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Theorem List for Metamath Proof Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisocnv2 6901 Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆(𝐴, 𝐵))

Theoremisocnv3 6902 Complementation law for isomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝐶 = ((𝐴 × 𝐴) ∖ 𝑅)    &   𝐷 = ((𝐵 × 𝐵) ∖ 𝑆)       (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝐶, 𝐷 (𝐴, 𝐵))

Theoremisores2 6903 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 6904 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 6905 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 6906 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 6907 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 6908 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 6909 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 6910* Lemma for isofr 6912. (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 6911* Lemma for isose 6913. (Contributed by Mario Carneiro, 23-Jun-2015.)
(𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑 → (𝐻𝑥) ∈ V)       (𝜑 → (𝑅 Se 𝐴𝑆 Se 𝐵))

Theoremisofr 6912 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 6913 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 6914 A weak form of isofr 6912 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 6915 Lemma for isopo 6916. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵𝑅 Po 𝐴))

Theoremisopo 6916 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 6917 Lemma for isoso 6918. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵𝑅 Or 𝐴))

Theoremisoso 6918 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 6919 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 6920* A weak form of isowe 6919 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 6921* 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 6922* 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 6923* 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 6924 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 6925 Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso 7480. (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 6926 Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso2 7481. (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 6927* 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 6928 No set 𝐴 is equinumerous to its power set (Cantor's theorem), i.e. no function can map 𝐴 it onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 8460. Note that 𝐴 must be a set: this theorem does not hold when 𝐴 is too large to be a set; see ncanth 6929 for a counterexample. (Use nex 1763 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 6929 Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 5070). Specifically, the identity function maps the universe onto its power class. Compare canth 6928 that works for sets. See also the remark in ru 3674 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.)
I :V–onto→𝒫 V

2.3.17  Restricted iota (description binder)

Syntaxcrio 6930 Extend class notation with restricted description binder.
class (𝑥𝐴 𝜑)

Definitiondf-riota 6931 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 6146. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 2-Sep-2018.)
(𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))

Theoremriotaeqdv 6932* Formula-building deduction for iota. (Contributed by NM, 15-Sep-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐵 𝜓))

Theoremriotabidv 6933* Formula-building deduction for restricted iota. (Contributed by NM, 15-Sep-2011.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))

Theoremriotaeqbidv 6934* Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐵 𝜒))

Theoremriotaex 6935 Restricted iota is a set. (Contributed by NM, 15-Sep-2011.)
(𝑥𝐴 𝜓) ∈ V

Theoremriotav 6936 An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
(𝑥 ∈ V 𝜑) = (℩𝑥𝜑)

Theoremriotauni 6937 Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)
(∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = {𝑥𝐴𝜑})

Theoremnfriota1 6938* 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)

Theoremnfriotad 6939 Deduction version of nfriota 6940. (Contributed by NM, 18-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑𝑥𝐴)       (𝜑𝑥(𝑦𝐴 𝜓))

Theoremnfriota 6940* 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 6941* Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)

Theoremcbvriotav 6942* Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)

Theoremcsbriota 6943* 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 6944 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 6945* Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
(∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)

Theoremriotasbc 6946 Substitution law for descriptions. Compare iotasbc 40168. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
(∃!𝑥𝐴 𝜑[(𝑥𝐴 𝜑) / 𝑥]𝜑)

Theoremriotabidva 6947* Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 3396 analog.) (Contributed by NM, 17-Jan-2012.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))

Theoremriotabiia 6948 Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 3392 analog.) (Contributed by NM, 16-Jan-2012.)
(𝑥𝐴 → (𝜑𝜓))       (𝑥𝐴 𝜑) = (𝑥𝐴 𝜓)

Theoremriota1 6949* Property of restricted iota. Compare iota1 6160. (Contributed by Mario Carneiro, 15-Oct-2016.)
(∃!𝑥𝐴 𝜑 → ((𝑥𝐴𝜑) ↔ (𝑥𝐴 𝜑) = 𝑥))

Theoremriota1a 6950 Property of iota. (Contributed by NM, 23-Aug-2011.)
((𝑥𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))

Theoremriota2df 6951* A deduction version of riota2f 6952. (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 6952* 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 6953* 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 6954* 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 6955* Properties of a restricted definite description operator. (Contributed by NM, 23-Nov-2013.)
𝑥𝜓    &   𝐵 = (𝑥𝐴 𝜑)    &   (𝑥 = 𝐵 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 → (𝐵𝐴𝜓))

Theoremriota5f 6956* 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 6957* 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 6958* 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 6959* 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 6960* 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 6961 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 6962* Change the variable 𝑥 in the expression for "the unique 𝑥 such that 𝜓 " to another variable 𝑦 contained in expression 𝐵. Use reuhypd 5170 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 6963* 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 6964* 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 6965* 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 6966* 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 6967* 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 6968* 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 6969* 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 6970 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 10665.)
class (𝐴𝐹𝐵)

Syntaxcoprab 6971 Extend class notation to include class abstraction (class builder) of nested ordered pairs.
class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Syntaxcmpo 6972 Extend the definition of a class to include maps-to notation for defining an operation via a rule.
class (𝑥𝐴, 𝑦𝐵𝐶)

Definitiondf-ov 6973 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 11592). 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 7008 and ovprc2 7009. On the other hand, we often find uses for this definition when 𝐹 is a proper class, such as +o in oav 7932. 𝐹 is normally equal to a class of nested ordered pairs of the form defined by df-oprab 6974. (Contributed by NM, 28-Feb-1995.)
(𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)

Definitiondf-oprab 6974* 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 6973 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 the most common operation class builder is given by ovmpo 7120. (Contributed by NM, 12-Mar-1995.)
{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}

Definitiondf-mpo 6975* 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 5003 for two arguments. (Contributed by NM, 17-Feb-2008.)
(𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}

Theoremoveq 6976 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(𝐹 = 𝐺 → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))

Theoremoveq1 6977 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(𝐴 = 𝐵 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶))

Theoremoveq2 6978 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
(𝐴 = 𝐵 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵))

Theoremoveq12 6979 Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))

Theoremoveq1i 6980 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
𝐴 = 𝐵       (𝐴𝐹𝐶) = (𝐵𝐹𝐶)

Theoremoveq2i 6981 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
𝐴 = 𝐵       (𝐶𝐹𝐴) = (𝐶𝐹𝐵)

Theoremoveq12i 6982 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 6983 Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
𝐴 = 𝐵       (𝐶𝐴𝐷) = (𝐶𝐵𝐷)

Theoremoveq123i 6984 Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)
𝐴 = 𝐶    &   𝐵 = 𝐷    &   𝐹 = 𝐺       (𝐴𝐹𝐵) = (𝐶𝐺𝐷)

Theoremoveq1d 6985 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐹𝐶))

Theoremoveq2d 6986 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐹𝐴) = (𝐶𝐹𝐵))

Theoremoveqd 6987 Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐷) = (𝐶𝐵𝐷))

Theoremoveq12d 6988 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 6989 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))

Theoremoveqan12rd 6990 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜓𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))

Theoremoveq123d 6991 Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐺𝐷))

Theoremfvoveq1d 6992 Equality deduction for nested function and operation value. (Contributed by AV, 23-Jul-2022.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹‘(𝐴𝑂𝐶)) = (𝐹‘(𝐵𝑂𝐶)))

Theoremfvoveq1 6993 Equality theorem for nested function and operation value. Closed form of fvoveq1d 6992. (Contributed by AV, 23-Jul-2022.)
(𝐴 = 𝐵 → (𝐹‘(𝐴𝑂𝐶)) = (𝐹‘(𝐵𝑂𝐶)))

Theoremovanraleqv 6994* 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 6995 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 6996* 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 6997* Restricted specialization of operands, using implicit substitution. (Contributed by Mario Carneiro, 6-Dec-2014.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))       ((𝜑 ∧ (𝑋𝐴𝑌𝐵)) → (𝑋𝐹𝑌) = (𝑋𝐺𝑌))

Theoremoveqdr 6998 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))

Theoremnfovd 6999 Deduction version of bound-variable hypothesis builder nfov 7000. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐹)    &   (𝜑𝑥𝐵)       (𝜑𝑥(𝐴𝐹𝐵))

Theoremnfov 7000 Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.)
𝑥𝐴    &   𝑥𝐹    &   𝑥𝐵       𝑥(𝐴𝐹𝐵)

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