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Theorem List for Metamath Proof Explorer - 39101-39200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrexanuz3 39101* 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 39102* Equality theorem for restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝑥𝐴𝜒} = {𝑥𝐵𝜒})

Theoremcbvmpt22 39103* 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 39104* 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 39105* Indexed union of indexed intersections. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴 = 𝑥𝐵 𝑦𝐶 𝐷       (𝑍𝑉 → (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷))

Theoremssabf 39106 Subclass of a class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴       (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))

Theoremuniexd 39107 Deduction version of the ZF Axiom of Union in class notation. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)       (𝜑 𝐴 ∈ V)

Theorempwexd 39108 Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)       (𝜑 → 𝒫 𝐴 ∈ V)

Theorempssnssi 39109 A proper subclass does not include the other class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴𝐵        ¬ 𝐵𝐴

Theoremrabidim2 39110 Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝑥 ∈ {𝑥𝐴𝜑} → 𝜑)

Theoremxpexd 39111 The Cartesian product of two sets is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐴 × 𝐵) ∈ V)

Theoremeluni2f 39112* Membership in class union. Restricted quantifier version. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴    &   𝑥𝐵       (𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)

Theoremeliin2f 39113* Membership in indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐵       (𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))

Theoremnssd 39114 Negation of subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑋𝐴)    &   (𝜑 → ¬ 𝑋𝐵)       (𝜑 → ¬ 𝐴𝐵)

Theoremiineq12dv 39115* Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐵) → 𝐶 = 𝐷)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)

Theoremsupxrcld 39116 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 39117 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 39118* Counterexample to show that the additional conditions in eliuniin 39105 and eliuniin2 39129 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 39119* Counterexample to show that the additional conditions in eliin 4523 and eliin2 39125 are actually needed. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴 = V    &   𝐵 = ∅        ¬ (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶)

Theoremeliinid 39120* Membership in an indexed intersection implies membership in any intersected set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
((𝐴 𝑥𝐵 𝐶𝑥𝐵) → 𝐴𝐶)

Theoremabssf 39121 Class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴       ({𝑥𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))

Theoremfexd 39122 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 39123 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 39124 Subclass of a restricted class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐵    &   𝑥𝐴       (𝐵 ⊆ {𝑥𝐴𝜑} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))

Theoremeliin2 39125* Membership in indexed intersection. See eliincex 39119 for a counterexample showing that the precondition 𝐵 ≠ ∅ cannot be simply dropped. eliin 4523 uses an alternative precondition (and it doesn't have a disjoint var constraint between 𝐵 and 𝑥; see eliin2f 39113). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝐵 ≠ ∅ → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))

Theoremssrab2f 39126 Subclass relation for a restricted class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴       {𝑥𝐴𝜑} ⊆ 𝐴

Theoremrestuni3 39127 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 39128 Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐵       ({𝑥𝐴𝜑} ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝜑𝑥𝐵))

Theoremeliuniin2 39129* Indexed union of indexed intersections. See eliincex 39119 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 39130 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 39131 The underlying set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 (𝐴t 𝐵) = ( 𝐴𝐵))

Theoremrestuni5 39132 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 39133 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)

Theoremne0d 39134 If a set has elements, then it is not empty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐵𝐴)       (𝜑𝐴 ≠ ∅)

Theoreminiin1 39135* Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐴 ≠ ∅ → ( 𝑥𝐴 𝐶𝐵) = 𝑥𝐴 (𝐶𝐵))

Theoreminiin2 39136* Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐴 ≠ ∅ → (𝐵 𝑥𝐴 𝐶) = 𝑥𝐴 (𝐵𝐶))

Theoremcbvrabv2 39137* A more general version of cbvrabv 3197. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑦𝐵𝜓}

Theoremiinssiin 39138 Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 𝑥𝐴 𝐵 𝑥𝐴 𝐶)

Theoremeliind2 39139* Membership in indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐵) → 𝐴𝐶)       (𝜑𝐴 𝑥𝐵 𝐶)

Theoremiinssd 39140* Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑋𝐴)    &   (𝑥 = 𝑋𝐵 = 𝐷)    &   (𝜑𝐷𝐶)       (𝜑 𝑥𝐴 𝐵𝐶)

Theoremralrimia 39141 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝜓)       (𝜑 → ∀𝑥𝐴 𝜓)

Theoremtpid2g 39142 Closed theorem form of tpid2 4302. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐴𝐵𝐴 ∈ {𝐶, 𝐴, 𝐷})

Theoremrabbida2 39143 Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Theoremiinexd 39144* The existence of an indexed union. 𝑥 is normally a free-variable parameter in 𝐵, which should be read 𝐵(𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐴 ≠ ∅)    &   (𝜑 → ∀𝑥𝐴 𝐵𝐶)       (𝜑 𝑥𝐴 𝐵 ∈ V)

Theoremrabexf 39145 Separation Scheme in terms of a restricted class abstraction. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐴    &   𝐴𝑉       {𝑥𝐴𝜑} ∈ V

Theoremrabbida3 39146 Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Theoremresexd 39147 The restriction of a set is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐴𝑉)       (𝜑 → (𝐴𝐵) ∈ V)

Theoremtpid1g 39148 Closed theorem form of tpid1 4301. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐴𝐵𝐴 ∈ {𝐴, 𝐶, 𝐷})

Theoremfnexd 39149 If the domain of a function is a set, the function is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐴𝑉)       (𝜑𝐹 ∈ V)

Theoremr19.36vf 39150 Restricted quantifier version of one direction of 19.36 2097. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜓       (∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑𝜓))

Theoremraleqd 39151 Equality deduction for restricted universal quantifier. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐴    &   𝑥𝐵    &   (𝜑𝐴 = 𝐵)       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))

Theoremralimda 39152 Deduction quantifying both antecedent and consequent. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))

Theoremiinssf 39153 Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐶       (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)

Theoremiinssdf 39154 Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐴    &   𝑥𝑋    &   𝑥𝐶    &   𝑥𝐷    &   (𝜑𝑋𝐴)    &   (𝑥 = 𝑋𝐵 = 𝐷)    &   (𝜑𝐷𝐶)       (𝜑 𝑥𝐴 𝐵𝐶)

Theoremifcli 39155 Membership (closure) of a conditional operator. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐴𝐶    &   𝐵𝐶       if(𝜑, 𝐴, 𝐵) ∈ 𝐶

Theoremresabs2i 39156 Absorption law for restriction. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐵𝐶       ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐵)

Theoremssdf2 39157 A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   ((𝜑𝑥𝐴) → 𝑥𝐵)       (𝜑𝐴𝐵)

Theoremrabssd 39158 Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   𝑥𝐵    &   ((𝜑𝑥𝐴𝜒) → 𝑥𝐵)       (𝜑 → {𝑥𝐴𝜒} ⊆ 𝐵)

Theoremssrind 39159 Add right intersection to subclass relation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴𝐵)       (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremrexnegd 39160 Minus a real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → -𝑒𝐴 = -𝐴)

Theoremrexlimd3 39161 * Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   𝑥𝜒    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)       (𝜑 → (∃𝑥𝐴 𝜓𝜒))

Theoremresabs1i 39162 Absorption law for restriction. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐵𝐶       ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵)

Theoremnel1nelin 39163 Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐴𝐵 → ¬ 𝐴 ∈ (𝐵𝐶))

Theoremnel2nelin 39164 Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐴𝐶 → ¬ 𝐴 ∈ (𝐵𝐶))

Theoremrexlimdva2 39165* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)       (𝜑 → (∃𝑥𝐴 𝜓𝜒))

Theoremnel1nelini 39166 Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
¬ 𝐴𝐵        ¬ 𝐴 ∈ (𝐵𝐶)

Theoremnel2nelini 39167 Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
¬ 𝐴𝐶        ¬ 𝐴 ∈ (𝐵𝐶)

20.32.2  Functions

Theoremunima 39168 Image of a union. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ (𝐵𝐶)) = ((𝐹𝐵) ∪ (𝐹𝐶)))

Theoremfeq1dd 39169 Equality deduction for functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐹:𝐴𝐵)       (𝜑𝐺:𝐴𝐵)

Theoremfnresdmss 39170 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 39171* Maps-to notation and cross product for a singleton function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴𝑉𝐵𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) = ({𝐴} × {𝐵}))

Theoremfvmpt2bd 39172* 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 39173* 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 39174 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 39175* 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 39176 A function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐹:𝐴𝐵𝐴 ∈ Fin) → 𝐹 ∈ Fin)

Theoremsuprnmpt 39177* 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 39178 The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐹:𝐴𝐵𝐴 ∈ Fin) → ran 𝐹 ∈ Fin)

Theoremmptelpm 39179* 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 39180* 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 39181* Restriction of the mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐵𝐴       ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶)

Theoremfouniiun 39182* 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 39183 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 39184 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 39185 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 39186* An onto mapping expressed in terms of function values. As dffo3 6372 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝐹       (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))

Theoremrnresss 39187 The range of a restriction is a subset of the whole range. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
ran (𝐴𝐵) ⊆ ran 𝐴

Theoremelrnmptd 39188* 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 39189 The range of a function in maps-to notation. Same as elrnmpt 5370, 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 39190* 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 39191* 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 39192 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 39193* 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 39194* 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 39195* Property of a surjective function. As foelrn 6376 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝐹       ((𝐹:𝐴onto𝐵𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))

Theoremnelrnres 39196 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 39197* 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 39198* Elementhood in an image set. Same as elrnmpt1s 5371, 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 39199* 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 39200* 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)

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