 28-Jan-2020.)
⊢ 𝐷 = dom 𝐵    ⇒   ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐴 ↾ 𝐷) = 𝐵)
Theorem	fnfvelrn 7104	A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹‘𝐵) ∈ ran 𝐹)
Theorem	ffvelcdm 7105	A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐵)
Theorem	fnfvelrnd 7106	A function's value belongs to its range. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹‘𝐵) ∈ ran 𝐹)
Theorem	ffvelcdmi 7107	A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)
⊢ 𝐹:𝐴⟶𝐵    ⇒   ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) ∈ 𝐵)
Theorem	ffvelcdmda 7108	A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐵)
Theorem	ffvelcdmd 7109	A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝐵)
Theorem	feldmfvelcdm 7110	A class is an element of the domain iff it's function value is an element of the codomain of a function. (Contributed by AV, 22-Apr-2025.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ ∅ ∉ 𝐵) → (𝑋 ∈ 𝐴 ↔ (𝐹‘𝑋) ∈ 𝐵))
Theorem	rexrn 7111*	Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓))
Theorem	ralrn 7112*	Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐹 Fn 𝐴 → (∀𝑥 ∈ ran 𝐹𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓))
Theorem	elrnrexdm 7113*	For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥)))
Theorem	elrnrexdmb 7114*	For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥)))
Theorem	eldmrexrn 7115*	For any element in the domain of a function there is an element in the range of the function which is the function value for the element of the domain. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
⊢ (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌)))
Theorem	eldmrexrnb 7116*	For any element in the domain of a function, there is an element in the range of the function which is the value of the function at that element. Because of the definition df-fv 6566 of the value of a function, the theorem is only valid in general if the empty set is not contained in the range of the function (the implication "to the right" is always valid). Indeed, with the definition df-fv 6566 of the value of a function, (𝐹‘𝑌) = ∅ may mean that the value of 𝐹 at 𝑌 is the empty set or that 𝐹 is not defined at 𝑌. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
⊢ ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 ↔ ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌)))
Theorem	fvcofneq 7117*	The values of two function compositions are equal if the values of the composed functions are pairwise equal. (Contributed by AV, 26-Jan-2019.)
⊢ ((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) → ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥)) → ((𝐹 ∘ 𝐺)‘𝑋) = ((𝐻 ∘ 𝐾)‘𝑋)))
Theorem	ralrnmptw 7118*	A restricted quantifier over an image set. Version of ralrnmpt 7120 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2404. (Revised by GG, 26-Jan-2024.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))
Theorem	rexrnmptw 7119*	A restricted quantifier over an image set. Version of rexrnmpt 7121 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2404. (Revised by GG, 26-Jan-2024.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))
Theorem	ralrnmpt 7120*	A restricted quantifier over an image set. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker ralrnmptw 7118 when possible. (Contributed by Mario Carneiro, 20-Aug-2015.) (New usage is discouraged.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))
Theorem	rexrnmpt 7121*	A restricted quantifier over an image set. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker rexrnmptw 7119 when possible. (Contributed by Mario Carneiro, 20-Aug-2015.) (New usage is discouraged.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))
Theorem	f0cli 7122	Unconditional closure of a function when the codomain includes the empty set. (Contributed by Mario Carneiro, 12-Sep-2013.)
⊢ 𝐹:𝐴⟶𝐵    &   ⊢ ∅ ∈ 𝐵    ⇒   ⊢ (𝐹‘𝐶) ∈ 𝐵
Theorem	dff2 7123	Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵)))
Theorem	dff3 7124*	Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦))
Theorem	dff4 7125*	Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦))
Theorem	dffo3 7126*	An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)
⊢ (𝐹:𝐴⟶onto𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
Theorem	dffo4 7127*	Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
⊢ (𝐹:𝐴⟶onto𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))
Theorem	dffo5 7128*	Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
⊢ (𝐹:𝐴⟶onto𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 𝑥𝐹𝑦))
Theorem	exfo 7129*	A relation equivalent to the existence of an onto mapping. The right-hand 𝑓 is not necessarily a function. (Contributed by NM, 20-Mar-2007.)
⊢ (∃𝑓 𝑓:𝐴⟶onto𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥))
Theorem	dffo3f 7130*	An onto mapping expressed in terms of function values. As dffo3 7126 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝐹    ⇒   ⊢ (𝐹:𝐴⟶onto𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
Theorem	foelrn 7131*	Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.)
⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 = (𝐹‘𝑥))
Theorem	foelrnf 7132*	Property of a surjective function. As foelrn 7131 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝐹    ⇒   ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 = (𝐹‘𝑥))
Theorem	foco2 7133	If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.)
⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴⟶onto𝐶) → 𝐹:𝐵⟶onto𝐶)
Theorem	fmpt 7134*	Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
Theorem	f1ompt 7135*	Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)    ⇒   ⊢ (𝐹:𝐴⟶1-1onto𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶))
Theorem	fmpti 7136*	Functionality of the mapping operation. (Contributed by NM, 19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)    &   ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)    ⇒   ⊢ 𝐹:𝐴⟶𝐵
Theorem	fvmptelcdm 7137*	The value of a function at a point of its domain belongs to its codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
Theorem	fmptd 7138*	Domain and codomain of the mapping operation; deduction form. (Contributed by Mario Carneiro, 13-Jan-2013.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
Theorem	fmpttd 7139*	Version of fmptd 7138 with inlined definition. Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Proof shortened by BJ, 16-Aug-2022.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)
Theorem	fmpt3d 7140*	Domain and codomain of the mapping operation; deduction form. (Contributed by Thierry Arnoux, 4-Jun-2017.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
Theorem	fmptdf 7141*	A version of fmptd 7138 using bound-variable hypothesis instead of a distinct variable condition for 𝜑. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
Theorem	fompt 7142*	Express being onto for a mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)    ⇒   ⊢ (𝐹:𝐴⟶onto𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))
Theorem	ffnfv 7143*	A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
Theorem	ffnfvf 7144	A function maps to a class to which all values belong. This version of ffnfv 7143 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝐹    ⇒   ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
Theorem	fnfvrnss 7145*	An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → ran 𝐹 ⊆ 𝐵)
Theorem	fcdmssb 7146*	A function is a function into a subset of its codomain if all of its values are elements of this subset. (Contributed by AV, 7-Feb-2021.)
⊢ ((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) → (𝐹:𝐴⟶𝑊 ↔ 𝐹:𝐴⟶𝑉))
Theorem	rnmptss 7147*	The range of an operation given by the maps-to notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)
Theorem	rnmptssd 7148*	The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → ran 𝐹 ⊆ 𝐶)
Theorem	fmpt2d 7149*	Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
Theorem	ffvresb 7150*	A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)
⊢ (Fun 𝐹 → ((𝐹 ↾ 𝐴):𝐴⟶𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)))
Theorem	fssrescdmd 7151	Restriction of a function to a subclass of its domain as a function with domain and codomain. (Contributed by AV, 13-May-2025.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → (𝐹 “ 𝐶) ⊆ 𝐷)    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐷)
Theorem	f1oresrab 7152*	Build a bijection between restricted abstract builders, given a bijection between the base classes, deduction version. (Contributed by Thierry Arnoux, 17-Aug-2018.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)    &   ⊢ (𝜑 → 𝐹:𝐴⟶1-1onto𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → (𝜒 ↔ 𝜓))    ⇒   ⊢ (𝜑 → (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}):{𝑥 ∈ 𝐴 ∣ 𝜓}⟶1-1onto{𝑦 ∈ 𝐵 ∣ 𝜒})
Theorem	f1ossf1o 7153*	Restricting a bijection, which is a mapping from a restricted class abstraction, to a subset is a bijection. (Contributed by AV, 7-Aug-2022.)
⊢ 𝑋 = {𝑤 ∈ 𝐴 ∣ (𝜓 ∧ 𝜒)}    &   ⊢ 𝑌 = {𝑤 ∈ 𝐴 ∣ 𝜓}    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 𝐵)    &   ⊢ 𝐺 = (𝑥 ∈ 𝑌 ↦ 𝐵)    &   ⊢ (𝜑 → 𝐺:𝑌⟶1-1onto𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌 ∧ 𝑦 = 𝐵) → (𝜏 ↔ [𝑥 / 𝑤]𝜒))    ⇒   ⊢ (𝜑 → 𝐹:𝑋⟶1-1onto{𝑦 ∈ 𝐶 ∣ 𝜏})
Theorem	fmptco 7154*	Composition of two functions expressed as ordered-pair class abstractions. If 𝐹 has the equation (𝑥 + 2) and 𝐺 the equation (3∗𝑧) then (𝐺 ∘ 𝐹) has the equation (3∗(𝑥 + 2)). (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅))    &   ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))    &   ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇))
Theorem	fmptcof 7155*	Version of fmptco 7154 where 𝜑 needn't be distinct from 𝑥. (Contributed by NM, 27-Dec-2014.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅))    &   ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))    &   ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇))
Theorem	fmptcos 7156*	Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅))    &   ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆))
Theorem	cofmpt 7157*	Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 29-Jun-2017.)
⊢ (𝜑 → 𝐹:𝐶⟶𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)))
Theorem	fcompt 7158*	Express composition of two functions as a maps-to applying both in sequence. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝐴 ∘ 𝐵) = (𝑥 ∈ 𝐶 ↦ (𝐴‘(𝐵‘𝑥))))
Theorem	fcoconst 7159	Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹‘𝑌)}))
Theorem	fsn 7160	A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩})
Theorem	fsn2 7161	A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}))
Theorem	fsng 7162	A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩}))
Theorem	fsn2g 7163	A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by Thierry Arnoux, 11-Jul-2020.)
⊢ (𝐴 ∈ 𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})))
Theorem	xpsng 7164	The Cartesian product of two singletons is the singleton consisting in the associated ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
Theorem	xpsn 7165	The Cartesian product of two singletons is the singleton consisting in the associated ordered pair. (Contributed by NM, 4-Nov-2006.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩}
Theorem	xpprsng 7166	The Cartesian product of an unordered pair and a singleton. (Contributed by AV, 20-May-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → ({𝐴, 𝐵} × {𝐶}) = {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐶⟩})
Theorem	xpsnprg 7167	The Cartesian product of a singleton and an unordered pair. (Contributed by AV, 21-Aug-2026.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → ({𝐴} × {𝐵, 𝐶}) = {⟨𝐴, 𝐵⟩, ⟨𝐴, 𝐶⟩})
Theorem	xpsntpg 7168	The Cartesian product of a singleton and an unordered triple. (Contributed by AV, 18-Aug-2026.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑇)) → ({𝐴} × {𝐵, 𝐶, 𝐷}) = {⟨𝐴, 𝐵⟩, ⟨𝐴, 𝐶⟩, ⟨𝐴, 𝐷⟩})
Theorem	f1o2sn 7169	A singleton consisting in a nested ordered pair is a one-to-one function from the cartesian product of two singletons onto a singleton (case where the two singletons are equal). (Contributed by AV, 15-Aug-2019.)
⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:({𝐸} × {𝐸})⟶1-1onto{𝑋})
Theorem	residpr 7170	Restriction of the identity to a pair. (Contributed by AV, 11-Dec-2018.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ( I ↾ {𝐴, 𝐵}) = {⟨𝐴, 𝐴⟩, ⟨𝐵, 𝐵⟩})
Theorem	dfmpt 7171	Alternate definition for the maps-to notation df-mpt 5190 (although it requires that 𝐵 be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}
Theorem	fnasrn 7172	A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩)
Theorem	idref 7173*	Two ways to state that a relation is reflexive on a class. (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Revised by NM, 30-Mar-2016.)
⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
Theorem	funiun 7174*	A function is a union of singletons of ordered pairs indexed by its domain. (Contributed by AV, 18-Sep-2020.)
⊢ (Fun 𝐹 → 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩})
Theorem	funopsn 7175*	If a function is an ordered pair then it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) (Proof shortened by AV, 15-Jul-2021.) (Proof shortened by Eric Schmidt, 9-May-2026.) A function is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a function is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is funsng 6613, as relsnopg 5786 is to relop 5834. (New usage is discouraged.)
⊢ 𝑋 ∈ V    &   ⊢ 𝑌 ∈ V    ⇒   ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
Theorem	funopsnOLD 7176*	Obsolete version of funopsn 7175 as of 9-May-2026. (Contributed by AV, 20-Sep-2020.) (Proof shortened by AV, 15-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ V    &   ⊢ 𝑌 ∈ V    ⇒   ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
Theorem	funop 7177*	An ordered pair is a function iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) A function is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a function is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is funsng 6613, as relsnopg 5786 is to relop 5834. (New usage is discouraged.)
⊢ 𝑋 ∈ V    &   ⊢ 𝑌 ∈ V    ⇒   ⊢ (Fun ⟨𝑋, 𝑌⟩ ↔ ∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}))
Theorem	funopdmsn 7178	The domain of a function which is an ordered pair is a singleton. (Contributed by AV, 15-Nov-2021.) (Avoid depending on this detail.)
⊢ 𝐺 = ⟨𝑋, 𝑌⟩    &   ⊢ 𝑋 ∈ 𝑉    &   ⊢ 𝑌 ∈ 𝑊    ⇒   ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵)
Theorem	funsndifnop 7179	A singleton of an ordered pair is not an ordered pair if the components are different. (Contributed by AV, 23-Sep-2020.) (Avoid depending on this detail.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐺 = {⟨𝐴, 𝐵⟩}    ⇒   ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐺 ∈ (V × V))
Theorem	funsneqopb 7180	A singleton of an ordered pair is an ordered pair iff the components are equal. (Contributed by AV, 24-Sep-2020.) (Avoid depending on this detail.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐺 = {⟨𝐴, 𝐵⟩}    ⇒   ⊢ (𝐴 = 𝐵 ↔ 𝐺 ∈ (V × V))
Theorem	ressnop0 7181	If 𝐴 is not in 𝐶, then the restriction of a singleton of ⟨𝐴, 𝐵⟩ to 𝐶 is null. (Contributed by Scott Fenton, 15-Apr-2011.)
⊢ (¬ 𝐴 ∈ 𝐶 → ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅)
Theorem	fpr 7182	A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ (𝐴 ≠ 𝐵 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}⟶{𝐶, 𝐷})
Theorem	fprg 7183	A function with a domain of two elements. (Contributed by FL, 2-Feb-2014.)
⊢ (((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐹) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐻) ∧ 𝐴 ≠ 𝐵) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}⟶{𝐶, 𝐷})
Theorem	ftpg 7184	A function with a domain of three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩}:{𝑋, 𝑌, 𝑍}⟶{𝐴, 𝐵, 𝐶})
Theorem	ftp 7185	A function with a domain of three elements. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by Alexander van der Vekens, 23-Jan-2018.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝑋 ∈ V    &   ⊢ 𝑌 ∈ V    &   ⊢ 𝑍 ∈ V    &   ⊢ 𝐴 ≠ 𝐵    &   ⊢ 𝐴 ≠ 𝐶    &   ⊢ 𝐵 ≠ 𝐶    ⇒   ⊢ {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩, ⟨𝐶, 𝑍⟩}:{𝐴, 𝐵, 𝐶}⟶{𝑋, 𝑌, 𝑍}
Theorem	fnressn 7186	A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹‘𝐵)⟩})
Theorem	funressn 7187	A function restricted to a singleton. (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ (Fun 𝐹 → (𝐹 ↾ {𝐵}) ⊆ {⟨𝐵, (𝐹‘𝐵)⟩})
Theorem	fressnfv 7188	The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶))
Theorem	fvrnressn 7189	If the value of a function is in the range of the function restricted to the singleton containing the argument, then the value of the function is in the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹))
Theorem	fvressn 7190	The value of a function restricted to the singleton containing the argument equals the value of the function for this argument. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
⊢ (𝑋 ∈ 𝑉 → ((𝐹 ↾ {𝑋})‘𝑋) = (𝐹‘𝑋))
Theorem	fvconst 7191	The value of a constant function. (Contributed by NM, 30-May-1999.)
⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) = 𝐵)
Theorem	fnsnr 7192	If a class belongs to a function on a singleton, then that class is the obvious ordered pair. Note that this theorem also holds when 𝐴 is a proper class, but its meaning is then different. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.)
⊢ (𝐹 Fn {𝐴} → (𝐵 ∈ 𝐹 → 𝐵 = ⟨𝐴, (𝐹‘𝐴)⟩))
Theorem	fnsnbg 7193	A function's domain is a singleton iff the function is a singleton. (Contributed by Steven Nguyen, 18-Aug-2023.) Relax condition for being in the universal class. (Revised by Zhi Wang, 21-Oct-2025.)
⊢ (𝐴 ∈ 𝑉 → (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}))
Theorem	fnsnb 7194	A function whose domain is a singleton can be represented as a singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Revised to add reverse implication. (Revised by NM, 29-Dec-2018.) (Proof shortened by Zhi Wang, 21-Oct-2025.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩})
Theorem	fmptsn 7195*	Express a singleton function in maps-to notation. (Contributed by NM, 6-Jun-2006.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 28-Feb-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
Theorem	fmptsng 7196*	Express a singleton function in maps-to notation. Version of fmptsn 7195 allowing the value 𝐵 to depend on the variable 𝑥. (Contributed by AV, 27-Feb-2019.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
Theorem	fmptsnd 7197*	Express a singleton function in maps-to notation. Deduction form of fmptsng 7196. (Contributed by AV, 4-Aug-2019.)
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    ⇒   ⊢ (𝜑 → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
Theorem	fmptap 7198*	Append an additional value to a function. (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝑅 ∪ {𝐴}) = 𝑆    &   ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐵)    ⇒   ⊢ ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶)
Theorem	fmptapd 7199*	Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.) (Revised by AV, 10-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶))
Theorem	fmptpr 7200*	Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐸 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐸 = 𝐷)    ⇒   ⊢ (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸))