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Theorem List for Metamath Proof Explorer - 5701-5800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiunexg 5701* The existence of an indexed union. is normally a free-variable parameter in . (Contributed by NM, 23-Mar-2006.)

Theoremabrexex2g 5702* Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremopabex3 5703* Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiunex 5704* The existence of an indexed union. is normally a free-variable parameter in the class expression substituted for , which can be read informally as . (Contributed by NM, 13-Oct-2003.)

Theoremimaiun 5705* The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremimauni 5706* The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)

Theoremfniunfv 5707* The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.)

Theoremfuniunfv 5708* The indexed union of a function's values is the union of its image under the index class.

Note: This theorem depends on the fact that our function value is the empty set outside of its domain. If the antecedent is changed to , the theorem can be proved without this dependency. (Contributed by NM, 26-Mar-2006.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremfuniunfvf 5709* The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 5708 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.) (Revised by David Abernethy, 15-Apr-2013.)

Theoremeluniima 5710* Membership in the union of an image of a function. (Contributed by NM, 28-Sep-2006.)

Theoremelunirn 5711* Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.)

Theoremfnunirn 5712* Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)

TheoremelunirnALT 5713* Membership in the union of the range of a function, proved directly. Unlike elunirn 5711, it doesn't appeal to ndmfv 5486 (via funiunfv 5708). (Contributed by NM, 24-Sep-2006.) (Proof modification is discouraged.)

Theoremabrexex2 5714* Existence of an existentially restricted class abstraction. is normally has free-variable parameters and . See also abrexex 5697. (Contributed by NM, 12-Sep-2004.)

Theoremabexssex 5715* Existence of a class abstraction with an existentially quantified expression. Both and can be free in . (Contributed by NM, 29-Jul-2006.)

Theoremabexex 5716* A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)

Theoremdff13 5717* A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 29-Oct-1996.)

Theoremdff13f 5718* A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.)

Theoremf1mpt 5719* Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremf1fveq 5720 Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.)

Theoremf1elima 5721 Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremf1imass 5722 Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremf1imaeq 5723 Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremf1imapss 5724 Taking images under a one-to-one function preserves proper subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremdff1o6 5725* A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)

Theoremf1ocnvfv1 5726 The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvfv2 5727 The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvfv 5728 Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremf1ocnvfvb 5729 Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvdm 5730 The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.)

Theoremfcof1 5731 An application is injective if a retraction exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 11-Nov-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremfcofo 5732 An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Theoremcbvfo 5733* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theoremcbvexfo 5734* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)

Theoremcocan1 5735 An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)

Theoremcocan2 5736 A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theoremfcof1o 5737 Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremfoeqcnvco 5738 Condition for function equality in terms of vanishing of the composition with the converse. EDITORIAL: Is there a relation-algebraic proof of this? (Contributed by Stefan O'Rear, 12-Feb-2015.)

Theoremf1eqcocnv 5739 Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.)

Theoremfveqf1o 5740 Given a bijection , produce another bijection which additionally maps two specified points. (Contributed by Mario Carneiro, 30-May-2015.)

Theoremfliftrel 5741* , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftel 5742* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftel1 5743* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftcnv 5744* Converse of the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfun 5745* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfund 5746* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfuns 5747* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftf 5748* The domain and range of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftval 5749* The value of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremisoeq1 5750 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq2 5751 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq3 5752 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq4 5753 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq5 5754 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremnfiso 5755 Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremisof1o 5756 An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.)

Theoremisorel 5757 An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)

Theoremsoisores 5758* Express the condition of isomorphism on two strict orders for a function's restriction. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremsoisoi 5759* Infer isomorphism from one direction of an order proof for isomorphisms between strict orders. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Theoremisoid 5760 Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)

Theoremisocnv 5761 Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)

Theoremisocnv2 5762 Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)

Theoremisocnv3 5763 Complementation law for isomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremisores2 5764 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremisores1 5765 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremisores3 5766 Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisotr 5767 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.)

Theoremisomin 5768 Isomorphisms preserve minimal elements. Note that is Takeuti and Zaring's idiom for the initial segment . Proposition 6.31(1) of [TakeutiZaring] p. 33. (Contributed by NM, 19-Apr-2004.)

Theoremisoini 5769 Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)

Theoremisoini2 5770 Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)

Theoremisofrlem 5771* Lemma for isofr 5773. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)

Theoremisoselem 5772* Lemma for isose 5774. (Contributed by Mario Carneiro, 23-Jun-2015.)
Se Se

Theoremisofr 5773 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.)

Theoremisose 5774 An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)
Se Se

Theoremisofr2 5775 A weak form of isofr 5773 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)

Theoremisopolem 5776 Lemma for isopo 5777. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisopo 5777 An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisosolem 5778 Lemma for isoso 5779. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisoso 5779 An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisowe 5780 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.)

Theoremisowe2 5781* A weak form of isowe 5780 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)

Theoremf1oiso 5782* Any one-to-one onto function determines an isomorphism with an induced relation . Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by NM, 30-Apr-2004.)

Theoremf1oiso2 5783* Any one-to-one onto function determines an isomorphism with an induced relation . (Contributed by Mario Carneiro, 9-Mar-2013.)

Theoremf1owe 5784* Well-ordering of isomorphic relations. (Contributed by NM, 4-Mar-1997.)

Theoremf1oweALT 5785* Well-ordering of isomorphic relations. (This version is proved directly instead of with the isomorphism predicate.) (Contributed by NM, 4-Mar-1997.) (Proof modification is discouraged.)

Theoremweniso 5786 A set-like well-ordering has no nontrivial automorphisms. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
Se

Theoremweisoeq 5787 Thus there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso 5789. (Contributed by Mario Carneiro, 25-Jun-2015.)
Se

Theoremweisoeq2 5788 Thus there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso2 5790. (Contributed by Mario Carneiro, 25-Jun-2015.)
Se

Theoremwemoiso 5789* Thus there is at most one isomorphism between any two well-ordered sets. TODO: Shorten finnisoeu 7708. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)

Theoremwemoiso2 5790* Thus there is at most one isomorphism between any two well-ordered sets. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)

Theoremknatar 5791* 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 . (Contributed by Mario Carneiro, 11-Jun-2015.)

2.4.8  Operations

Syntaxco 5792 Extend class notation to include the value of an operation (such as ) for two arguments and . 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 9006.)

Syntaxcoprab 5793 Extend class notation to include class abstraction (class builder) of nested ordered pairs.

Syntaxcmpt2 5794 Extend the definition of a class to include maps-to notation for defining an operation via a rule.

Definitiondf-ov 5795 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 and its arguments and - will be useful for proving meaningful theorems. For example, if class is the operation and arguments and are and , the expression can be proved to equal (see 3p2e5 9823). This definition is well-defined, although not very meaningful, when classes and/or are proper classes (i.e. are not sets); see ovprc1 5820 and ovprc2 5821. On the other hand, we often find uses for this definition when is a proper class, such as in oav 6478. is normally equal to a class of nested ordered pairs of the form defined by df-oprab 5796. (Contributed by NM, 28-Feb-1995.)

Definitiondf-oprab 5796* 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 , , and are distinct, although the definition doesn't strictly require it. See df-ov 5795 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 ovmpt2 5917. (Contributed by NM, 12-Mar-1995.)

Definitiondf-mpt2 5797* Define maps-to notation for defining an operation via a rule. Read as "the operation defined by the map from (in ) to ." An extension of df-mpt 4053 for two arguments. (Contributed by NM, 17-Feb-2008.)

Theoremoveq 5798 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)

Theoremoveq1 5799 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)

Theoremoveq2 5800 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)

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