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Theorem List for Metamath Proof Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfvreseq0 6801* Equality of restricted functions is determined by their values (for functions with different domains). (Contributed by AV, 6-Jan-2019.)
(((𝐹 Fn 𝐴𝐺 Fn 𝐶) ∧ (𝐵𝐴𝐵𝐶)) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
 
Theoremfvreseq1 6802* Equality of a function restricted to the domain of another function. (Contributed by AV, 6-Jan-2019.)
(((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → ((𝐹𝐵) = 𝐺 ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
 
Theoremfvreseq 6803* Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.) (Proof shortened by AV, 4-Mar-2019.)
(((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐵𝐴) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
 
Theoremfnmptfvd 6804* A function with a given domain is a mapping defined by its function values. (Contributed by AV, 1-Mar-2019.)
(𝜑𝑀 Fn 𝐴)    &   (𝑖 = 𝑎𝐷 = 𝐶)    &   ((𝜑𝑖𝐴) → 𝐷𝑈)    &   ((𝜑𝑎𝐴) → 𝐶𝑉)       (𝜑 → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = 𝐷))
 
Theoremfndmdif 6805* Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
 
Theoremfndmdifcom 6806 The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = dom (𝐺𝐹))
 
Theoremfndmdifeq0 6807 The difference set of two functions is empty if and only if the functions are equal. (Contributed by Stefan O'Rear, 17-Jan-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom (𝐹𝐺) = ∅ ↔ 𝐹 = 𝐺))
 
Theoremfndmin 6808* Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐺𝑥)})
 
Theoremfneqeql 6809 Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ dom (𝐹𝐺) = 𝐴))
 
Theoremfneqeql2 6810 Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺𝐴 ⊆ dom (𝐹𝐺)))
 
Theoremfnreseql 6811 Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = (𝐺𝑋) ↔ 𝑋 ⊆ dom (𝐹𝐺)))
 
Theoremchfnrn 6812* 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 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → ran 𝐹 𝐴)
 
Theoremfunfvop 6813 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
 
Theoremfunfvbrb 6814 Two ways to say that 𝐴 is in the domain of 𝐹. (Contributed by Mario Carneiro, 1-May-2014.)
(Fun 𝐹 → (𝐴 ∈ dom 𝐹𝐴𝐹(𝐹𝐴)))
 
Theoremfvimacnvi 6815 A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)
((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → (𝐹𝐴) ∈ 𝐵)
 
Theoremfvimacnv 6816 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 6431 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
 
Theoremfunimass3 6817 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 6816 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 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))
 
Theoremfunimass5 6818* A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐴 ⊆ (𝐹𝐵) ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremfunconstss 6819* Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∀𝑥𝐴 (𝐹𝑥) = 𝐵𝐴 ⊆ (𝐹 “ {𝐵})))
 
TheoremfvimacnvALT 6820 Alternate proof of fvimacnv 6816, based on funimass3 6817. If funimass3 6817 is ever proved directly, as opposed to using funimacnv 6429 pointwise, then the proof of funimacnv 6429 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
 
Theoremelpreima 6821 Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐹 Fn 𝐴 → (𝐵 ∈ (𝐹𝐶) ↔ (𝐵𝐴 ∧ (𝐹𝐵) ∈ 𝐶)))
 
Theoremelpreimad 6822 Membership in the preimage of a set under a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐵𝐴)    &   (𝜑 → (𝐹𝐵) ∈ 𝐶)       (𝜑𝐵 ∈ (𝐹𝐶))
 
Theoremfniniseg 6823 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}) ↔ (𝐶𝐴 ∧ (𝐹𝐶) = 𝐵)))
 
Theoremfncnvima2 6824* Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐹 Fn 𝐴 → (𝐹𝐵) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ 𝐵})
 
Theoremfniniseg2 6825* Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) = {𝑥𝐴 ∣ (𝐹𝑥) = 𝐵})
 
Theoremunpreima 6826 Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∪ (𝐹𝐵)))
 
Theoreminpreima 6827 Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
 
Theoremdifpreima 6828 Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
 
Theoremrespreima 6829 The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)
(Fun 𝐹 → ((𝐹𝐵) “ 𝐴) = ((𝐹𝐴) ∩ 𝐵))
 
Theoremiinpreima 6830* Preimage of an intersection. (Contributed by FL, 16-Apr-2012.)
((Fun 𝐹𝐴 ≠ ∅) → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐹𝐵))
 
Theoremintpreima 6831* Preimage of an intersection. (Contributed by FL, 28-Apr-2012.)
((Fun 𝐹𝐴 ≠ ∅) → (𝐹 𝐴) = 𝑥𝐴 (𝐹𝑥))
 
Theoremfimacnv 6832 The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)
(𝐹:𝐴𝐵 → (𝐹𝐵) = 𝐴)
 
Theoremfimacnvinrn 6833 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 → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
 
Theoremfimacnvinrn2 6834 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 𝐹𝐵) → (𝐹𝐴) = (𝐹 “ (𝐴𝐵)))
 
Theoremfvn0ssdmfun 6835* 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 6702. (Contributed by AV, 27-Jan-2020.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
(∀𝑎𝐷 (𝐹𝑎) ≠ ∅ → (𝐷 ⊆ dom 𝐹 ∧ Fun (𝐹𝐷)))
 
Theoremfnopfv 6836 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.)
((𝐹 Fn 𝐴𝐵𝐴) → ⟨𝐵, (𝐹𝐵)⟩ ∈ 𝐹)
 
Theoremfvelrn 6837 A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)
 
Theoremnelrnfvne 6838 A function value cannot be any element not contained in the range of the function. (Contributed by AV, 28-Jan-2020.)
((Fun 𝐹𝑋 ∈ dom 𝐹𝑌 ∉ ran 𝐹) → (𝐹𝑋) ≠ 𝑌)
 
Theoremfveqdmss 6839* 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 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → 𝐷 ⊆ dom 𝐴)
 
Theoremfveqressseq 6840* 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 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (𝐴𝐷) = 𝐵)
 
Theoremfnfvelrn 6841 A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) ∈ ran 𝐹)
 
Theoremffvelrn 6842 A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)
((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐵)
 
Theoremffvelrni 6843 A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)
𝐹:𝐴𝐵       (𝐶𝐴 → (𝐹𝐶) ∈ 𝐵)
 
Theoremffvelrnda 6844 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐹:𝐴𝐵)       ((𝜑𝐶𝐴) → (𝐹𝐶) ∈ 𝐵)
 
Theoremffvelrnd 6845 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → (𝐹𝐶) ∈ 𝐵)
 
Theoremrexrn 6846* Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
(𝑥 = (𝐹𝑦) → (𝜑𝜓))       (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦𝐴 𝜓))
 
Theoremralrn 6847* Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
(𝑥 = (𝐹𝑦) → (𝜑𝜓))       (𝐹 Fn 𝐴 → (∀𝑥 ∈ ran 𝐹𝜑 ↔ ∀𝑦𝐴 𝜓))
 
Theoremelrnrexdm 6848* 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 = (𝐹𝑥)))
 
Theoremelrnrexdmb 6849* 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 = (𝐹𝑥)))
 
Theoremeldmrexrn 6850* 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 = (𝐹𝑌)))
 
Theoremeldmrexrnb 6851* 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 6357 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 6357 of the value of a function, (𝐹𝑌) = ∅ 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 = (𝐹𝑌)))
 
Theoremfvcofneq 6852* 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 𝐴𝐾 Fn 𝐵) → ((𝑋 ∈ (𝐴𝐵) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑥) = (𝐻𝑥)) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋)))
 
Theoremralrnmptw 6853* Version of ralrnmpt 6855 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑦 = 𝐵 → (𝜓𝜒))       (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 𝜒))
 
Theoremrexrnmptw 6854* Version of rexrnmpt 6856 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑦 = 𝐵 → (𝜓𝜒))       (∀𝑥𝐴 𝐵𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥𝐴 𝜒))
 
Theoremralrnmpt 6855* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑦 = 𝐵 → (𝜓𝜒))       (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 𝜒))
 
Theoremrexrnmpt 6856* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑦 = 𝐵 → (𝜓𝜒))       (∀𝑥𝐴 𝐵𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥𝐴 𝜒))
 
Theoremf0cli 6857 Unconditional closure of a function when the range includes the empty set. (Contributed by Mario Carneiro, 12-Sep-2013.)
𝐹:𝐴𝐵    &   ∅ ∈ 𝐵       (𝐹𝐶) ∈ 𝐵
 
Theoremdff2 6858 Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)))
 
Theoremdff3 6859* Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)
(𝐹:𝐴𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))
 
Theoremdff4 6860* Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)
(𝐹:𝐴𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝐹𝑦))
 
Theoremdffo3 6861* An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)
(𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
 
Theoremdffo4 6862* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
(𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
 
Theoremdffo5 6863* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
(𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦))
 
Theoremexfo 6864* 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:𝐴onto𝐵 ↔ ∃𝑓(∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
 
Theoremfoelrn 6865* Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.)
((𝐹:𝐴onto𝐵𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))
 
Theoremfoco2 6866 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:𝐵𝐶𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) → 𝐹:𝐵onto𝐶)
 
Theoremfmpt 6867* Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐹 = (𝑥𝐴𝐶)       (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
 
Theoremf1ompt 6868* Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.)
𝐹 = (𝑥𝐴𝐶)       (𝐹:𝐴1-1-onto𝐵 ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶))
 
Theoremfmpti 6869* Functionality of the mapping operation. (Contributed by NM, 19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
𝐹 = (𝑥𝐴𝐶)    &   (𝑥𝐴𝐶𝐵)       𝐹:𝐴𝐵
 
Theoremfvmptelrn 6870* The value of a function at a point of its domain belongs to its codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑 → (𝑥𝐴𝐵):𝐴𝐶)       ((𝜑𝑥𝐴) → 𝐵𝐶)
 
Theoremfmptd 6871* Domain and codomain of the mapping operation; deduction form. (Contributed by Mario Carneiro, 13-Jan-2013.)
((𝜑𝑥𝐴) → 𝐵𝐶)    &   𝐹 = (𝑥𝐴𝐵)       (𝜑𝐹:𝐴𝐶)
 
Theoremfmpttd 6872* Version of fmptd 6871 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.)
((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
 
Theoremfmpt3d 6873* Domain and codomain of the mapping operation; deduction form. (Contributed by Thierry Arnoux, 4-Jun-2017.)
(𝜑𝐹 = (𝑥𝐴𝐵))    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑𝐹:𝐴𝐶)
 
Theoremfmptdf 6874* A version of fmptd 6871 using bound-variable hypothesis instead of a distinct variable condition for 𝜑. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝐶)    &   𝐹 = (𝑥𝐴𝐵)       (𝜑𝐹:𝐴𝐶)
 
Theoremffnfv 6875* A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremffnfvf 6876 A function maps to a class to which all values belong. This version of ffnfv 6875 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝐹       (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremfnfvrnss 6877* An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)
((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → ran 𝐹𝐵)
 
Theoremfrnssb 6878* 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.)
((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → (𝐹:𝐴𝑊𝐹:𝐴𝑉))
 
Theoremrnmptss 6879* The range of an operation given by the maps-to notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.)
𝐹 = (𝑥𝐴𝐵)       (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
 
Theoremfmpt2d 6880* Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑𝐹 = (𝑥𝐴𝐵))    &   ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ 𝐶)       (𝜑𝐹:𝐴𝐶)
 
Theoremffvresb 6881* A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)
(Fun 𝐹 → ((𝐹𝐴):𝐴𝐵 ↔ ∀𝑥𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)))
 
Theoremf1oresrab 6882* Build a bijection between restricted abstract builders, given a bijection between the base classes, deduction version. (Contributed by Thierry Arnoux, 17-Aug-2018.)
𝐹 = (𝑥𝐴𝐶)    &   (𝜑𝐹:𝐴1-1-onto𝐵)    &   ((𝜑𝑥𝐴𝑦 = 𝐶) → (𝜒𝜓))       (𝜑 → (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
 
Theoremf1ossf1o 6883* 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 = {𝑤𝐴 ∣ (𝜓𝜒)}    &   𝑌 = {𝑤𝐴𝜓}    &   𝐹 = (𝑥𝑋𝐵)    &   𝐺 = (𝑥𝑌𝐵)    &   (𝜑𝐺:𝑌1-1-onto𝐶)    &   ((𝜑𝑥𝑌𝑦 = 𝐵) → (𝜏 ↔ [𝑥 / 𝑤]𝜒))       (𝜑𝐹:𝑋1-1-onto→{𝑦𝐶𝜏})
 
Theoremfmptco 6884* Composition of two functions expressed as ordered-pair class abstractions. If 𝐹 has the equation (𝑥 + 2) and 𝐺 the equation (3∗𝑧) then (𝐺𝐹) has the equation (3∗(𝑥 + 2)). (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.)
((𝜑𝑥𝐴) → 𝑅𝐵)    &   (𝜑𝐹 = (𝑥𝐴𝑅))    &   (𝜑𝐺 = (𝑦𝐵𝑆))    &   (𝑦 = 𝑅𝑆 = 𝑇)       (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))
 
Theoremfmptcof 6885* Version of fmptco 6884 where 𝜑 needn't be distinct from 𝑥. (Contributed by NM, 27-Dec-2014.)
(𝜑 → ∀𝑥𝐴 𝑅𝐵)    &   (𝜑𝐹 = (𝑥𝐴𝑅))    &   (𝜑𝐺 = (𝑦𝐵𝑆))    &   (𝑦 = 𝑅𝑆 = 𝑇)       (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))
 
Theoremfmptcos 6886* Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝜑 → ∀𝑥𝐴 𝑅𝐵)    &   (𝜑𝐹 = (𝑥𝐴𝑅))    &   (𝜑𝐺 = (𝑦𝐵𝑆))       (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑅 / 𝑦𝑆))
 
Theoremcofmpt 6887* Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 29-Jun-2017.)
(𝜑𝐹:𝐶𝐷)    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → (𝐹 ∘ (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ (𝐹𝐵)))
 
Theoremfcompt 6888* 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:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (𝐴‘(𝐵𝑥))))
 
Theoremfcoconst 6889 Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
((𝐹 Fn 𝑋𝑌𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹𝑌)}))
 
Theoremfsn 6890 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⟩})
 
Theoremfsn2 6891 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 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))
 
Theoremfsng 6892 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)
((𝐴𝐶𝐵𝐷) → (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩}))
 
Theoremfsn2g 6893 A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by Thierry Arnoux, 11-Jul-2020.)
(𝐴𝑉 → (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹𝐴) ∈ 𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩})))
 
Theoremxpsng 6894 The Cartesian product of two singletons is the singleton consisting in the associated ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝐴𝑉𝐵𝑊) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
 
Theoremxpprsng 6895 The Cartesian product of an unordered pair and a singleton. (Contributed by AV, 20-May-2019.)
((𝐴𝑉𝐵𝑊𝐶𝑈) → ({𝐴, 𝐵} × {𝐶}) = {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐶⟩})
 
Theoremxpsn 6896 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⟩}
 
Theoremf1o2sn 6897 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.)
((𝐸𝑉𝑋𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:({𝐸} × {𝐸})–1-1-onto→{𝑋})
 
Theoremresidpr 6898 Restriction of the identity to a pair. (Contributed by AV, 11-Dec-2018.)
((𝐴𝑉𝐵𝑊) → ( I ↾ {𝐴, 𝐵}) = {⟨𝐴, 𝐴⟩, ⟨𝐵, 𝐵⟩})
 
Theoremdfmpt 6899 Alternate definition for the maps-to notation df-mpt 5139 (although it requires that 𝐵 be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.)
𝐵 ∈ V       (𝑥𝐴𝐵) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
 
Theoremfnasrn 6900 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       (𝑥𝐴𝐵) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩)
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