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Theorem List for Metamath Proof Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisores2 7101 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, (𝑆 ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
 
Theoremisores1 7102 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵))
 
Theoremisores3 7103 Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)
((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾𝐴𝑋 = (𝐻𝐾)) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋))
 
Theoremisotr 7104 Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑆, 𝑇 (𝐵, 𝐶)) → (𝐺𝐻) Isom 𝑅, 𝑇 (𝐴, 𝐶))
 
Theoremisomin 7105 Isomorphisms preserve minimal elements. Note that (𝑅 “ {𝐷}) is Takeuti and Zaring's idiom for the initial segment {𝑥𝑥𝑅𝐷}. Proposition 6.31(1) of [TakeutiZaring] p. 33. (Contributed by NM, 19-Apr-2004.)
((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → ((𝐶 ∩ (𝑅 “ {𝐷})) = ∅ ↔ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅))
 
Theoremisoini 7106 Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)
((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝐷}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})))
 
Theoremisoini2 7107 Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)
𝐶 = (𝐴 ∩ (𝑅 “ {𝑋}))    &   𝐷 = (𝐵 ∩ (𝑆 “ {(𝐻𝑋)}))       ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷))
 
Theoremisofrlem 7108* Lemma for isofr 7110. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
(𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑 → (𝐻𝑥) ∈ V)       (𝜑 → (𝑆 Fr 𝐵𝑅 Fr 𝐴))
 
Theoremisoselem 7109* Lemma for isose 7111. (Contributed by Mario Carneiro, 23-Jun-2015.)
(𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑 → (𝐻𝑥) ∈ V)       (𝜑 → (𝑅 Se 𝐴𝑆 Se 𝐵))
 
Theoremisofr 7110 An isomorphism preserves well-foundedness. Proposition 6.32(1) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Fr 𝐴𝑆 Fr 𝐵))
 
Theoremisose 7111 An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴𝑆 Se 𝐵))
 
Theoremisofr2 7112 A weak form of isofr 7110 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)
((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵𝑉) → (𝑆 Fr 𝐵𝑅 Fr 𝐴))
 
Theoremisopolem 7113 Lemma for isopo 7114. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵𝑅 Po 𝐴))
 
Theoremisopo 7114 An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Po 𝐴𝑆 Po 𝐵))
 
Theoremisosolem 7115 Lemma for isoso 7116. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵𝑅 Or 𝐴))
 
Theoremisoso 7116 An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Or 𝐴𝑆 Or 𝐵))
 
Theoremisowe 7117 An isomorphism preserves well-ordering. Proposition 6.32(3) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 We 𝐴𝑆 We 𝐵))
 
Theoremisowe2 7118* A weak form of isowe 7117 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)
((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → (𝑆 We 𝐵𝑅 We 𝐴))
 
Theoremf1oiso 7119* Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by NM, 30-Apr-2004.)
((𝐻:𝐴1-1-onto𝐵𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
 
Theoremf1oiso2 7120* Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. (Contributed by Mario Carneiro, 9-Mar-2013.)
𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦))}       (𝐻:𝐴1-1-onto𝐵𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
 
Theoremf1owe 7121* Well-ordering of isomorphic relations. (Contributed by NM, 4-Mar-1997.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹𝑥)𝑆(𝐹𝑦)}       (𝐹:𝐴1-1-onto𝐵 → (𝑆 We 𝐵𝑅 We 𝐴))
 
Theoremweniso 7122 A set-like well-ordering has no nontrivial automorphisms. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
((𝑅 We 𝐴𝑅 Se 𝐴𝐹 Isom 𝑅, 𝑅 (𝐴, 𝐴)) → 𝐹 = ( I ↾ 𝐴))
 
Theoremweisoeq 7123 Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso 7701. (Contributed by Mario Carneiro, 25-Jun-2015.)
(((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺)
 
Theoremweisoeq2 7124 Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso2 7702. (Contributed by Mario Carneiro, 25-Jun-2015.)
(((𝑆 We 𝐵𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺)
 
Theoremknatar 7125* The Knaster-Tarski theorem says that every monotone function over a complete lattice has a (least) fixpoint. Here we specialize this theorem to the case when the lattice is the powerset lattice 𝒫 𝐴. (Contributed by Mario Carneiro, 11-Jun-2015.)
𝑋 = {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹𝑧) ⊆ 𝑧}       ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝑋𝐴 ∧ (𝐹𝑋) = 𝑋))
 
2.3.16  Cantor's Theorem
 
Theoremcanth 7126 No set 𝐴 is equinumerous to its power set (Cantor's theorem), i.e., no function can map 𝐴 onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 8722. Note that 𝐴 must be a set: this theorem does not hold when 𝐴 is too large to be a set; see ncanth 7127 for a counterexample. (Use nex 1807 if you want the form ¬ ∃𝑓𝑓:𝐴onto→𝒫 𝐴.) (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
𝐴 ∈ V        ¬ 𝐹:𝐴onto→𝒫 𝐴
 
Theoremncanth 7127 Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 5183). Specifically, the identity function maps the universe onto its power class. Compare canth 7126 that works for sets.

This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3679): 𝒫 V, being a class, cannot contain proper classes, so it is no larger than V, which is why the identity function "succeeds" in being surjective onto 𝒫 V (see pwv 4793). See also the remark in ru 3679 about NF, in which Cantor's theorem fails for sets that are "too large". This theorem gives some intuition behind that failure: in NF the universal class is a set, and it equals its own power set. (Contributed by NM, 29-Jun-2004.) (Proof shortened by BJ, 29-Dec-2023.)

I :V–onto→𝒫 V
 
2.3.17  Restricted iota (description binder)
 
Syntaxcrio 7128 Extend class notation with restricted description binder.
class (𝑥𝐴 𝜑)
 
Definitiondf-riota 7129 Define restricted description binder. In case there is no unique 𝑥 such that (𝑥𝐴𝜑) holds, it evaluates to the empty set. See also comments for df-iota 6297. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 2-Sep-2018.)
(𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
 
Theoremriotaeqdv 7130* Formula-building deduction for iota. (Contributed by NM, 15-Sep-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐵 𝜓))
 
Theoremriotabidv 7131* Formula-building deduction for restricted iota. (Contributed by NM, 15-Sep-2011.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
 
Theoremriotaeqbidv 7132* Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐵 𝜒))
 
Theoremriotaex 7133 Restricted iota is a set. (Contributed by NM, 15-Sep-2011.)
(𝑥𝐴 𝜓) ∈ V
 
Theoremriotav 7134 An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
(𝑥 ∈ V 𝜑) = (℩𝑥𝜑)
 
Theoremriotauni 7135 Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)
(∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = {𝑥𝐴𝜑})
 
Theoremnfriota1 7136* The abstraction variable in a restricted iota descriptor isn't free. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥(𝑥𝐴 𝜑)
 
Theoremnfriotadw 7137* Deduction version of nfriota 7142 with a disjoint variable condition, which contrary to nfriotad 7141 does not require ax-13 2372. (Contributed by NM, 18-Feb-2013.) (Revised by Gino Giotto, 26-Jan-2024.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑𝑥𝐴)       (𝜑𝑥(𝑦𝐴 𝜓))
 
Theoremcbvriotaw 7138* Change bound variable in a restricted description binder. Version of cbvriota 7143 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 18-Mar-2013.) (Revised by Gino Giotto, 26-Jan-2024.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
 
Theoremcbvriotavw 7139* Change bound variable in a restricted description binder. Version of cbvriotav 7144 with a disjoint variable condition, which requires fewer axioms . (Contributed by NM, 18-Mar-2013.) (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
 
TheoremcbvriotavwOLD 7140* Obsolete version of cbvriotavw 7139 as of 30-Sep-2024. (Contributed by NM, 18-Mar-2013.) (Revised by Gino Giotto, 26-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
 
Theoremnfriotad 7141 Deduction version of nfriota 7142. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfriotadw 7137 when possible. (Contributed by NM, 18-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑𝑥𝐴)       (𝜑𝑥(𝑦𝐴 𝜓))
 
Theoremnfriota 7142* A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.)
𝑥𝜑    &   𝑥𝐴       𝑥(𝑦𝐴 𝜑)
 
Theoremcbvriota 7143* Change bound variable in a restricted description binder. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbvriotaw 7138 when possible. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
 
Theoremcbvriotav 7144* Change bound variable in a restricted description binder. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbvriotavw 7139 when possible. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
 
Theoremcsbriota 7145* Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 2-Sep-2018.)
𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑)
 
Theoremriotacl2 7146 Membership law for "the unique element in 𝐴 such that 𝜑". (Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
(∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
 
Theoremriotacl 7147* Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
(∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
 
Theoremriotasbc 7148 Substitution law for descriptions. Compare iotasbc 41597. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
(∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)
 
Theoremriotabidva 7149* Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 3379 analog.) (Contributed by NM, 17-Jan-2012.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
 
Theoremriotabiia 7150 Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 3373 analog.) (Contributed by NM, 16-Jan-2012.)
(𝑥𝐴 → (𝜑𝜓))       (𝑥𝐴 𝜑) = (𝑥𝐴 𝜓)
 
Theoremriota1 7151* Property of restricted iota. Compare iota1 6316. (Contributed by Mario Carneiro, 15-Oct-2016.)
(∃!𝑥𝐴 𝜑 → ((𝑥𝐴𝜑) ↔ (𝑥𝐴 𝜑) = 𝑥))
 
Theoremriota1a 7152 Property of iota. (Contributed by NM, 23-Aug-2011.)
((𝑥𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
 
Theoremriota2df 7153* A deduction version of riota2f 7154. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝜑    &   (𝜑𝑥𝐵)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑𝐵𝐴)    &   ((𝜑𝑥 = 𝐵) → (𝜓𝜒))       ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝜒 ↔ (𝑥𝐴 𝜓) = 𝐵))
 
Theoremriota2f 7154* This theorem shows a condition that allows us to represent a descriptor with a class expression 𝐵. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐵    &   𝑥𝜓    &   (𝑥 = 𝐵 → (𝜑𝜓))       ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
 
Theoremriota2 7155* This theorem shows a condition that allows us to represent a descriptor with a class expression 𝐵. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 10-Dec-2016.)
(𝑥 = 𝐵 → (𝜑𝜓))       ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
 
Theoremriotaeqimp 7156* If two restricted iota descriptors for an equality are equal, then the terms of the equality are equal. (Contributed by AV, 6-Dec-2020.)
𝐼 = (𝑎𝑉 𝑋 = 𝐴)    &   𝐽 = (𝑎𝑉 𝑌 = 𝐴)    &   (𝜑 → ∃!𝑎𝑉 𝑋 = 𝐴)    &   (𝜑 → ∃!𝑎𝑉 𝑌 = 𝐴)       ((𝜑𝐼 = 𝐽) → 𝑋 = 𝑌)
 
Theoremriotaprop 7157* Properties of a restricted definite description operator. (Contributed by NM, 23-Nov-2013.)
𝑥𝜓    &   𝐵 = (𝑥𝐴 𝜑)    &   (𝑥 = 𝐵 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 → (𝐵𝐴𝜓))
 
Theoremriota5f 7158* A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝑥𝐵)    &   (𝜑𝐵𝐴)    &   ((𝜑𝑥𝐴) → (𝜓𝑥 = 𝐵))       (𝜑 → (𝑥𝐴 𝜓) = 𝐵)
 
Theoremriota5 7159* A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.)
(𝜑𝐵𝐴)    &   ((𝜑𝑥𝐴) → (𝜓𝑥 = 𝐵))       (𝜑 → (𝑥𝐴 𝜓) = 𝐵)
 
Theoremriotass2 7160* Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)
(((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜓))
 
Theoremriotass 7161* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜑))
 
Theoremmoriotass 7162* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.)
((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝐵 𝜑) → (𝑥𝐴 𝜑) = (𝑥𝐵 𝜑))
 
Theoremsnriota 7163 A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)
(∃!𝑥𝐴 𝜑 → {𝑥𝐴𝜑} = {(𝑥𝐴 𝜑)})
 
Theoremriotaxfrd 7164* Change the variable 𝑥 in the expression for "the unique 𝑥 such that 𝜓 " to another variable 𝑦 contained in expression 𝐵. Use reuhypd 5286 to eliminate the last hypothesis. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑦𝐶    &   ((𝜑𝑦𝐴) → 𝐵𝐴)    &   ((𝜑 ∧ (𝑦𝐴 𝜒) ∈ 𝐴) → 𝐶𝐴)    &   (𝑥 = 𝐵 → (𝜓𝜒))    &   (𝑦 = (𝑦𝐴 𝜒) → 𝐵 = 𝐶)    &   ((𝜑𝑥𝐴) → ∃!𝑦𝐴 𝑥 = 𝐵)       ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝑥𝐴 𝜓) = 𝐶)
 
Theoremeusvobj2 7165* Specify the same property in two ways when class 𝐵(𝑦) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
𝐵 ∈ V       (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑥 = 𝐵))
 
Theoremeusvobj1 7166* Specify the same object in two ways when class 𝐵(𝑦) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
𝐵 ∈ V       (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (℩𝑥𝑦𝐴 𝑥 = 𝐵) = (℩𝑥𝑦𝐴 𝑥 = 𝐵))
 
Theoremf1ofveu 7167* There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.)
((𝐹:𝐴1-1-onto𝐵𝐶𝐵) → ∃!𝑥𝐴 (𝐹𝑥) = 𝐶)
 
Theoremf1ocnvfv3 7168* Value of the converse of a one-to-one onto function. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
((𝐹:𝐴1-1-onto𝐵𝐶𝐵) → (𝐹𝐶) = (𝑥𝐴 (𝐹𝑥) = 𝐶))
 
Theoremriotaund 7169* Restricted iota equals the empty set when not meaningful. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 13-Sep-2018.)
(¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = ∅)
 
Theoremriotassuni 7170* The restricted iota class is limited in size by the base set. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝑥𝐴 𝜑) ⊆ (𝒫 𝐴 𝐴)
 
Theoremriotaclb 7171* Bidirectional closure of restricted iota when domain is not empty. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.) (Revised by NM, 13-Sep-2018.)
(¬ ∅ ∈ 𝐴 → (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴))
 
2.3.18  Operations
 
Syntaxco 7172 Extend class notation to include the value of an operation 𝐹 (such as +) for two arguments 𝐴 and 𝐵. Note that the syntax is simply three class symbols in a row surrounded by parentheses. Since operation values are the only possible class expressions consisting of three class expressions in a row surrounded by parentheses, the syntax is unambiguous. (For an example of how syntax could become ambiguous if we are not careful, see the comment in cneg 10951.)
class (𝐴𝐹𝐵)
 
Syntaxcoprab 7173 Extend class notation to include class abstraction (class builder) of nested ordered pairs.
class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
 
Syntaxcmpo 7174 Extend the definition of a class to include maps-to notation for defining an operation via a rule.
class (𝑥𝐴, 𝑦𝐵𝐶)
 
Definitiondf-ov 7175 Define the value of an operation. Definition of operation value in [Enderton] p. 79. Note that the syntax is simply three class expressions in a row bracketed by parentheses. There are no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation 𝐹 and its arguments 𝐴 and 𝐵- will be useful for proving meaningful theorems. For example, if class 𝐹 is the operation + and arguments 𝐴 and 𝐵 are 3 and 2, the expression (3 + 2) can be proved to equal 5 (see 3p2e5 11869). This definition is well-defined, although not very meaningful, when classes 𝐴 and/or 𝐵 are proper classes (i.e. are not sets); see ovprc1 7211 and ovprc2 7212. On the other hand, we often find uses for this definition when 𝐹 is a proper class, such as +o in oav 8169. 𝐹 is normally equal to a class of nested ordered pairs of the form defined by df-oprab 7176. (Contributed by NM, 28-Feb-1995.)
(𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
 
Definitiondf-oprab 7176* Define the class abstraction (class builder) of a collection of nested ordered pairs (for use in defining operations). This is a special case of Definition 4.16 of [TakeutiZaring] p. 14. Normally 𝑥, 𝑦, and 𝑧 are distinct, although the definition doesn't strictly require it. See df-ov 7175 for the value of an operation. The brace notation is called "class abstraction" by Quine; it is also called a "class builder" in the literature. The value of an operation given by a class abstraction is given by ovmpo 7327. (Contributed by NM, 12-Mar-1995.)
{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
 
Definitiondf-mpo 7177* Define maps-to notation for defining an operation via a rule. Read as "the operation defined by the map from 𝑥, 𝑦 (in 𝐴 × 𝐵) to 𝐶(𝑥, 𝑦)". An extension of df-mpt 5111 for two arguments. (Contributed by NM, 17-Feb-2008.)
(𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
 
Theoremoveq 7178 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(𝐹 = 𝐺 → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))
 
Theoremoveq1 7179 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(𝐴 = 𝐵 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶))
 
Theoremoveq2 7180 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(𝐴 = 𝐵 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵))
 
Theoremoveq12 7181 Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveq1i 7182 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
𝐴 = 𝐵       (𝐴𝐹𝐶) = (𝐵𝐹𝐶)
 
Theoremoveq2i 7183 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
𝐴 = 𝐵       (𝐶𝐹𝐴) = (𝐶𝐹𝐵)
 
Theoremoveq12i 7184 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐹𝐶) = (𝐵𝐹𝐷)
 
Theoremoveqi 7185 Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
𝐴 = 𝐵       (𝐶𝐴𝐷) = (𝐶𝐵𝐷)
 
Theoremoveq123i 7186 Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)
𝐴 = 𝐶    &   𝐵 = 𝐷    &   𝐹 = 𝐺       (𝐴𝐹𝐵) = (𝐶𝐺𝐷)
 
Theoremoveq1d 7187 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶))
 
Theoremoveq2d 7188 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵))
 
Theoremoveqd 7189 Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐷) = (𝐶𝐵𝐷))
 
Theoremoveq12d 7190 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveqan12d 7191 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveqan12rd 7192 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜓𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
 
Theoremoveq123d 7193 Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐺𝐷))
 
Theoremfvoveq1d 7194 Equality deduction for nested function and operation value. (Contributed by AV, 23-Jul-2022.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹‘(𝐴𝑂𝐶)) = (𝐹‘(𝐵𝑂𝐶)))
 
Theoremfvoveq1 7195 Equality theorem for nested function and operation value. Closed form of fvoveq1d 7194. (Contributed by AV, 23-Jul-2022.)
(𝐴 = 𝐵 → (𝐹‘(𝐴𝑂𝐶)) = (𝐹‘(𝐵𝑂𝐶)))
 
Theoremovanraleqv 7196* Equality theorem for a conjunction with an operation values within a restricted universal quantification. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)
(𝐵 = 𝑋 → (𝜑𝜓))       (𝐵 = 𝑋 → (∀𝑥𝑉 (𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ ∀𝑥𝑉 (𝜓 ∧ (𝐴 · 𝑋) = 𝐶)))
 
Theoremimbrov2fvoveq 7197 Equality theorem for nested function and operation value in an implication for a binary relation. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 17-Aug-2022.)
(𝑋 = 𝑌 → (𝜑𝜓))       (𝑋 = 𝑌 → ((𝜑 → (𝐹‘((𝐺𝑋) · 𝑂))𝑅𝐴) ↔ (𝜓 → (𝐹‘((𝐺𝑌) · 𝑂))𝑅𝐴)))
 
Theoremovrspc2v 7198* If an operation value is element of a class for all operands of two classes, then the operation value is an element of the class for specific operands of the two classes. (Contributed by Mario Carneiro, 6-Dec-2014.)
(((𝑋𝐴𝑌𝐵) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶) → (𝑋𝐹𝑌) ∈ 𝐶)
 
Theoremoveqrspc2v 7199* Restricted specialization of operands, using implicit substitution. (Contributed by Mario Carneiro, 6-Dec-2014.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))       ((𝜑 ∧ (𝑋𝐴𝑌𝐵)) → (𝑋𝐹𝑌) = (𝑋𝐺𝑌))
 
Theoremoveqdr 7200 Equality of two operations for any two operands. Useful in proofs using *propd theorems. (Contributed by Mario Carneiro, 29-Jun-2015.)
(𝜑𝐹 = 𝐺)       ((𝜑𝜓) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
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