P. 71 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 7001-7100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fvimacnv 7001	The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 6575 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))
Theorem	funimass3 7002	A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv 7001 would be the special case of 𝐴 being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵)))
Theorem	funimass5 7003*	A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐴 ⊆ (◡𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
Theorem	funconstss 7004*	Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ {𝐵})))
Theorem	fvimacnvALT 7005	Alternate proof of fvimacnv 7001, based on funimass3 7002. If funimass3 7002 is ever proved directly, as opposed to using funimacnv 6573 pointwise, then the proof of funimacnv 6573 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))
Theorem	elpreima 7006	Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ (◡𝐹 “ 𝐶) ↔ (𝐵 ∈ 𝐴 ∧ (𝐹‘𝐵) ∈ 𝐶)))
Theorem	elpreimad 7007	Membership in the preimage of a set under a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐹‘𝐵) ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 ∈ (◡𝐹 “ 𝐶))
Theorem	fniniseg 7008	Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014.) (Proof shortened by Mario Carneiro , 28-Apr-2015.)
⊢ (𝐹 Fn 𝐴 → (𝐶 ∈ (◡𝐹 “ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) = 𝐵)))
Theorem	fncnvima2 7009*	Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ 𝐵) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ 𝐵})
Theorem	fniniseg2 7010*	Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {𝐵}) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝐵})
Theorem	unpreima 7011	Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∪ 𝐵)) = ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵)))
Theorem	inpreima 7012	Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)
⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵)))
Theorem	difpreima 7013	Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∖ 𝐵)) = ((◡𝐹 “ 𝐴) ∖ (◡𝐹 “ 𝐵)))
Theorem	respreima 7014	The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐵) “ 𝐴) = ((◡𝐹 “ 𝐴) ∩ 𝐵))
Theorem	cnvimainrn 7015	The preimage of the intersection of the range of a class and a class 𝐴 is the preimage of the class 𝐴. (Contributed by AV, 17-Sep-2024.)
⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐴)) = (◡𝐹 “ 𝐴))
Theorem	sspreima 7016	The preimage of a subset is a subset of the preimage. (Contributed by Brendan Leahy, 23-Sep-2017.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ 𝐵) → (◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ 𝐵))
Theorem	iinpreima 7017*	Preimage of an intersection. (Contributed by FL, 16-Apr-2012.)
⊢ ((Fun 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵))
Theorem	intpreima 7018*	Preimage of an intersection. (Contributed by FL, 28-Apr-2012.)
⊢ ((Fun 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ ∩ 𝐴) = ∩ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥))
Theorem	fimacnvinrn 7019	Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 21-Jan-2017.)
⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))
Theorem	fimacnvinrn2 7020	Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 17-Feb-2017.)
⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ 𝐵)))
Theorem	rescnvimafod 7021	The restriction of a function to a preimage of a class is a function onto the intersection of this class and the range of the function. (Contributed by AV, 13-Sep-2024.) (Revised by AV, 29-Sep-2024.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐵))    &   ⊢ (𝜑 → 𝐷 = (◡𝐹 “ 𝐵))    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→𝐸)
Theorem	fvn0ssdmfun 7022*	If a class' function values for certain arguments is not the empty set, the arguments are contained in the domain of the class, and the class restricted to the arguments is a function, analogous to fvfundmfvn0 6874. (Contributed by AV, 27-Jan-2020.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ (∀𝑎 ∈ 𝐷 (𝐹‘𝑎) ≠ ∅ → (𝐷 ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ 𝐷)))
Theorem	fnopfv 7023	Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ⟨𝐵, (𝐹‘𝐵)⟩ ∈ 𝐹)
Theorem	fvelrn 7024	A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)
Theorem	nelrnfvne 7025	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 7026*	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 7027*	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 7028	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 7029	A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐵)
Theorem	fnfvelrnd 7030	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 7031	A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)
⊢ 𝐹:𝐴⟶𝐵    ⇒   ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) ∈ 𝐵)
Theorem	ffvelcdmda 7032	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 7033	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 7034	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 7035*	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 7036*	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 7037*	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 7038*	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 7039*	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 7040*	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 6500 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 6500 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 7041*	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 7042*	A restricted quantifier over an image set. Version of ralrnmpt 7044 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2380. (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 7043*	A restricted quantifier over an image set. Version of rexrnmpt 7045 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2380. (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 7044*	A restricted quantifier over an image set. Usage of this theorem is discouraged because it depends on ax-13 2380. Use the weaker ralrnmptw 7042 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 7045*	A restricted quantifier over an image set. Usage of this theorem is discouraged because it depends on ax-13 2380. Use the weaker rexrnmptw 7043 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 7046	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 7047	Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵)))
Theorem	dff3 7048*	Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦))
Theorem	dff4 7049*	Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦))
Theorem	dffo3 7050*	An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)
⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
Theorem	dffo4 7051*	Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))
Theorem	dffo5 7052*	Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 𝑥𝐹𝑦))
Theorem	exfo 7053*	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–onto→𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦𝑓𝑥))
Theorem	dffo3f 7054*	An onto mapping expressed in terms of function values. As dffo3 7050 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝐹    ⇒   ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
Theorem	foelrn 7055*	Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.)
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 = (𝐹‘𝑥))
Theorem	foelrnf 7056*	Property of a surjective function. As foelrn 7055 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝐹    ⇒   ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 = (𝐹‘𝑥))
Theorem	foco2 7057	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–onto→𝐶) → 𝐹:𝐵–onto→𝐶)
Theorem	fmpt 7058*	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 7059*	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–1-1-onto→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐶))
Theorem	fmpti 7060*	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 7061*	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 7062*	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 7063*	Version of fmptd 7062 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 7064*	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 7065*	A version of fmptd 7062 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 7066*	Express being onto for a mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)    ⇒   ⊢ (𝐹:𝐴–onto→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))
Theorem	ffnfv 7067*	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 7068	A function maps to a class to which all values belong. This version of ffnfv 7067 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 7069*	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 7070*	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 7071*	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 7072*	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 7073*	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 7074*	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 7075	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 7076*	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–1-1-onto→𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → (𝜒 ↔ 𝜓))    ⇒   ⊢ (𝜑 → (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}):{𝑥 ∈ 𝐴 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})
Theorem	f1ossf1o 7077*	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–1-1-onto→𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌 ∧ 𝑦 = 𝐵) → (𝜏 ↔ [𝑥 / 𝑤]𝜒))    ⇒   ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→{𝑦 ∈ 𝐶 ∣ 𝜏})
Theorem	fmptco 7078*	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 7079*	Version of fmptco 7078 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 7080*	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 7081*	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 7082*	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 7083	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 7084	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 7085	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 7086	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 7087	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 7088	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	xpprsng 7089	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	xpsn 7090	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	f1o2sn 7091	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})–1-1-onto→{𝑋})
Theorem	residpr 7092	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 7093	Alternate definition for the maps-to notation df-mpt 5161 (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 7094	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 7095*	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 7096*	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 7097*	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 6543, as relsnopg 5753 is to relop 5799. (New usage is discouraged.)
⊢ 𝑋 ∈ V    &   ⊢ 𝑌 ∈ V    ⇒   ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
Theorem	funopsnOLD 7098*	Obsolete version of funopsn 7097 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 7099*	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 6543, as relsnopg 5753 is to relop 5799. (New usage is discouraged.)
⊢ 𝑋 ∈ V    &   ⊢ 𝑌 ∈ V    ⇒   ⊢ (Fun ⟨𝑋, 𝑌⟩ ↔ ∃𝑎(𝑋 = {𝑎} ∧ ⟨𝑋, 𝑌⟩ = {⟨𝑎, 𝑎⟩}))
Theorem	funopdmsn 7100	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)