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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fliftrel 5701* |
![]() ![]() ![]() ![]() |
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Theorem | fliftel 5702* |
Elementhood in the relation ![]() |
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Theorem | fliftel1 5703* |
Elementhood in the relation ![]() |
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Theorem | fliftcnv 5704* |
Converse of the relation ![]() |
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Theorem | fliftfun 5705* |
The function ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | fliftfund 5706* |
The function ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | fliftfuns 5707* |
The function ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | fliftf 5708* |
The domain and range of the function ![]() |
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Theorem | fliftval 5709* |
The value of the function ![]() |
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Theorem | isoeq1 5710 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
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Theorem | isoeq2 5711 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
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Theorem | isoeq3 5712 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
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Theorem | isoeq4 5713 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
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Theorem | isoeq5 5714 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
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Theorem | nfiso 5715 | Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
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Theorem | isof1o 5716 | An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.) |
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Theorem | isorel 5717 | An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.) |
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Theorem | isoresbr 5718* | A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.) |
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Theorem | isoid 5719 | Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
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Theorem | isocnv 5720 | Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
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Theorem | isocnv2 5721 | Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.) |
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Theorem | isores2 5722 | An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) |
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Theorem | isores1 5723 | An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) |
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Theorem | isores3 5724 | Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
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Theorem | isotr 5725 | 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.) |
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Theorem | iso0 5726 |
The empty set is an ![]() ![]() ![]() |
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Theorem | isoini 5727 | Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.) |
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Theorem | isoini2 5728 | Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.) |
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Theorem | isoselem 5729* | Lemma for isose 5730. (Contributed by Mario Carneiro, 23-Jun-2015.) |
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Theorem | isose 5730 | An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.) |
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Theorem | isopolem 5731 | Lemma for isopo 5732. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
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Theorem | isopo 5732 | An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
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Theorem | isosolem 5733 | Lemma for isoso 5734. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
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Theorem | isoso 5734 | An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
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Theorem | f1oiso 5735* |
Any one-to-one onto function determines an isomorphism with an induced
relation ![]() |
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Theorem | f1oiso2 5736* |
Any one-to-one onto function determines an isomorphism with an induced
relation ![]() |
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Syntax | crio 5737 | Extend class notation with restricted description binder. |
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Definition | df-riota 5738 |
Define restricted description binder. In case there is no unique ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | riotaeqdv 5739* | Formula-building deduction for iota. (Contributed by NM, 15-Sep-2011.) |
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Theorem | riotabidv 5740* | Formula-building deduction for restricted iota. (Contributed by NM, 15-Sep-2011.) |
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Theorem | riotaeqbidv 5741* | Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.) |
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Theorem | riotaexg 5742* | Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.) |
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Theorem | riotav 5743 | An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.) |
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Theorem | riotauni 5744 | Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.) |
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Theorem | nfriota1 5745* | The abstraction variable in a restricted iota descriptor isn't free. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
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Theorem | nfriotadxy 5746* | Deduction version of nfriota 5747. (Contributed by Jim Kingdon, 12-Jan-2019.) |
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Theorem | nfriota 5747* | A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.) |
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Theorem | cbvriota 5748* | Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
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Theorem | cbvriotav 5749* | Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
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Theorem | csbriotag 5750* | Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.) |
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Theorem | riotacl2 5751 |
Membership law for "the unique element in ![]() ![]() (Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
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Theorem | riotacl 5752* | Closure of restricted iota. (Contributed by NM, 21-Aug-2011.) |
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Theorem | riotasbc 5753 | Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
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Theorem | riotabidva 5754* | Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 2677 analog.) (Contributed by NM, 17-Jan-2012.) |
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Theorem | riotabiia 5755 | Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 2674 analog.) (Contributed by NM, 16-Jan-2012.) |
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Theorem | riota1 5756* | Property of restricted iota. Compare iota1 5110. (Contributed by Mario Carneiro, 15-Oct-2016.) |
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Theorem | riota1a 5757 | Property of iota. (Contributed by NM, 23-Aug-2011.) |
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Theorem | riota2df 5758* | A deduction version of riota2f 5759. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
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Theorem | riota2f 5759* |
This theorem shows a condition that allows us to represent a descriptor
with a class expression ![]() |
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Theorem | riota2 5760* |
This theorem shows a condition that allows us to represent a descriptor
with a class expression ![]() |
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Theorem | riotaprop 5761* | Properties of a restricted definite description operator. Todo (df-riota 5738 update): can some uses of riota2f 5759 be shortened with this? (Contributed by NM, 23-Nov-2013.) |
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Theorem | riota5f 5762* | A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
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Theorem | riota5 5763* | A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.) |
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Theorem | riotass2 5764* | Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.) |
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Theorem | riotass 5765* | Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.) |
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Theorem | moriotass 5766* | Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.) |
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Theorem | snriota 5767 | A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.) |
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Theorem | eusvobj2 5768* |
Specify the same property in two ways when class ![]() ![]() ![]() ![]() |
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Theorem | eusvobj1 5769* |
Specify the same object in two ways when class ![]() ![]() ![]() ![]() |
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Theorem | f1ofveu 5770* | There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.) |
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Theorem | f1ocnvfv3 5771* | 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.) |
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Theorem | riotaund 5772* | 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.) |
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Theorem | acexmidlema 5773* | Lemma for acexmid 5781. (Contributed by Jim Kingdon, 6-Aug-2019.) |
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Theorem | acexmidlemb 5774* | Lemma for acexmid 5781. (Contributed by Jim Kingdon, 6-Aug-2019.) |
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Theorem | acexmidlemph 5775* | Lemma for acexmid 5781. (Contributed by Jim Kingdon, 6-Aug-2019.) |
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Theorem | acexmidlemab 5776* | Lemma for acexmid 5781. (Contributed by Jim Kingdon, 6-Aug-2019.) |
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Theorem | acexmidlemcase 5777* |
Lemma for acexmid 5781. Here we divide the proof into cases (based
on the
disjunction implicit in an unordered pair, not the sort of case
elimination which relies on excluded middle).
The cases are (1) the choice function evaluated at
Because of the way we represent the choice function
Although it isn't exactly about the division into cases, it is also
convenient for this lemma to also include the step that if the choice
function evaluated at (Contributed by Jim Kingdon, 7-Aug-2019.) |
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Theorem | acexmidlem1 5778* | Lemma for acexmid 5781. List the cases identified in acexmidlemcase 5777 and hook them up to the lemmas which handle each case. (Contributed by Jim Kingdon, 7-Aug-2019.) |
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Theorem | acexmidlem2 5779* |
Lemma for acexmid 5781. This builds on acexmidlem1 5778 by noting that every
element of ![]()
(Note that
The set (Contributed by Jim Kingdon, 5-Aug-2019.) |
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Theorem | acexmidlemv 5780* |
Lemma for acexmid 5781.
This is acexmid 5781 with additional distinct variable
constraints, most
notably between (Contributed by Jim Kingdon, 6-Aug-2019.) |
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Theorem | acexmid 5781* |
The axiom of choice implies excluded middle. Theorem 1.3 in [Bauer]
p. 483.
The statement of the axiom of choice given here is ac2 in the Metamath
Proof Explorer (version of 3-Aug-2019). In particular, note that the
choice function Essentially the same proof can also be found at "The axiom of choice implies instances of EM", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". Often referred to as Diaconescu's theorem, or Diaconescu-Goodman-Myhill theorem, after Radu Diaconescu who discovered it in 1975 in the framework of topos theory and N. D. Goodman and John Myhill in 1978 in the framework of set theory (although it already appeared as an exercise in Errett Bishop's book Foundations of Constructive Analysis from 1967). For this theorem stated using the df-ac 7079 and df-exmid 4127 syntaxes, see exmidac 7082. (Contributed by Jim Kingdon, 4-Aug-2019.) |
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Syntax | co 5782 |
Extend class notation to include the value of an operation ![]() ![]() ![]() |
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Syntax | coprab 5783 | Extend class notation to include class abstraction (class builder) of nested ordered pairs. |
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Syntax | cmpo 5784 | Extend the definition of a class to include maps-to notation for defining an operation via a rule. |
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Definition | df-ov 5785 |
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 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Definition | df-oprab 5786* |
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 ![]() ![]() ![]() |
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Definition | df-mpo 5787* |
Define maps-to notation for defining an operation via a rule. Read as
"the operation defined by the map from ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | oveq 5788 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
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Theorem | oveq1 5789 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
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Theorem | oveq2 5790 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
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Theorem | oveq12 5791 | Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.) |
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Theorem | oveq1i 5792 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) |
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Theorem | oveq2i 5793 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) |
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Theorem | oveq12i 5794 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
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Theorem | oveqi 5795 | Equality inference for operation value. (Contributed by NM, 24-Nov-2007.) |
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Theorem | oveq123i 5796 | Equality inference for operation value. (Contributed by FL, 11-Jul-2010.) |
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Theorem | oveq1d 5797 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) |
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Theorem | oveq2d 5798 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) |
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Theorem | oveqd 5799 | Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.) |
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Theorem | oveq12d 5800 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
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