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Theorem List for Metamath Proof Explorer - 6001-6100   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremelunirnALT 6001* Membership in the union of the range of a function, proved directly. Unlike elunirn 5999, it doesn't appeal to ndmfv 5756 (via funiunfv 5996). (Contributed by NM, 24-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremabrexex2 6002* Existence of an existentially restricted class abstraction. is normally has free-variable parameters and . See also abrexex 5984. (Contributed by NM, 12-Sep-2004.)

Theoremabexssex 6003* Existence of a class abstraction with an existentially quantified expression. Both and can be free in . (Contributed by NM, 29-Jul-2006.)

Theoremabexex 6004* A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)

Theoremdff13 6005* 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.)

Theoremf1veqaeq 6006 If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)

Theoremdff13f 6007* 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 6008* Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremf1fveq 6009 Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.)

Theoremf1elima 6010 Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremf1imass 6011 Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremf1imaeq 6012 Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremf1imapss 6013 Taking images under a one-to-one function preserves proper subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremdff1o6 6014* A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)

Theoremf1ocnvfv1 6015 The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvfv2 6016 The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvfv 6017 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 6018 Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvdm 6019 The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.)

Theoremf1ocnvfvrneq 6020 If the values of a one-to-one function for two arguments from the range of the function are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)

Theoremfcof1 6021 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 6022 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 6023* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theoremcbvexfo 6024* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)

Theoremcocan1 6025 An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)

Theoremcocan2 6026 A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theoremfcof1o 6027 Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremfoeqcnvco 6028 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 6029 Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.)

Theoremfveqf1o 6030 Given a bijection , produce another bijection which additionally maps two specified points. (Contributed by Mario Carneiro, 30-May-2015.)

Theoremfliftrel 6031* , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftel 6032* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftel1 6033* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftcnv 6034* Converse of the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfun 6035* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfund 6036* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfuns 6037* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftf 6038* The domain and range of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftval 6039* The value of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremisoeq1 6040 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq2 6041 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq3 6042 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq4 6043 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq5 6044 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremnfiso 6045 Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremisof1o 6046 An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.)

Theoremisorel 6047 An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)

Theoremsoisores 6048* Express the condition of isomorphism on two strict orders for a function's restriction. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremsoisoi 6049* Infer isomorphism from one direction of an order proof for isomorphisms between strict orders. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Theoremisoid 6050 Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)

Theoremisocnv 6051 Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)

Theoremisocnv2 6052 Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)

Theoremisocnv3 6053 Complementation law for isomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremisores2 6054 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremisores1 6055 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremisores3 6056 Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisotr 6057 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 6058 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 6059 Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)

Theoremisoini2 6060 Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)

Theoremisofrlem 6061* Lemma for isofr 6063. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)

Theoremisoselem 6062* Lemma for isose 6064. (Contributed by Mario Carneiro, 23-Jun-2015.)
Se Se

Theoremisofr 6063 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 6064 An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)
Se Se

Theoremisofr2 6065 A weak form of isofr 6063 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)

Theoremisopolem 6066 Lemma for isopo 6067. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisopo 6067 An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisosolem 6068 Lemma for isoso 6069. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisoso 6069 An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisowe 6070 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 6071* A weak form of isowe 6070 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)

Theoremf1oiso 6072* 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 6073* Any one-to-one onto function determines an isomorphism with an induced relation . (Contributed by Mario Carneiro, 9-Mar-2013.)

Theoremf1owe 6074* Well-ordering of isomorphic relations. (Contributed by NM, 4-Mar-1997.)

Theoremf1oweALT 6075* 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.) (New usage is discouraged.)

Theoremweniso 6076 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 6077 Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso 6079. (Contributed by Mario Carneiro, 25-Jun-2015.)
Se

Theoremweisoeq2 6078 Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso2 6080. (Contributed by Mario Carneiro, 25-Jun-2015.)
Se

Theoremwemoiso 6079* Thus, there is at most one isomorphism between any two well-ordered sets. TODO: Shorten finnisoeu 7995. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)

Theoremwemoiso2 6080* 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 6081* 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.)

2.4.10  Operations

Syntaxco 6082 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 9293.)

Syntaxcoprab 6083 Extend class notation to include class abstraction (class builder) of nested ordered pairs.

Syntaxcmpt2 6084 Extend the definition of a class to include maps-to notation for defining an operation via a rule.

Definitiondf-ov 6085 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 10112). This definition is well-defined, although not very meaningful, when classes and/or are proper classes (i.e. are not sets); see ovprc1 6110 and ovprc2 6111. On the other hand, we often find uses for this definition when is a proper class, such as in oav 6756. is normally equal to a class of nested ordered pairs of the form defined by df-oprab 6086. (Contributed by NM, 28-Feb-1995.)

Definitiondf-oprab 6086* 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 6085 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 6210. (Contributed by NM, 12-Mar-1995.)

Definitiondf-mpt2 6087* 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 4269 for two arguments. (Contributed by NM, 17-Feb-2008.)

Theoremoveq 6088 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)

Theoremoveq1 6089 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)

Theoremoveq2 6090 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)

Theoremoveq12 6091 Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)

Theoremoveq1i 6092 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)

Theoremoveq2i 6093 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)

Theoremoveq12i 6094 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremoveqi 6095 Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)

Theoremoveq123i 6096 Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)

Theoremoveq1d 6097 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)

Theoremoveq2d 6098 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)

Theoremoveqd 6099 Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)

Theoremoveq12d 6100 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

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