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Theorem List for Metamath Proof Explorer - 4601-4700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremonsuci 4601 The successor of an ordinal number is an ordinal number. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)
 |-  A  e.  On   =>    |-  suc  A  e.  On
 
Theoremonuniorsuci 4602 An ordinal number is either its own union (if zero or a limit ordinal) or the successor of its union. (Contributed by NM, 13-Jun-1994.)
 |-  A  e.  On   =>    |-  ( A  =  U. A  \/  A  =  suc  U. A )
 
Theoremonuninsuci 4603* A limit ordinal is not a successor ordinal. (Contributed by NM, 18-Feb-2004.)
 |-  A  e.  On   =>    |-  ( A  =  U. A  <->  -.  E. x  e. 
 On  A  =  suc  x )
 
Theoremonsucssi 4604 A set belongs to an ordinal number iff its successor is a subset of the ordinal number. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 16-Sep-1995.)
 |-  A  e.  On   &    |-  B  e.  On   =>    |-  ( A  e.  B  <->  suc 
 A  C_  B )
 
Theoremnlimsucg 4605 A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  e.  V  ->  -.  Lim  suc  A )
 
Theoremorduninsuc 4606* An ordinal equal to its union is not a successor. (Contributed by NM, 18-Feb-2004.)
 |-  ( Ord  A  ->  ( A  =  U. A  <->  -. 
 E. x  e.  On  A  =  suc  x ) )
 
Theoremordunisuc2 4607* An ordinal equal to its union contains the successor of each of its members. (Contributed by NM, 1-Feb-2005.)
 |-  ( Ord  A  ->  ( A  =  U. A  <->  A. x  e.  A  suc  x  e.  A ) )
 
Theoremordzsl 4608* An ordinal is zero, a successor ordinal, or a limit ordinal. (Contributed by NM, 1-Oct-2003.)
 |-  ( Ord  A  <->  ( A  =  (/) 
 \/  E. x  e.  On  A  =  suc  x  \/  Lim 
 A ) )
 
Theoremonzsl 4609* An ordinal number is zero, a successor ordinal, or a limit ordinal number. (Contributed by NM, 1-Oct-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  e.  On  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
 
Theoremdflim3 4610* An alternate definition of a limit ordinal, which is any ordinal that is neither zero nor a successor. (Contributed by NM, 1-Nov-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( Lim  A  <->  ( Ord  A  /\  -.  ( A  =  (/) 
 \/  E. x  e.  On  A  =  suc  x ) ) )
 
Theoremdflim4 4611* An alternate definition of a limit ordinal. (Contributed by NM, 1-Feb-2005.)
 |-  ( Lim  A  <->  ( Ord  A  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A ) )
 
Theoremlimsuc 4612 The successor of a member of a limit ordinal is also a member. (Contributed by NM, 3-Sep-2003.)
 |-  ( Lim  A  ->  ( B  e.  A  <->  suc  B  e.  A ) )
 
Theoremlimsssuc 4613 A class includes a limit ordinal iff the successor of the class includes it. (Contributed by NM, 30-Oct-2003.)
 |-  ( Lim  A  ->  ( A  C_  B  <->  A  C_  suc  B ) )
 
Theoremnlimon 4614* Two ways to express the class of non-limit ordinal numbers. Part of Definition 7.27 of [TakeutiZaring] p. 42, who use the symbol KI for this class. (Contributed by NM, 1-Nov-2004.)
 |- 
 { x  e.  On  |  ( x  =  (/)  \/ 
 E. y  e.  On  x  =  suc  y ) }  =  { x  e.  On  |  -.  Lim  x }
 
Theoremlimuni3 4615* The union of a nonempty class of limit ordinals is a limit ordinal. (Contributed by NM, 1-Feb-2005.)
 |-  ( ( A  =/=  (/)  /\  A. x  e.  A  Lim  x )  ->  Lim  U. A )
 
2.4.3  Transfinite induction
 
Theoremtfi 4616* The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring] p. 39. This principle states that if  A is a class of ordinal numbers with the property that every ordinal number included in  A also belongs to  A, then every ordinal number is in  A.

See theorem tfindes 4625 or tfinds 4622 for the version involving basis and induction hypotheses. (Contributed by NM, 18-Feb-2004.)

 |-  ( ( A  C_  On  /\  A. x  e. 
 On  ( x  C_  A  ->  x  e.  A ) )  ->  A  =  On )
 
Theoremtfis 4617* Transfinite Induction Schema. If all ordinal numbers less than a given number  x have a property (induction hypothesis), then all ordinal numbers have the property (conclusion). Exercise 25 of [Enderton] p. 200. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 20-Nov-2016.)
 |-  ( x  e.  On  ->  ( A. y  e.  x  [ y  /  x ] ph  ->  ph )
 )   =>    |-  ( x  e.  On  -> 
 ph )
 
Theoremtfis2f 4618* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  e.  On  ->  (
 A. y  e.  x  ps  ->  ph ) )   =>    |-  ( x  e. 
 On  ->  ph )
 
Theoremtfis2 4619* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  e.  On  ->  (
 A. y  e.  x  ps  ->  ph ) )   =>    |-  ( x  e. 
 On  ->  ph )
 
Theoremtfis3 4620* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  e.  On  ->  (
 A. y  e.  x  ps  ->  ph ) )   =>    |-  ( A  e.  On  ->  ch )
 
Theoremtfisi 4621* A transfinite induction scheme in "implicit" form where the induction is done on an object derived from the object of interest. (Contributed by Stefan O'Rear, 24-Aug-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  T  e.  On )   &    |-  (
 ( ph  /\  ( R  e.  On  /\  R  C_  T )  /\  A. y ( S  e.  R  ->  ch ) )  ->  ps )   &    |-  ( x  =  y  ->  ( ps  <->  ch ) )   &    |-  ( x  =  A  ->  ( ps  <->  th ) )   &    |-  ( x  =  y  ->  R  =  S )   &    |-  ( x  =  A  ->  R  =  T )   =>    |-  ( ph  ->  th )
 
Theoremtfinds 4622* Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 16-Apr-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  On  ->  ( ch  ->  th )
 )   &    |-  ( Lim  x  ->  ( A. y  e.  x  ch  ->  ph ) )   =>    |-  ( A  e.  On  ->  ta )
 
Theoremtfindsg 4623* Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. The basis of this version is an arbitrary ordinal  B instead of zero. Remark in [TakeutiZaring] p. 57. (Contributed by NM, 5-Mar-2004.)
 |-  ( x  =  B  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ( B  e.  On  ->  ps )   &    |-  ( ( ( y  e.  On  /\  B  e.  On )  /\  B  C_  y )  ->  ( ch  ->  th )
 )   &    |-  ( ( ( Lim 
 x  /\  B  e.  On )  /\  B  C_  x )  ->  ( A. y  e.  x  ( B  C_  y  ->  ch )  -> 
 ph ) )   =>    |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  B  C_  A )  ->  ta )
 
Theoremtfindsg2 4624* Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. The basis of this version is an arbitrary ordinal  suc  B instead of zero. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 5-Jan-2005.)
 |-  ( x  =  suc  B 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ( B  e.  On  ->  ps )   &    |-  ( ( y  e.  On  /\  B  e.  y )  ->  ( ch  ->  th ) )   &    |-  (
 ( Lim  x  /\  B  e.  x )  ->  ( A. y  e.  x  ( B  e.  y  ->  ch )  ->  ph )
 )   =>    |-  ( ( A  e.  On  /\  B  e.  A )  ->  ta )
 
Theoremtfindes 4625* Transfinite Induction with explicit substitution. The first hypothesis is the basis, the second is the induction hypothesis for successors, and the third is the induction hypothesis for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 5-Mar-2004.)
 |-  [. (/)  /  x ]. ph   &    |-  ( x  e.  On  ->  ( ph  ->  [. suc  x 
 /  x ]. ph )
 )   &    |-  ( Lim  y  ->  ( A. x  e.  y  ph 
 ->  [. y  /  x ].
 ph ) )   =>    |-  ( x  e. 
 On  ->  ph )
 
Theoremtfinds2 4626* Transfinite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last three are the basis and the induction hypotheses (for successor and limit ordinals respectively). Theorem Schema 4 of [Suppes] p. 197. The wff  ta is an auxiliary antecedent to help shorten proofs using this theorem. (Contributed by NM, 4-Sep-2004.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( ta  ->  ps )   &    |-  ( y  e. 
 On  ->  ( ta  ->  ( ch  ->  th )
 ) )   &    |-  ( Lim  x  ->  ( ta  ->  ( A. y  e.  x  ch  ->  ph ) ) )   =>    |-  ( x  e.  On  ->  ( ta  ->  ph )
 )
 
Theoremtfinds3 4627* Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. (Contributed by NM, 6-Jan-2005.) (Revised by David Abernethy, 21-Jun-2011.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ( et  ->  ps )   &    |-  ( y  e. 
 On  ->  ( et  ->  ( ch  ->  th )
 ) )   &    |-  ( Lim  x  ->  ( et  ->  ( A. y  e.  x  ch  ->  ph ) ) )   =>    |-  ( A  e.  On  ->  ( et  ->  ta )
 )
 
2.4.4  The natural numbers (i.e. finite ordinals)
 
Syntaxcom 4628 Extend class notation to include the class of natural numbers.
 class  om
 
Definitiondf-om 4629* Define the class of natural numbers, which are all ordinal numbers that are less than every limit ordinal, i.e. all finite ordinals. Our definition is a variant of the Definition of N of [BellMachover] p. 471. See dfom2 4630 for an alternate definition. Later, when we assume the Axiom of Infinity, we show  om is a set in omex 7312, and  om can then be defined per dfom3 7316 (the smallest inductive set) and dfom4 7318.

Note: the natural numbers  om are a subset of the ordinal numbers df-on 4368. They are completely different from the natural numbers  NN (df-n 9715) that are a subset of the complex numbers defined much later in our development, although the two sets have analogous properties and operations defined on them. (Contributed by NM, 15-May-1994.)

 |- 
 om  =  { x  e.  On  |  A. y
 ( Lim  y  ->  x  e.  y ) }
 
Theoremdfom2 4630 An alternate definition of the set of natural numbers  om. Definition 7.28 of [TakeutiZaring] p. 42, who use the symbol KI for the inner class builder of non-limit ordinal numbers (see nlimon 4614). (Contributed by NM, 1-Nov-2004.)
 |- 
 om  =  { x  e.  On  |  suc  x  C_ 
 { y  e.  On  |  -.  Lim  y } }
 
Theoremelom 4631* Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 7317. (Contributed by NM, 15-May-1994.)
 |-  ( A  e.  om  <->  ( A  e.  On  /\  A. x ( Lim  x  ->  A  e.  x ) ) )
 
Theoremomsson 4632 Omega is a subset of  On. (Contributed by NM, 13-Jun-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |- 
 om  C_  On
 
Theoremlimomss 4633 The class of natural numbers is a subclass of any (infinite) limit ordinal. Exercise 1 of [TakeutiZaring] p. 44. Remarkably, our proof does not require the Axiom of Infinity. (Contributed by NM, 30-Oct-2003.)
 |-  ( Lim  A  ->  om  C_  A )
 
Theoremnnon 4634 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
 |-  ( A  e.  om  ->  A  e.  On )
 
Theoremnnoni 4635 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
 |-  A  e.  om   =>    |-  A  e.  On
 
Theoremnnord 4636 A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
 |-  ( A  e.  om  ->  Ord  A )
 
Theoremordom 4637 Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |- 
 Ord  om
 
Theoremelnn 4638 A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)
 |-  ( ( A  e.  B  /\  B  e.  om )  ->  A  e.  om )
 
Theoremomon 4639 The class of natural numbers  om is either an ordinal number (if we accept the Axiom of Infinity) or the proper class of all ordinal numbers (if we deny the Axiom of Infinity). Remark in [TakeutiZaring] p. 43. (Contributed by NM, 10-May-1998.)
 |-  ( om  e.  On  \/  om  =  On )
 
Theoremomelon2 4640 Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
 |-  ( om  e.  _V  ->  om  e.  On )
 
Theoremnnlim 4641 A natural number is not a limit ordinal. (Contributed by NM, 18-Oct-1995.)
 |-  ( A  e.  om  ->  -.  Lim  A )
 
Theoremomssnlim 4642 The class of natural numbers is a subclass of the class of non-limit ordinal numbers. Exercise 4 of [TakeutiZaring] p. 42. (Contributed by NM, 2-Nov-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |- 
 om  C_  { x  e. 
 On  |  -.  Lim  x }
 
Theoremlimom 4643 Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. Our proof, however, does not require the Axiom of Infinity. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
 |- 
 Lim  om
 
Theorempeano2b 4644 A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
 |-  ( A  e.  om  <->  suc  A  e.  om )
 
Theoremnnsuc 4645* A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
 |-  ( ( A  e.  om 
 /\  A  =/=  (/) )  ->  E. x  e.  om  A  =  suc  x )
 
Theoremssnlim 4646* An ordinal subclass of non-limit ordinals is a class of natural numbers. Exercise 7 of [TakeutiZaring] p. 42. (Contributed by NM, 2-Nov-2004.)
 |-  ( ( Ord  A  /\  A  C_  { x  e.  On  |  -.  Lim  x } )  ->  A  C_ 
 om )
 
2.4.5  Peano's postulates
 
Theorempeano1 4647 Zero is a natural number. One of Peano's 5 postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. Note: Unlike most textbooks, our proofs of peano1 4647 through peano5 4651 do not use the Axiom of Infinity. Unlike Takeuti and Zaring, they also do not use the Axiom of Regularity. (Contributed by NM, 15-May-1994.)
 |-  (/)  e.  om
 
Theorempeano2 4648 The successor of any natural number is a natural number. One of Peano's 5 postulates for arithmetic. Proposition 7.30(2) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)
 |-  ( A  e.  om  ->  suc  A  e.  om )
 
Theorempeano3 4649 The successor of any natural number is not zero. One of Peano's 5 postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)
 |-  ( A  e.  om  ->  suc  A  =/=  (/) )
 
Theorempeano4 4650 Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's 5 postulates for arithmetic. Proposition 7.30(4) of [TakeutiZaring] p. 43. (Contributed by NM, 3-Sep-2003.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
 
Theorempeano5 4651* The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's 5 postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43, except our proof does not require the Axiom of Infinity. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction hypothesis, is derived from this theorem as theorem findes 4658. (Contributed by NM, 18-Feb-2004.)
 |-  ( ( (/)  e.  A  /\  A. x  e.  om  ( x  e.  A  ->  suc  x  e.  A ) )  ->  om  C_  A )
 
Theoremnn0suc 4652* A natural number is either 0 or a successor. (Contributed by NM, 27-May-1998.)
 |-  ( A  e.  om  ->  ( A  =  (/)  \/ 
 E. x  e.  om  A  =  suc  x ) )
 
2.4.6  Finite induction (for finite ordinals)
 
Theoremfind 4653* The Principle of Finite Induction (mathematical induction). Corollary 7.31 of [TakeutiZaring] p. 43. The simpler hypothesis shown here was suggested in an email from "Colin" on 1-Oct-2001. The hypothesis states that  A is a set of natural numbers, zero belongs to 
A, and given any member of  A the member's successor also belongs to  A. The conclusion is that every natural number is in  A. (Contributed by NM, 22-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  C_  om  /\  (/) 
 e.  A  /\  A. x  e.  A  suc  x  e.  A )   =>    |-  A  =  om
 
Theoremfinds 4654* Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 14-Apr-1995.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  om  ->  ( ch  ->  th )
 )   =>    |-  ( A  e.  om  ->  ta )
 
Theoremfindsg 4655* Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. The basis of this version is an arbitrary natural number  B instead of zero. (Contributed by NM, 16-Sep-1995.)
 |-  ( x  =  B  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ( B  e.  om  ->  ps )   &    |-  (
 ( ( y  e. 
 om  /\  B  e.  om )  /\  B  C_  y )  ->  ( ch 
 ->  th ) )   =>    |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  B  C_  A )  ->  ta )
 
Theoremfinds2 4656* Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( ta  ->  ps )   &    |-  ( y  e. 
 om  ->  ( ta  ->  ( ch  ->  th )
 ) )   =>    |-  ( x  e.  om  ->  ( ta  ->  ph )
 )
 
Theoremfinds1 4657* Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 22-Mar-2006.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ps   &    |-  (
 y  e.  om  ->  ( ch  ->  th )
 )   =>    |-  ( x  e.  om  -> 
 ph )
 
Theoremfindes 4658 Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. See tfindes 4625 for the transfinite version. (Contributed by Raph Levien, 9-Jul-2003.)
 |-  [. (/)  /  x ]. ph   &    |-  ( x  e.  om  ->  (
 ph  ->  [. suc  x  /  x ]. ph ) )   =>    |-  ( x  e.  om  ->  ph )
 
2.4.7  Functions and relations
 
Syntaxcxp 4659 Extend the definition of a class to include the cross product.
 class  ( A  X.  B )
 
Syntaxccnv 4660 Extend the definition of a class to include the converse of a class.
 class  `' A
 
Syntaxcdm 4661 Extend the definition of a class to include the domain of a class.
 class  dom  A
 
Syntaxcrn 4662 Extend the definition of a class to include the range of a class.
 class  ran  A
 
Syntaxcres 4663 Extend the definition of a class to include the restriction of a class. (Read: The restriction of  A to  B.)
 class  ( A  |`  B )
 
Syntaxcima 4664 Extend the definition of a class to include the image of a class. (Read: The image of  B under  A.)
 class  ( A " B )
 
Syntaxccom 4665 Extend the definition of a class to include the composition of two classes. (Read: The composition of  A and  B.)
 class  ( A  o.  B )
 
Syntaxwrel 4666 Extend the definition of a wff to include the relation predicate. (Read:  A is a relation.)
 wff  Rel  A
 
Syntaxwfun 4667 Extend the definition of a wff to include the function predicate. (Read:  A is a function.)
 wff  Fun  A
 
Syntaxwfn 4668 Extend the definition of a wff to include the function predicate with a domain. (Read:  A is a function on  B.)
 wff  A  Fn  B
 
Syntaxwf 4669 Extend the definition of a wff to include the function predicate with domain and codomain. (Read: 
F maps  A into  B.)
 wff  F : A --> B
 
Syntaxwf1 4670 Extend the definition of a wff to include one-to-one functions. (Read:  F maps  A one-to-one into  B.) The notation ("1-1" above the arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27.
 wff  F : A -1-1-> B
 
Syntaxwfo 4671 Extend the definition of a wff to include onto functions. (Read:  F maps  A onto  B.) The notation ("onto" below the arrow) is from Definition 6.15(4) of [TakeutiZaring] p. 27.
 wff  F : A -onto-> B
 
Syntaxwf1o 4672 Extend the definition of a wff to include one-to-one onto functions. (Read:  F maps  A one-to-one onto  B.) The notation ("1-1" above the arrow and "onto" below the arrow) is from Definition 6.15(6) of [TakeutiZaring] p. 27.
 wff  F : A -1-1-onto-> B
 
Syntaxcfv 4673 Extend the definition of a class to include the value of a function. (Read: The value of  F at  A, or " F of  A.")
 class  ( F `  A )
 
Syntaxwiso 4674 Extend the definition of a wff to include the isomorphism property. (Read:  H is an  R,  S isomorphism of  A onto  B.)
 wff  H  Isom  R ,  S  ( A ,  B )
 
Definitiondf-xp 4675* Define the cross product of two classes. Definition 9.11 of [Quine] p. 64. For example,  ( { 1 ,  5 }  X.  {
2 ,  7 } )  =  ( { <. 1 ,  2 >. , 
<. 1 ,  7
>. }  u.  { <. 5 ,  2 >. , 
<. 5 ,  7
>. } ) (ex-xp 20766). Another example is that the set of rational numbers are defined in df-q 10284 using the cross-product  ( ZZ  X.  NN ); the left- and right-hand sides of the cross-product represent the top (integer) and bottom (natural) numbers of a fraction. (Contributed by NM, 4-Jul-1994.)
 |-  ( A  X.  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }
 
Definitiondf-rel 4676 Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5112 and dfrel3 5118. (Contributed by NM, 1-Aug-1994.)
 |-  ( Rel  A  <->  A  C_  ( _V 
 X.  _V ) )
 
Definitiondf-cnv 4677* Define the converse of a class. Definition 9.12 of [Quine] p. 64. The converse of a binary relation swaps its arguments, i.e., if  A  e. 
_V and  B  e.  _V then  ( A `' R B  <-> 
B R A ), as proven in brcnv 4852 (see df-br 3998 and df-rel 4676 for more on relations). For example,  `' { <. 2 ,  6 >. , 
<. 3 ,  9
>. }  =  { <. 6 ,  2 >. , 
<. 9 ,  3
>. } (ex-cnv 20767). We use Quine's breve accent (smile) notation. Like Quine, we use it as a prefix, which eliminates the need for parentheses. Many authors use the postfix superscript "to the minus one." "Converse" is Quine's terminology; some authors call it "inverse," especially when the argument is a function. (Contributed by NM, 4-Jul-1994.)
 |-  `' A  =  { <. x ,  y >.  |  y A x }
 
Definitiondf-co 4678* Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. For example,  ( ( exp 
o.  cos ) `  0
)  =  _e (ex-co 20768) because  ( cos `  0 )  =  1 (see cos0 12392) and  ( exp `  1
)  =  _e (see df-e 12312). Note that Definition 7 of [Suppes] p. 63 reverses  A and  B, uses  /. instead of  o., and calls the operation "relative product." (Contributed by NM, 4-Jul-1994.)
 |-  ( A  o.  B )  =  { <. x ,  y >.  |  E. z
 ( x B z 
 /\  z A y ) }
 
Definitiondf-dm 4679* Define the domain of a class. Definition 3 of [Suppes] p. 59. For example,  F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
>. }  ->  dom  F  =  { 2 ,  3 } (ex-dm 20769). Another example is the domain of the complex arctangent,  ( A  e. 
dom arctan 
<->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/=  _i ) ) (for proof see atandm 20134). Contrast with range (defined in df-rn 4680). For alternate definitions see dfdm2 5191, dfdm3 4855, and dfdm4 4860. The notation " dom " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994.)
 |- 
 dom  A  =  { x  |  E. y  x A y }
 
Definitiondf-rn 4680 Define the range of a class. For example,  F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
>. }  ->  ran  F  =  { 6 ,  9 } (ex-rn 20770). Contrast with domain (defined in df-dm 4679). For alternate definitions, see dfrn2 4856, dfrn3 4857, and dfrn4 5121. The notation " ran " is used by Enderton; other authors sometimes use script R or script W. (Contributed by NM, 1-Aug-1994.)
 |- 
 ran  A  =  dom  `'  A
 
Definitiondf-res 4681 Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring] p. 24. For example, the expression  ( exp  |`  RR ) (used in reeff1 12362) means "the exponential function e to the x, but the exponent x must be in the reals" (df-ef 12311 defines the exponential function, which normally allows the exponent to be a complex number). Another example is that  ( F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
>. }  /\  B  =  { 1 ,  2 } )  ->  ( F  |`  B )  =  { <. 2 ,  6
>. } (ex-res 20771). (Contributed by NM, 2-Aug-1994.)
 |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V )
 )
 
Definitiondf-ima 4682 Define the image of a class (as restricted by another class). Definition 6.6(2) of [TakeutiZaring] p. 24. For example,  ( F  =  { <. 2 ,  6
>. ,  <. 3 ,  9 >. }  /\  B  =  { 1 ,  2 } )  ->  ( F " B )  =  { 6 } (ex-ima 20772). Contrast with restriction (df-res 4681) and range (df-rn 4680). For an alternate definition, see dfima2 5002. (Contributed by NM, 2-Aug-1994.)
 |-  ( A " B )  =  ran  (  A  |`  B )
 
Definitiondf-fun 4683 Define predicate that determines if some class  A is a function. Definition 10.1 of [Quine] p. 65. For example, the expression  Fun  cos is true once we define cosine (df-cos 12314). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4051 with the maps-to notation (see df-mpt 4053 and df-mpt2 5797). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 4684), a function with a given domain and codomain (df-f 4685), a one-to-one function (df-f1 4686), an onto function (df-fo 4687), or a one-to-one onto function (df-f1o 4688). For alternate definitions, see dffun2 5204, dffun3 5205, dffun4 5206, dffun5 5207, dffun6 5209, dffun7 5219, dffun8 5220, and dffun9 5221. (Contributed by NM, 1-Aug-1994.)
 |-  ( Fun  A  <->  ( Rel  A  /\  ( A  o.  `' A )  C_  _I  )
 )
 
Definitiondf-fn 4684 Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 5328, dffn3 5334, dffn4 5395, and dffn5 5502. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  Fn  B  <->  ( Fun  A  /\  dom  A  =  B ) )
 
Definitiondf-f 4685 Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. For alternate definitions, see dff2 5606, dff3 5607, and dff4 5608. (Contributed by NM, 1-Aug-1994.)
 |-  ( F : A --> B 
 <->  ( F  Fn  A  /\  ran  F  C_  B ) )
 
Definitiondf-f1 4686 Define a one-to-one function. For equivalent definitions see dff12 5374 and dff13 5717. Compare Definition 6.15(5) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow). (Contributed by NM, 1-Aug-1994.)
 |-  ( F : A -1-1-> B  <-> 
 ( F : A --> B  /\  Fun  `' F ) )
 
Definitiondf-fo 4687 Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27. We use their notation ("onto" under the arrow). For alternate definitions, see dffo2 5393, dffo3 5609, dffo4 5610, and dffo5 5611. (Contributed by NM, 1-Aug-1994.)
 |-  ( F : A -onto-> B 
 <->  ( F  Fn  A  /\  ran  F  =  B ) )
 
Definitiondf-f1o 4688 Define a one-to-one onto function. For equivalent definitions see dff1o2 5415, dff1o3 5416, dff1o4 5418, and dff1o5 5419. Compare Definition 6.15(6) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow and "onto" below the arrow). (Contributed by NM, 1-Aug-1994.)
 |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
 
Definitiondf-fv 4689* Define the value of a function,  ( F `  A
), also known as function application. For example,  ( cos `  0
)  =  1 (we prove this in cos0 12392 after we define cosine in df-cos 12314). Typically function  F is defined using maps-to notation (see df-mpt 4053 and df-mpt2 5797), but this is not required. For example,  F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
>. }  ->  ( F `  3 )  =  9 (ex-fv 20773). Note that df-ov 5795 will define two-argument functions using ordered pairs as  ( A F B )  =  ( F `  <. A ,  B >. ). Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 5029), our definition apparently does not appear in the literature. However, it is quite convenient: it can be applied to any class and evaluates to the empty set when it is not meaningful (as shown by ndmfv 5486 and fvprc 5455). The left apostrophe notation originated with Peano and was adopted in Definition *30.01 of [WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and Definition 6.11 of [TakeutiZaring] p. 26. It means the same thing as the more familiar  F ( A ) notation for a function's value at  A, i.e. " F of  A," but without context-dependent notational ambiguity. Alternate definitions are dffv2 5526 and dffv3 6253. For other alternate definitions (that fail to evaluate to the empty set for proper classes), see fv2 5454, fv3 5474, and fv4 6254. Restricted equivalents that require  F to be a function are shown in funfv 5520 and funfv2 5521. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 5500. (Contributed by NM, 1-Aug-1994.)
 |-  ( F `  A )  =  U. { x  |  ( F " { A } )  =  { x } }
 
Definitiondf-isom 4690* Define the isomorphism predicate. We read this as " H is an  R,  S isomorphism of  A onto  B." Normally,  R and  S are ordering relations on  A and  B respectively. Definition 6.28 of [TakeutiZaring] p. 32, whose notation is the same as ours except that  R and  S are subscripts. (Contributed by NM, 4-Mar-1997.)
 |-  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  ( H : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <->  ( H `  x ) S ( H `  y ) ) ) )
 
Theoremxpeq1 4691 Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)
 |-  ( A  =  B  ->  ( A  X.  C )  =  ( B  X.  C ) )
 
Theoremxpeq2 4692 Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.)
 |-  ( A  =  B  ->  ( C  X.  A )  =  ( C  X.  B ) )
 
Theoremelxpi 4693* Membership in a cross product. Uses fewer axioms than elxp 4694. (Contributed by NM, 4-Jul-1994.)
 |-  ( A  e.  ( B  X.  C )  ->  E. x E. y ( A  =  <. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C ) ) )
 
Theoremelxp 4694* Membership in a cross product. (Contributed by NM, 4-Jul-1994.)
 |-  ( A  e.  ( B  X.  C )  <->  E. x E. y
 ( A  =  <. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C )
 ) )
 
Theoremelxp2 4695* Membership in a cross product. (Contributed by NM, 23-Feb-2004.)
 |-  ( A  e.  ( B  X.  C )  <->  E. x  e.  B  E. y  e.  C  A  =  <. x ,  y >. )
 
Theoremxpeq12 4696 Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  X.  C )  =  ( B  X.  D ) )
 
Theoremxpeq1i 4697 Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( A  X.  C )  =  ( B  X.  C )
 
Theoremxpeq2i 4698 Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( C  X.  A )  =  ( C  X.  B )
 
Theoremxpeq12i 4699 Equality inference for cross product. (Contributed by FL, 31-Aug-2009.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  X.  C )  =  ( B  X.  D )
 
Theoremxpeq1d 4700 Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  X.  C )  =  ( B  X.  C ) )
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