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Theorem List for Metamath Proof Explorer - 38301-38400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnsstr 38301 If it's not a subclass, it's not a subclass of a smaller one. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
((¬ 𝐴𝐵𝐶𝐵) → ¬ 𝐴𝐶)

Theoremrabbida 38302 Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Theoremrexanuz3 38303* Combine two different upper integer properties into one, for a single integer. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑗𝜑    &   𝑍 = (ℤ𝑀)    &   (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜒)    &   (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓)    &   (𝑘 = 𝑗 → (𝜒𝜃))    &   (𝑘 = 𝑗 → (𝜓𝜏))       (𝜑 → ∃𝑗𝑍 (𝜃𝜏))

Theoremrabeqd 38304* Equality theorem for restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝑥𝐴𝜒} = {𝑥𝐵𝜒})

Theoremcbvmpt22 38305* Rule to change the second bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑦𝐴    &   𝑤𝐴    &   𝑤𝐶    &   𝑦𝐸    &   (𝑦 = 𝑤𝐶 = 𝐸)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑤𝐵𝐸)

Theoremcbvmpt21 38306* Rule to change the first bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐵    &   𝑧𝐵    &   𝑧𝐶    &   𝑥𝐸    &   (𝑥 = 𝑧𝐶 = 𝐸)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑦𝐵𝐸)

Theoremeliuniin 38307* Indexed union of indexed intersections. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴 = 𝑥𝐵 𝑦𝐶 𝐷       (𝑍𝑉 → (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷))

Theoremssabf 38308 Subclass of a class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴       (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))

Theoremrabbia2 38309 Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))       {𝑥𝐴𝜓} = {𝑥𝐵𝜒}

Theoremuniexd 38310 Deduction version of the ZF Axiom of Union in class notation. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)       (𝜑 𝐴 ∈ V)

Theorempwexd 38311 Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)       (𝜑 → 𝒫 𝐴 ∈ V)

Theorempssnssi 38312 A proper subclass does not include the other class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴𝐵        ¬ 𝐵𝐴

Theoremrabidim2 38313 Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝑥 ∈ {𝑥𝐴𝜑} → 𝜑)

Theoremxpexd 38314 The Cartesian product of two sets is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐴 × 𝐵) ∈ V)

Theoremeluni2f 38315* Membership in class union. Restricted quantifier version. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴    &   𝑥𝐵       (𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)

Theoremeliin2f 38316* Membership in indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐵       (𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))

Theoremnssd 38317 Negation of subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑋𝐴)    &   (𝜑 → ¬ 𝑋𝐵)       (𝜑 → ¬ 𝐴𝐵)

Theoremrabidim1 38318 Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴)

Theoremiineq12dv 38319* Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐵) → 𝐶 = 𝐷)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)

Theoremrabeqif 38320 Equality theorem for restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴    &   𝑥𝐵    &   𝐴 = 𝐵       {𝑥𝐴𝜑} = {𝑥𝐵𝜑}

Theoremsupxrcld 38321 The supremum of an arbitrary set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ⊆ ℝ*)       (𝜑 → sup(𝐴, ℝ*, < ) ∈ ℝ*)

Theoremelrestd 38322 A sufficient condition for being an open set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐽𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝑋𝐽)    &   𝐴 = (𝑋𝐵)       (𝜑𝐴 ∈ (𝐽t 𝐵))

Theoremeliuniincex 38323* Counterexample to show that the additional conditions in eliuniin 38307 and eliuniin2 38335 are actually needed. Notice that the definition of 𝐴 is not even needed (it can be any class). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐵 = {∅}    &   𝐶 = ∅    &   𝐷 = ∅    &   𝑍 = V        ¬ (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷)

Theoremeliincex 38324* Counterexample to show that the additional conditions in eliin 4461 and eliin2 38330 are actually needed. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴 = V    &   𝐵 = ∅        ¬ (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶)

Theoremeliinid 38325* Membership in an indexed intersection implies membership in any intersected set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
((𝐴 𝑥𝐵 𝐶𝑥𝐵) → 𝐴𝐶)

Theoremabssf 38326 Class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴       ({𝑥𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))

Theoremfexd 38327 If the domain of a mapping is a set, the function is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝐶)       (𝜑𝐹 ∈ V)

Theoremsupxrubd 38328 A member of a set of extended reals is less than or equal to the set's supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ⊆ ℝ*)    &   (𝜑𝐵𝐴)    &   𝑆 = sup(𝐴, ℝ*, < )       (𝜑𝐵𝑆)

Theoremssrabf 38329 Subclass of a restricted class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐵    &   𝑥𝐴       (𝐵 ⊆ {𝑥𝐴𝜑} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))

Theoremeliin2 38330* Membership in indexed intersection. See eliincex 38324 for a counterexample showing that the precondition 𝐵 ≠ ∅ cannot be simply dropped. eliin 4461 uses an alternative precondition (and it doesn't have a disjoint var constraint between 𝐵 and 𝑥; see eliin2f 38316). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))

Theoremssrab2f 38331 Subclass relation for a restricted class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴       {𝑥𝐴𝜑} ⊆ 𝐴

Theoremrabeqi 38332* Equality theorem for restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴 = 𝐵       {𝑥𝐴𝜑} = {𝑥𝐵𝜑}

Theoremrestuni3 38333 The underlying set of a subspace induced by the subspace operator t. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 (𝐴t 𝐵) = ( 𝐴𝐵))

Theoremrabssf 38334 Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐵       ({𝑥𝐴𝜑} ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝜑𝑥𝐵))

Theoremeliuniin2 38335* Indexed union of indexed intersections. See eliincex 38324 for a counterexample showing that the precondition 𝐶 ≠ ∅ cannot be simply dropped. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐶    &   𝐴 = 𝑥𝐵 𝑦𝐶 𝐷       (𝐶 ≠ ∅ → (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷))

Theoremrestuni4 38336 The underlying set of a subspace induced by the t operator. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵 𝐴)       (𝜑 (𝐴t 𝐵) = 𝐵)

Theoremrestuni6 38337 The underlying set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 (𝐴t 𝐵) = ( 𝐴𝐵))

Theoremrestuni5 38338 The underlying set of a subspace induced by the t operator. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑋 = 𝐽       ((𝐽𝑉𝐴𝑋) → 𝐴 = (𝐽t 𝐴))

Theoremunirestss 38339 The union of an elementwise intersection is a subset of the underlying set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 (𝐴t 𝐵) ⊆ 𝐴)

21.31.2  Functions

Theoremunima 38340 Image of a union. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ (𝐵𝐶)) = ((𝐹𝐵) ∪ (𝐹𝐶)))

Theoremfeq1dd 38341 Equality deduction for functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐹:𝐴𝐵)       (𝜑𝐺:𝐴𝐵)

Theoremfnresdmss 38342 A function does not change when restricted to a set that contains its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐹 Fn 𝐴𝐴𝐵) → (𝐹𝐵) = 𝐹)

Theoremfmptsnxp 38343* Maps-to notation and cross product for a singleton function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴𝑉𝐵𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) = ({𝐴} × {𝐵}))

Theoremmptex2 38344* If a class given as a map-to notation is a set, it's image values are set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝑡𝐴𝐵):𝐴𝐶)       ((𝜑𝑡𝐴) → 𝐵𝐶)

Theoremfvmpt2bd 38345* Value of a function given by the "maps to" notation. Deduction version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹 = (𝑥𝐴𝐵))       ((𝜑𝑥𝐴𝐵𝐶) → (𝐹𝑥) = 𝐵)

Theoremrnmptfi 38346* The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴 = (𝑥𝐵𝐶)       (𝐵 ∈ Fin → ran 𝐴 ∈ Fin)

Theoremfresin2 38347 Restriction of a function with respect to the intersection with its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐹:𝐴𝐵 → (𝐹 ↾ (𝐶𝐴)) = (𝐹𝐶))

Theoremrnmptc 38348* Range of a constant function in map to notation. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝐶)    &   (𝜑𝐴 ≠ ∅)       (𝜑 → ran 𝐹 = {𝐵})

Theoremffi 38349 A function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐹:𝐴𝐵𝐴 ∈ Fin) → 𝐹 ∈ Fin)

Theoremsuprnmpt 38350* An explicit bound for the range of a bounded function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ≠ ∅)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)    &   𝐹 = (𝑥𝐴𝐵)    &   𝐶 = sup(ran 𝐹, ℝ, < )       (𝜑 → (𝐶 ∈ ℝ ∧ ∀𝑥𝐴 𝐵𝐶))

Theoremrnffi 38351 The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐹:𝐴𝐵𝐴 ∈ Fin) → ran 𝐹 ∈ Fin)

Theoremmptelpm 38352* A function in maps-to notation is a partial map . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
((𝜑𝑥𝐴) → 𝐵𝐶)    &   (𝜑𝐴𝐷)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)       (𝜑 → (𝑥𝐴𝐵) ∈ (𝐶pm 𝐷))

Theoremrnmptpr 38353* Range of a function defined on an unordered pair. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶)    &   (𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑥 = 𝐵𝐶 = 𝐸)       (𝜑 → ran 𝐹 = {𝐷, 𝐸})

Theoremresmpti 38354* Restriction of the mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐵𝐴       ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶)

Theoremfouniiun 38355* Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐹:𝐴onto𝐵 𝐵 = 𝑥𝐴 (𝐹𝑥))

Theoremf1oeq2d 38356 Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐶))

Theoremrnresun 38357 Distribution law for range of a restriction over a union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
ran (𝐹 ↾ (𝐴𝐵)) = (ran (𝐹𝐴) ∪ ran (𝐹𝐵))

Theoremf1oeq1d 38358 Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐹 = 𝐺)       (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐺:𝐴1-1-onto𝐵))

Theoremdffo3f 38359* An onto mapping expressed in terms of function values. As dffo3 6282 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝐹       (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))

Theoremrnresss 38360 The range of a restriction is a subset of the whole range. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
ran (𝐴𝐵) ⊆ ran 𝐴

Theoremelrnmptd 38361* The range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑 → ∃𝑥𝐴 𝐶 = 𝐵)    &   (𝜑𝐶𝑉)       (𝜑𝐶 ∈ ran 𝐹)

Theoremelrnmptf 38362 The range of a function in maps-to notation. Same as elrnmpt 5293, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝐶    &   𝐹 = (𝑥𝐴𝐵)       (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))

Theoremrnmptssrn 38363* Inclusion relation for two ranges expressed in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑥𝐴) → ∃𝑦𝐶 𝐵 = 𝐷)       (𝜑 → ran (𝑥𝐴𝐵) ⊆ ran (𝑦𝐶𝐷))

Theoremdisjf1 38364* A 1 to 1 mapping built from disjoint, nonempty sets . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)    &   (𝜑Disj 𝑥𝐴 𝐵)       (𝜑𝐹:𝐴1-1𝑉)

Theoremrnsnf 38365 The range of a function whose domain is a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:{𝐴}⟶𝐵)       (𝜑 → ran 𝐹 = {(𝐹𝐴)})

Theoremwessf1ornlem 38366* Given a function 𝐹 on a well ordered domain 𝐴 there exists a subset of 𝐴 such that 𝐹 restricted to such subset is injective and onto the range of 𝐹 (without using the axiom of choice). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐴𝑉)    &   (𝜑𝑅 We 𝐴)    &   𝐺 = (𝑦 ∈ ran 𝐹 ↦ (𝑥 ∈ (𝐹 “ {𝑦})∀𝑧 ∈ (𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥))       (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto→ran 𝐹)

Theoremwessf1orn 38367* Given a function 𝐹 on a well ordered domain 𝐴 there exists a subset of 𝐴 such that 𝐹 restricted to such subset is injective and onto the range of 𝐹 (without using the axiom of choice). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐴𝑉)    &   (𝜑𝑅 We 𝐴)       (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto→ran 𝐹)

Theoremfoelrnf 38368* Property of a surjective function. As foelrn 6286 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝐹       ((𝐹:𝐴onto𝐵𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))

Theoremnelrnres 38369 If 𝐴 is not in the range, it is not in the range of any restriction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐴 ∈ ran 𝐵 → ¬ 𝐴 ∈ ran (𝐵𝐶))

Theoremdisjrnmpt2 38370* Disjointness of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐹 = (𝑥𝐴𝐵)       (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ ran 𝐹 𝑦)

Theoremelrnmpt1sf 38371* Elementhood in an image set. Same as elrnmpt1s 5294, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝐶    &   𝐹 = (𝑥𝐴𝐵)    &   (𝑥 = 𝐷𝐵 = 𝐶)       ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)

Theoremfouniiun0 38372* Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐹:𝐴onto→(𝐵 ∪ {∅}) → 𝐵 = 𝑥𝐴 (𝐹𝑥))

Theoremdisjf1o 38373* A bijection built from disjoint sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑Disj 𝑥𝐴 𝐵)    &   𝐶 = {𝑥𝐴𝐵 ≠ ∅}    &   𝐷 = (ran 𝐹 ∖ {∅})       (𝜑 → (𝐹𝐶):𝐶1-1-onto𝐷)

Theoremfompt 38374* Express being onto for a mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐹 = (𝑥𝐴𝐶)       (𝐹:𝐴onto𝐵 ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶))

Theoremdisjinfi 38375* Only a finite number of disjoint sets can have a non empty intersection with a finite set 𝐶 (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑Disj 𝑥𝐴 𝐵)    &   (𝜑𝐶 ∈ Fin)       (𝜑 → {𝑥𝐴 ∣ (𝐵𝐶) ≠ ∅} ∈ Fin)

Theoremfvovco 38376 Value of the composition of an operator, with a given function. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐹:𝑋⟶(𝑉 × 𝑊))    &   (𝜑𝑌𝑋)       (𝜑 → ((𝑂𝐹)‘𝑌) = ((1st ‘(𝐹𝑌))𝑂(2nd ‘(𝐹𝑌))))

Theoremssnnf1octb 38377* There exists a bijection between a subset of and a given nonempty countable set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓1-1-onto𝐴))

Theoremmapdm0 38378 The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝐴𝑉 → (𝐴𝑚 ∅) = {∅})

Theoremnnf1oxpnn 38379 There is a bijection between the set of positive integers and the Cartesian product of the set of positive integers with itself. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑓 𝑓:ℕ–1-1-onto→(ℕ × ℕ)

Theoremrnmptssd 38380* The range of an operation given by the "maps to" notation as a subset. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑥𝜑    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → ran 𝐹𝐶)

Theoremprojf1o 38381* A biijection from a set to a projection in a two dimensional space. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)    &   𝐹 = (𝑥𝐵 ↦ ⟨𝐴, 𝑥⟩)       (𝜑𝐹:𝐵1-1-onto→({𝐴} × 𝐵))

Theoremfvmap 38382 Function value for a member of a set exponentiation. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐹 ∈ (𝐴𝑚 𝐵))    &   (𝜑𝐶𝐵)       (𝜑 → (𝐹𝐶) ∈ 𝐴)

Theoremmapsnd 38383* The value of set exponentiation with a singleton exponent. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐴𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}})

Theoremfvixp2 38384* Projection of a factor of an indexed Cartesian product. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
((𝐹X𝑥𝐴 𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)

Theoremfidmfisupp 38385 A function with a finite domain is finitely supported. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐹:𝐷𝑅)    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑍𝑉)       (𝜑𝐹 finSupp 𝑍)

Theoremmapsnend 38386 Set exponentiation to a singleton exponent is equinumerous to its base. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐴𝑚 {𝐵}) ≈ 𝐴)

Theoremchoicefi 38387* For a finite set, a choice function exists, without using the axiom of choice. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)       (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))

Theoremmpct 38388 The exponentiation of a countable set to a finite set is countable. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ≼ ω)    &   (𝜑𝐵 ∈ Fin)       (𝜑 → (𝐴𝑚 𝐵) ≼ ω)

Theoremcnmetcoval 38389 Value of the distance function of the metric space of complex numbers, composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝐷 = (abs ∘ − )    &   (𝜑𝐹:𝐴⟶(ℂ × ℂ))    &   (𝜑𝐵𝐴)       (𝜑 → ((𝐷𝐹)‘𝐵) = (abs‘((1st ‘(𝐹𝐵)) − (2nd ‘(𝐹𝐵)))))

Theoremfcomptss 38390* Express composition of two functions as a maps-to applying both in sequence. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐵𝐶)    &   (𝜑𝐺:𝐶𝐷)       (𝜑 → (𝐺𝐹) = (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))))

Theoremelmapsnd 38391 Membership in a set exponentiated to a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹 Fn {𝐴})    &   (𝜑𝐵𝑉)    &   (𝜑 → (𝐹𝐴) ∈ 𝐵)       (𝜑𝐹 ∈ (𝐵𝑚 {𝐴}))

Theoremmapss2 38392 Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑍)    &   (𝜑𝐶 ≠ ∅)       (𝜑 → (𝐴𝐵 ↔ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶)))

Theoremfsneq 38393 Equality condition for two functions defined on a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   𝐵 = {𝐴}    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑𝐺 Fn 𝐵)       (𝜑 → (𝐹 = 𝐺 ↔ (𝐹𝐴) = (𝐺𝐴)))

Theoremdifmap 38394 Difference of two sets exponentiations. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑍)    &   (𝜑𝐶 ≠ ∅)       (𝜑 → ((𝐴𝐵) ↑𝑚 𝐶) ⊆ ((𝐴𝑚 𝐶) ∖ (𝐵𝑚 𝐶)))

Theoremunirnmap 38395 Given a subset of a set exponentiation, the base set can be restricted. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝑋 ⊆ (𝐵𝑚 𝐴))       (𝜑𝑋 ⊆ (ran 𝑋𝑚 𝐴))

Theoreminmap 38396 Intersection of two sets exponentiations. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑍)       (𝜑 → ((𝐴𝑚 𝐶) ∩ (𝐵𝑚 𝐶)) = ((𝐴𝐵) ↑𝑚 𝐶))

Theoremfcoss 38397 Composition of two mappings. Similar to fco 5971, but with a weaker condition on the domain of 𝐹. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)    &   (𝜑𝐺:𝐷𝐶)       (𝜑 → (𝐹𝐺):𝐷𝐵)

Theoremfsneqrn 38398 Equality condition for two functions defined on a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   𝐵 = {𝐴}    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑𝐺 Fn 𝐵)       (𝜑 → (𝐹 = 𝐺 ↔ (𝐹𝐴) ∈ ran 𝐺))

Theoremdifmapsn 38399 Difference of two sets exponentiatiated to a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑍)       (𝜑 → ((𝐴𝑚 {𝐶}) ∖ (𝐵𝑚 {𝐶})) = ((𝐴𝐵) ↑𝑚 {𝐶}))

Theoremmapssbi 38400 Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑍)    &   (𝜑𝐶 ≠ ∅)       (𝜑 → (𝐴𝐵 ↔ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶)))

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