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Theorem List for Metamath Proof Explorer - 6201-6300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfvmptss 6201* If all the values of the mapping are subsets of a class 𝐶, then so is any evaluation of the mapping, even if 𝐷 is not in the base set 𝐴. (Contributed by Mario Carneiro, 13-Feb-2015.)
𝐹 = (𝑥𝐴𝐵)       (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)

Theoremfvmpt2d 6202* Deduction version of fvmpt2 6200. (Contributed by Thierry Arnoux, 8-Dec-2016.)
(𝜑𝐹 = (𝑥𝐴𝐵))    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)

Theoremfvmptex 6203* Express a function 𝐹 whose value 𝐵 may not always be a set in terms of another function 𝐺 for which sethood is guaranteed. (Note that ( I ‘𝐵) is just shorthand for if(𝐵 ∈ V, 𝐵, ∅), and it is always a set by fvex 6113.) Note also that these functions are not the same; wherever 𝐵(𝐶) is not a set, 𝐶 is not in the domain of 𝐹 (so it evaluates to the empty set), but 𝐶 is in the domain of 𝐺, and 𝐺(𝐶) is defined to be the empty set. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴 ↦ ( I ‘𝐵))       (𝐹𝐶) = (𝐺𝐶)

Theoremfvmptdf 6204* Alternate deduction version of fvmpt 6191, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))    &   𝑥𝐹    &   𝑥𝜓       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))

Theoremfvmptdv 6205* Alternate deduction version of fvmpt 6191, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))

Theoremfvmptdv2 6206* Alternate deduction version of fvmpt 6191, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = 𝐶))

Theoremmpteqb 6207* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 6219. (Contributed by Mario Carneiro, 14-Nov-2014.)
(∀𝑥𝐴 𝐵𝑉 → ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ↔ ∀𝑥𝐴 𝐵 = 𝐶))

Theoremfvmptt 6208* Closed theorem form of fvmpt 6191. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)
((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → (𝐹𝐴) = 𝐶)

Theoremfvmptf 6209* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 6189 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴    &   𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)       ((𝐴𝐷𝐶𝑉) → (𝐹𝐴) = 𝐶)

Theoremfvmptnf 6210* The value of a function given by an ordered-pair class abstraction is the empty set when the class it would otherwise map to is a proper class. This version of fvmptn 6211 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
𝑥𝐴    &   𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)       𝐶 ∈ V → (𝐹𝐴) = ∅)

Theoremfvmptn 6211* This somewhat non-intuitive theorem tells us the value of its function is the empty set when the class 𝐶 it would otherwise map to is a proper class. This is a technical lemma that can help eliminate redundant sethood antecedents otherwise required by fvmptg 6189. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 9-Sep-2013.)
(𝑥 = 𝐷𝐵 = 𝐶)    &   𝐹 = (𝑥𝐴𝐵)       𝐶 ∈ V → (𝐹𝐷) = ∅)

Theoremfvmptss2 6212* A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.)
(𝑥 = 𝐷𝐵 = 𝐶)    &   𝐹 = (𝑥𝐴𝐵)       (𝐹𝐷) ⊆ 𝐶

Theoremelfvmptrab1 6213* Implications for the value of a function 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 argument of the function. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})    &   (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)       (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))

Theoremelfvmptrab 6214* Implications for the value of a function 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.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝑀𝜑})    &   (𝑋𝑉𝑀 ∈ V)       (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑀))

Theoremfvopab4ndm 6215* Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)
𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}       𝐵𝐴 → (𝐹𝐵) = ∅)

Theoremfvmptndm 6216* Value of a function given by the "maps to" notation, outside of its domain. (Contributed by AV, 31-Dec-2020.)
𝐹 = (𝑥𝐴𝐵)       𝑋𝐴 → (𝐹𝑋) = ∅)

Theoremfvopab5 6217* The value of a function that is expressed as an ordered pair abstraction. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 11-Sep-2015.)
𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (𝐹𝐴) = (℩𝑦𝜓))

Theoremfvopab6 6218* Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐵)}    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐴𝐵 = 𝐶)       ((𝐴𝐷𝐶𝑅𝜓) → (𝐹𝐴) = 𝐶)

Theoremeqfnfv 6219* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))

Theoremeqfnfv2 6220* Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))

Theoremeqfnfv3 6221* Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥)))))

Theoremeqfnfvd 6222* Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐴)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))       (𝜑𝐹 = 𝐺)

Theoremeqfnfv2f 6223* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 6219 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)
𝑥𝐹    &   𝑥𝐺       ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))

Theoremeqfunfv 6224* Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)
((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 = 𝐺 ↔ (dom 𝐹 = dom 𝐺 ∧ ∀𝑥 ∈ dom 𝐹(𝐹𝑥) = (𝐺𝑥))))

Theoremfvreseq0 6225* 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 6226* 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 6227* Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.) (Prove shortened by AV, 4-Mar-2019.)
(((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐵𝐴) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))

Theoremfnmptfvd 6228* 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 6229* 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 6230 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 6231 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 6232* 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 6233 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 6234 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 6235 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 6236* 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 6237 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 6238 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 6239 A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)
((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → (𝐹𝐴) ∈ 𝐵)

Theoremfvimacnv 6240 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 5886 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))

Theoremfunimass3 6241 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 6240 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 6242* 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 6243* 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 6244 Alternate proof of fvimacnv 6240, based on funimass3 6241. If funimass3 6241 is ever proved directly, as opposed to using funimacnv 5884 pointwise, then the proof of funimacnv 5884 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 6245 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)))

Theoremfniniseg 6246 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 6247* Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐹 Fn 𝐴 → (𝐹𝐵) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ 𝐵})

Theoremfniniseg2 6248* 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 6249 Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∪ (𝐹𝐵)))

Theoreminpreima 6250 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 6251 Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))

Theoremrespreima 6252 The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)
(Fun 𝐹 → ((𝐹𝐵) “ 𝐴) = ((𝐹𝐴) ∩ 𝐵))

Theoremiinpreima 6253* Preimage of an intersection. (Contributed by FL, 16-Apr-2012.)
((Fun 𝐹𝐴 ≠ ∅) → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐹𝐵))

Theoremintpreima 6254* Preimage of an intersection. (Contributed by FL, 28-Apr-2012.)
((Fun 𝐹𝐴 ≠ ∅) → (𝐹 𝐴) = 𝑥𝐴 (𝐹𝑥))

Theoremfimacnv 6255 The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)
(𝐹:𝐴𝐵 → (𝐹𝐵) = 𝐴)

Theoremfimacnvinrn 6256 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 6257 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 6258* 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 6136. (Contributed by AV, 27-Jan-2020.)
(∀𝑎𝐷 (𝐹𝑎) ≠ ∅ → (𝐷 ⊆ dom 𝐹 ∧ Fun (𝐹𝐷)))

Theoremfnopfv 6259 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 6260 A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)

Theoremnelrnfvne 6261 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 6262* 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 6263* 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 6264 A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) ∈ ran 𝐹)

Theoremffvelrn 6265 A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)
((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐵)

Theoremffvelrni 6266 A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)
𝐹:𝐴𝐵       (𝐶𝐴 → (𝐹𝐶) ∈ 𝐵)

Theoremffvelrnda 6267 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐹:𝐴𝐵)       ((𝜑𝐶𝐴) → (𝐹𝐶) ∈ 𝐵)

Theoremffvelrnd 6268 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → (𝐹𝐶) ∈ 𝐵)

Theoremrexrn 6269* 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 6270* 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 6271* 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 6272* 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 6273* 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 6274* 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 5812 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 5812 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 6275* 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)))

Theoremralrnmpt 6276* 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 6277* 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 6278 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 6279 Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)))

Theoremdff3 6280* Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)
(𝐹:𝐴𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))

Theoremdff4 6281* Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)
(𝐹:𝐴𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝐹𝑦))

Theoremdffo3 6282* An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)
(𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))

Theoremdffo4 6283* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
(𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))

Theoremdffo5 6284* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
(𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦))

Theoremexfo 6285* 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 6286* Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.)
((𝐹:𝐴onto𝐵𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))

Theoremfoco2 6287 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)

Theoremfoco2OLD 6288 Obsolete proof of foco2 6287 as of 14-Jul-2021. (Contributed by Jeff Madsen, 16-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐹:𝐵𝐶𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) → 𝐹:𝐵onto𝐶)

Theoremfmpt 6289* 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 6290* 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 6291* Functionality of the mapping operation. (Contributed by NM, 19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
𝐹 = (𝑥𝐴𝐶)    &   (𝑥𝐴𝐶𝐵)       𝐹:𝐴𝐵

Theoremfmptd 6292* Domain and codomain of the mapping operation; deduction form. (Contributed by Mario Carneiro, 13-Jan-2013.)
((𝜑𝑥𝐴) → 𝐵𝐶)    &   𝐹 = (𝑥𝐴𝐵)       (𝜑𝐹:𝐴𝐶)

Theoremfmpt3d 6293* Domain and co-domain of the mapping operation; deduction form. (Contributed by Thierry Arnoux, 4-Jun-2017.)
(𝜑𝐹 = (𝑥𝐴𝐵))    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑𝐹:𝐴𝐶)

Theoremfmptdf 6294* A version of fmptd 6292 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 6295* 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 6296 A function maps to a class to which all values belong. This version of ffnfv 6295 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 6297* An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)
((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → ran 𝐹𝐵)

Theoremfrnssb 6298* 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 6299* 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 6300* Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑𝐹 = (𝑥𝐴𝐵))    &   ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ 𝐶)       (𝜑𝐹:𝐴𝐶)

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