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	fneqeql2 7001	Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ 𝐴 ⊆ dom (𝐹 ∩ 𝐺)))
Theorem	fnreseql 7002	Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ 𝑋 ⊆ dom (𝐹 ∩ 𝐺)))
Theorem	chfnrn 7003*	The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain. (Contributed by NM, 31-Aug-1999.)
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥) → ran 𝐹 ⊆ ∪ 𝐴)
Theorem	funfvop 7004	Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
Theorem	funfvbrb 7005	Two ways to say that 𝐴 is in the domain of 𝐹. (Contributed by Mario Carneiro, 1-May-2014.)
⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 ↔ 𝐴𝐹(𝐹‘𝐴)))
Theorem	fvimacnvi 7006	A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (◡𝐹 “ 𝐵)) → (𝐹‘𝐴) ∈ 𝐵)
Theorem	fvimacnv 7007	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 6583 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 7008	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 7007 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 7009*	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 7010*	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 7011	Alternate proof of fvimacnv 7007, based on funimass3 7008. If funimass3 7008 is ever proved directly, as opposed to using funimacnv 6581 pointwise, then the proof of funimacnv 6581 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 7012	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 7013	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 7014	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 7015*	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 7016*	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 7017	Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∪ 𝐵)) = ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵)))
Theorem	inpreima 7018	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 7019	Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∖ 𝐵)) = ((◡𝐹 “ 𝐴) ∖ (◡𝐹 “ 𝐵)))
Theorem	respreima 7020	The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐵) “ 𝐴) = ((◡𝐹 “ 𝐴) ∩ 𝐵))
Theorem	cnvimainrn 7021	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 7022	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 7023*	Preimage of an intersection. (Contributed by FL, 16-Apr-2012.)
⊢ ((Fun 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵))
Theorem	intpreima 7024*	Preimage of an intersection. (Contributed by FL, 28-Apr-2012.)
⊢ ((Fun 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ ∩ 𝐴) = ∩ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥))
Theorem	fimacnvinrn 7025	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 7026	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 7027	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 7028*	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 6882. (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 7029	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 7030	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 7031	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 7032*	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 7033*	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 7034	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 7035	A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐵)
Theorem	fnfvelrnd 7036	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 7037	A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)
⊢ 𝐹:𝐴⟶𝐵    ⇒   ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) ∈ 𝐵)
Theorem	ffvelcdmda 7038	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 7039	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 7040	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 7041*	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 7042*	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 7043*	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 7044*	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 7045*	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 7046*	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 6508 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 6508 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 7047*	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 7048*	A restricted quantifier over an image set. Version of ralrnmpt 7050 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2377. (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 7049*	A restricted quantifier over an image set. Version of rexrnmpt 7051 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2377. (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 7050*	A restricted quantifier over an image set. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker ralrnmptw 7048 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 7051*	A restricted quantifier over an image set. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker rexrnmptw 7049 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 7052	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 7053	Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵)))
Theorem	dff3 7054*	Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦))
Theorem	dff4 7055*	Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥𝐹𝑦))
Theorem	dffo3 7056*	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 7057*	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 7058*	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 7059*	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 7060*	An onto mapping expressed in terms of function values. As dffo3 7056 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 7061*	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 7062*	Property of a surjective function. As foelrn 7061 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 7063	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 7064*	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 7065*	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 7066*	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 7067*	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 7068*	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 7069*	Version of fmptd 7068 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 7070*	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 7071*	A version of fmptd 7068 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 7072*	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 7073*	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 7074	A function maps to a class to which all values belong. This version of ffnfv 7073 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 7075*	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 7076*	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 7077*	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 7078*	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 7079*	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 7080*	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 7081	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 7082*	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 7083*	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 7084*	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 7085*	Version of fmptco 7084 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 7086*	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 7087*	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 7088*	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 7089	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 7090	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 7091	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 7092	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 7093	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 7094	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 7095	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 7096	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 7097	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 7098	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 7099	Alternate definition for the maps-to notation df-mpt 5182 (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 7100	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⟩)