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Theorem List for Metamath Proof Explorer - 41301-41400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreximdd 41301 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑥𝜑    &   ((𝜑𝑥𝐴𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴 𝜓)       (𝜑 → ∃𝑥𝐴 𝜒)
 
Theoremunfid 41302 The union of two finite sets is finite. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)       (𝜑 → (𝐴𝐵) ∈ Fin)
 
20.36.2  Functions
 
Theoremfeq1dd 41303 Equality deduction for functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐹:𝐴𝐵)       (𝜑𝐺:𝐴𝐵)
 
Theoremfnresdmss 41304 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 41305* Maps-to notation and cross product for a singleton function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴𝑉𝐵𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) = ({𝐴} × {𝐵}))
 
Theoremfvmpt2bd 41306* 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 41307* 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 41308 Restriction of a function with respect to the intersection with its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐹:𝐴𝐵 → (𝐹 ↾ (𝐶𝐴)) = (𝐹𝐶))
 
Theoremffi 41309 A function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐹:𝐴𝐵𝐴 ∈ Fin) → 𝐹 ∈ Fin)
 
Theoremsuprnmpt 41310* 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 41311 The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐹:𝐴𝐵𝐴 ∈ Fin) → ran 𝐹 ∈ Fin)
 
Theoremmptelpm 41312* 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 41313* 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 41314* Restriction of the mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐵𝐴       ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶)
 
Theoremfouniiun 41315* Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐹:𝐴onto𝐵 𝐵 = 𝑥𝐴 (𝐹𝑥))
 
Theoremrnresun 41316 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 41317 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 41318* An onto mapping expressed in terms of function values. As dffo3 6861 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝐹       (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
 
Theoremrnresss 41319 The range of a restriction is a subset of the whole range. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
ran (𝐴𝐵) ⊆ ran 𝐴
 
Theoremelrnmptd 41320* 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 41321 The range of a function in maps-to notation. Same as elrnmpt 5822, 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 41322* Inclusion relation for two ranges expressed in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   ((𝜑𝑥𝐴) → ∃𝑦𝐶 𝐵 = 𝐷)       (𝜑 → ran (𝑥𝐴𝐵) ⊆ ran (𝑦𝐶𝐷))
 
Theoremdisjf1 41323* 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 41324 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 41325* 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 41326* 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 41327* Property of a surjective function. As foelrn 6865 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝐹       ((𝐹:𝐴onto𝐵𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))
 
Theoremnelrnres 41328 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 41329* Disjointness of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐹 = (𝑥𝐴𝐵)       (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ ran 𝐹 𝑦)
 
Theoremelrnmpt1sf 41330* Elementhood in an image set. Same as elrnmpt1s 5823, 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 41331* 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 41332* 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 41333* Express being onto for a mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐹 = (𝑥𝐴𝐶)       (𝐹:𝐴onto𝐵 ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶))
 
Theoremdisjinfi 41334* Only a finite number of disjoint sets can have a nonempty 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 41335 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 41336* 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))
 
Theoremnnf1oxpnn 41337 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 41338* 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 41339* 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 41340 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)
 
Theoremfvixp2 41341* Projection of a factor of an indexed Cartesian product. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
((𝐹X𝑥𝐴 𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
 
Theoremfidmfisupp 41342 A function with a finite domain is finitely supported. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐹:𝐷𝑅)    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑍𝑉)       (𝜑𝐹 finSupp 𝑍)
 
Theoremchoicefi 41343* 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 41344 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 41345 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 41346* 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 41347 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 41348 Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑍)    &   (𝜑𝐶 ≠ ∅)       (𝜑 → (𝐴𝐵 ↔ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)))
 
Theoremfsneq 41349 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 41350 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 41351 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 41352 Intersection of two sets exponentiations. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑍)       (𝜑 → ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) = ((𝐴𝐵) ↑m 𝐶))
 
Theoremfcoss 41353 Composition of two mappings. Similar to fco 6525, 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 41354 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 41355 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 41356 Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑍)    &   (𝜑𝐶 ≠ ∅)       (𝜑 → (𝐴𝐵 ↔ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)))
 
Theoremunirnmapsn 41357 Equality theorem for a subset of a set exponentiation, where the exponent is a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝐶 = {𝐴}    &   (𝜑𝑋 ⊆ (𝐵m 𝐶))       (𝜑𝑋 = (ran 𝑋m 𝐶))
 
Theoremiunmapss 41358* The indexed union of set exponentiations is a subset of the set exponentiation of the indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)       (𝜑 𝑥𝐴 (𝐵m 𝐶) ⊆ ( 𝑥𝐴 𝐵m 𝐶))
 
Theoremssmapsn 41359* A subset 𝐶 of a set exponentiation to a singleton, is its projection 𝐷 exponentiated to the singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑓𝐷    &   (𝜑𝐴𝑉)    &   (𝜑𝐶 ⊆ (𝐵m {𝐴}))    &   𝐷 = 𝑓𝐶 ran 𝑓       (𝜑𝐶 = (𝐷m {𝐴}))
 
Theoremiunmapsn 41360* The indexed union of set exponentiations to a singleton is equal to the set exponentiation of the indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   (𝜑𝐶𝑍)       (𝜑 𝑥𝐴 (𝐵m {𝐶}) = ( 𝑥𝐴 𝐵m {𝐶}))
 
Theoremabsfico 41361 Mapping domain and codomain of the absolute value function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
abs:ℂ⟶(0[,)+∞)
 
Theoremicof 41362 The set of left-closed right-open intervals of extended reals maps to subsets of extended reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
[,):(ℝ* × ℝ*)⟶𝒫 ℝ*
 
Theoremrnmpt0 41363* The range of a function in maps-to notation is empty if and only if its domain is empty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   𝐹 = (𝑥𝐴𝐵)       (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
 
Theoremrnmptn0 41364* The range of a function in maps-to notation is nonempty if the domain is nonempty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐴 ≠ ∅)       (𝜑 → ran 𝐹 ≠ ∅)
 
Theoremelpmrn 41365 The range of a partial function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝐹 ∈ (𝐴pm 𝐵) → ran 𝐹𝐴)
 
Theoremimaexi 41366 The image of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴𝑉       (𝐴𝐵) ∈ V
 
Theoremaxccdom 41367* Relax the constraint on ax-cc to dominance instead of equinumerosity. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑋 ≼ ω)    &   ((𝜑𝑧𝑋) → 𝑧 ≠ ∅)       (𝜑 → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧𝑋 (𝑓𝑧) ∈ 𝑧))
 
Theoremdmmptdf 41368* The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝐴 = (𝑥𝐵𝐶)    &   ((𝜑𝑥𝐵) → 𝐶𝑉)       (𝜑 → dom 𝐴 = 𝐵)
 
Theoremelpmi2 41369 The domain of a partial function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝐹 ∈ (𝐴pm 𝐵) → dom 𝐹𝐵)
 
Theoremdmrelrnrel 41370* A relation preserving function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))    &   (𝜑𝐵𝐴)    &   (𝜑𝐶𝐴)    &   (𝜑𝐵𝑅𝐶)       (𝜑 → (𝐹𝐵)𝑆(𝐹𝐶))
 
Theoremfco3 41371 Functionality of a composition. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑 → Fun 𝐹)    &   (𝜑 → Fun 𝐺)       (𝜑 → (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran 𝐹)
 
Theoremfvcod 41372 Value of a function composition. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑 → Fun 𝐺)    &   (𝜑𝐴 ∈ dom 𝐺)    &   𝐻 = (𝐹𝐺)       (𝜑 → (𝐻𝐴) = (𝐹‘(𝐺𝐴)))
 
Theoremfreld 41373 A mapping is a relation. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → Rel 𝐹)
 
Theoremelrnmpoid 41374* Membership in the range of an operation class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝑥𝐴𝑦𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝐶𝑉) → (𝑥𝐹𝑦) ∈ ran 𝐹)
 
Theoremaxccd 41375* An alternative version of the axiom of countable choice. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ≈ ω)    &   ((𝜑𝑥𝐴) → 𝑥 ≠ ∅)       (𝜑 → ∃𝑓𝑥𝐴 (𝑓𝑥) ∈ 𝑥)
 
Theoremaxccd2 41376* An alternative version of the axiom of countable choice. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ≼ ω)    &   ((𝜑𝑥𝐴) → 𝑥 ≠ ∅)       (𝜑 → ∃𝑓𝑥𝐴 (𝑓𝑥) ∈ 𝑥)
 
Theoremfunimassd 41377* Sufficient condition for the image of a function being a subclass. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑 → Fun 𝐹)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)       (𝜑 → (𝐹𝐴) ⊆ 𝐵)
 
Theoremfimassd 41378 The image of a class is a subset of its codomain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐹𝑋) ⊆ 𝐵)
 
Theoremfeqresmptf 41379* Express a restricted function as a mapping. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐹    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ (𝐹𝑥)))
 
Theoremelrnmpt1d 41380 Elementhood in an image set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝑥𝐴)    &   (𝜑𝐵𝑉)       (𝜑𝐵 ∈ ran 𝐹)
 
Theoremdmresss 41381 The domain of a restriction is a subset of the original domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
dom (𝐴𝐵) ⊆ dom 𝐴
 
Theoremdmmptssf 41382 The domain of a mapping is a subset of its base class. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐴    &   𝐹 = (𝑥𝐴𝐵)       dom 𝐹𝐴
 
Theoremdmmptdf2 41383 The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   𝑥𝐵    &   𝐴 = (𝑥𝐵𝐶)    &   ((𝜑𝑥𝐵) → 𝐶𝑉)       (𝜑 → dom 𝐴 = 𝐵)
 
Theoremdmuz 41384 Domain of the upper integers function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
dom ℤ = ℤ
 
Theoremfmptd2f 41385* Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
 
Theoremmpteq1df 41386 An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
 
Theoremmptexf 41387 If the domain of a function given by maps-to notation is a set, the function is a set. Inference version of mptexg 6976. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐴    &   𝐴 ∈ V       (𝑥𝐴𝐵) ∈ V
 
Theoremfvmpt4 41388* Value of a function given by the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
 
Theoremfmptf 41389* Functionality of the mapping operation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐵    &   𝐹 = (𝑥𝐴𝐶)       (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
 
Theoremresimass 41390 The image of a restriction is a subset of the original image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
((𝐴𝐵) “ 𝐶) ⊆ (𝐴𝐶)
 
Theoremmptssid 41391 The mapping operation expressed with its actual domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐴    &   𝐶 = {𝑥𝐴𝐵 ∈ V}       (𝑥𝐴𝐵) = (𝑥𝐶𝐵)
 
Theoremmptfnd 41392 The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)
𝑥𝐴    &   𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
 
Theoremmpteq12da 41393 An equality inference for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 = 𝐶)    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 
Theoremrnmptlb 41394* Boundness below of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
 
Theoremrnmptbddlem 41395* Boundness of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
 
Theoremrnmptbdd 41396* Boundness of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
 
Theoremmptima2 41397* Image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐶𝐴)       (𝜑 → ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥𝐶𝐵))
 
Theoremfunimaeq 41398* Membership relation for the values of a function whose image is a subclass. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑 → Fun 𝐹)    &   (𝜑 → Fun 𝐺)    &   (𝜑𝐴 ⊆ dom 𝐹)    &   (𝜑𝐴 ⊆ dom 𝐺)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))       (𝜑 → (𝐹𝐴) = (𝐺𝐴))
 
Theoremrnmptssf 41399* The range of an operation given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐶    &   𝐹 = (𝑥𝐴𝐵)       (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
 
Theoremrnmptbd2lem 41400* Boundness below of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))
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