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Theorem List for Metamath Proof Explorer - 6301-6400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremriotabidv 6301* Formula-building deduction rule for restricted iota. (Contributed by NM, 15-Sep-2011.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 iota_ x  e.  A ps )  =  ( iota_ x  e.  A ch ) )
 
Theoremriotaeqbidv 6302* Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 iota_ x  e.  A ps )  =  ( iota_ x  e.  B ch ) )
 
Theoremriotaex 6303 Restricted iota is a set. (Contributed by NM, 15-Sep-2011.)
 |-  ( iota_ x  e.  A ps )  e.  _V
 
Theoremriotav 6304 An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
 |-  ( iota_ x  e.  _V ph )  =  ( iota
 x ph )
 
Theoremriotaiota 6305 Restricted iota in terms of iota. (Contributed by NM, 15-Sep-2011.)
 |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph ) ) )
 
Theoremriotauni 6306 Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)
 |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  U. { x  e.  A  |  ph } )
 
Theoremnfriota1 6307* The abstraction variable in a restricted iota descriptor isn't free. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x ( iota_ x  e.  A ph )
 
Theoremnfriotad 6308 Deduction version of nfriota 6309. (Contributed by NM, 18-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/_ x (
 iota_ y  e.  A ps ) )
 
Theoremnfriota 6309* A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.)
 |- 
 F/ x ph   &    |-  F/_ x A   =>    |-  F/_ x ( iota_ y  e.  A ph )
 
Theoremcbvriota 6310* Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( iota_ x  e.  A ph )  =  ( iota_ y  e.  A ps )
 
Theoremcbvriotav 6311* Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( iota_ x  e.  A ph )  =  ( iota_ y  e.  A ps )
 
Theoremcsbriotag 6312* Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [. A  /  x ]. ph )
 )
 
Theoremriotacl2 6313 Membership law for "the unique element in  A such that  ph."

This can useful for expanding an iota-based definition (see df-iota 6252). If you have an unbounded iota, iotacl 6275 may be useful.

(Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

 |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  { x  e.  A  |  ph } )
 
Theoremriotacl 6314* Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
 |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  A )
 
Theoremriotasbc 6315 Substitution law for descriptions. Compare iotasbc 27018. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
 |-  ( E! x  e.  A  ph  ->  [. ( iota_ x  e.  A ph )  /  x ]. ph )
 
Theoremriotabidva 6316* Equivalent wff's yield equal restricted class abstractions (deduction rule). (rabbidva 2780 analog.) (Contributed by NM, 17-Jan-2012.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 iota_ x  e.  A ps )  =  ( iota_ x  e.  A ch ) )
 
Theoremriotabiia 6317 Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 2779 analog.) (Contributed by NM, 16-Jan-2012.)
 |-  ( x  e.  A  ->  ( ph  <->  ps ) )   =>    |-  ( iota_ x  e.  A ph )  =  ( iota_ x  e.  A ps )
 
Theoremriota1 6318* Property of restricted iota. Compare iota1 6266. (Contributed by Mario Carneiro, 15-Oct-2016.)
 |-  ( E! x  e.  A  ph  ->  ( ( x  e.  A  /\  ph )  <->  ( iota_ x  e.  A ph )  =  x ) )
 
Theoremriota1a 6319 Property of iota. (Contributed by NM, 23-Aug-2011.)
 |-  ( ( x  e.  A  /\  E! x  e.  A  ph )  ->  ( ph  <->  ( iota x ( x  e.  A  /\  ph ) )  =  x ) )
 
Theoremriota2df 6320* A deduction version of riota2f 6321. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/_ x B )   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ( ph  /\  x  =  B ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ( ph  /\ 
 E! x  e.  A  ps )  ->  ( ch  <->  (
 iota_ x  e.  A ps )  =  B ) )
 
Theoremriota2f 6321* 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.)
 |-  F/_ x B   &    |-  F/ x ps   &    |-  ( x  =  B  ->  (
 ph 
 <->  ps ) )   =>    |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ( ps  <->  ( iota_ x  e.  A ph )  =  B ) )
 
Theoremriota2 6322* 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  ->  ( ph  <->  ps ) )   =>    |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ( ps  <->  ( iota_ x  e.  A ph )  =  B ) )
 
Theoremriotaprop 6323* Properties of a restricted definite description operator. Todo: can some uses of riota2f 6321 be shortened with this? (Contributed by NM, 23-Nov-2013.)
 |- 
 F/ x ps   &    |-  B  =  ( iota_ x  e.  A ph )   &    |-  ( x  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( E! x  e.  A  ph  ->  ( B  e.  A  /\  ps ) )
 
Theoremriota5f 6324* A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( ph  ->  F/_ x B )   &    |-  ( ph  ->  B  e.  A )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( ps 
 <->  x  =  B ) )   =>    |-  ( ph  ->  ( iota_ x  e.  A ps )  =  B )
 
Theoremriota5 6325* A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.)
 |-  ( ph  ->  B  e.  A )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  x  =  B ) )   =>    |-  ( ph  ->  ( iota_ x  e.  A ps )  =  B )
 
Theoremriota5OLD 6326* A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (New usage is discouraged.)
 |-  ( ( ph  /\  B  e.  A  /\  x  e.  A )  ->  ( ps 
 <->  x  =  B ) )   =>    |-  ( ( ph  /\  B  e.  A )  ->  ( iota_ x  e.  A ps )  =  B )
 
Theoremriotass2 6327* Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)
 |-  ( ( ( A 
 C_  B  /\  A. x  e.  A  ( ph  ->  ps ) )  /\  ( E. x  e.  A  ph 
 /\  E! x  e.  B  ps ) )  ->  ( iota_ x  e.  A ph )  =  ( iota_ x  e.  B ps ) )
 
Theoremriotass 6328* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  A ph )  =  ( iota_ x  e.  B ph )
 )
 
Theoremmoriotass 6329* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.)
 |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E* x  e.  B ph )  ->  ( iota_ x  e.  A ph )  =  ( iota_ x  e.  B ph )
 )
 
Theoremsnriota 6330 A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)
 |-  ( E! x  e.  A  ph  ->  { x  e.  A  |  ph }  =  { ( iota_ x  e.  A ph ) }
 )
 
Theoremriotaxfrd 6331* Change the variable  x in the expression for "the unique 
x such that  ps " to another variable  y contained in expression  B. Use reuhypd 4560 to eliminate the last hypothesis. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ y C   &    |-  ( ( ph  /\  y  e.  A ) 
 ->  B  e.  A )   &    |-  ( ( ph  /\  ( iota_
 y  e.  A ch )  e.  A )  ->  C  e.  A )   &    |-  ( x  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 y  =  ( iota_ y  e.  A ch )  ->  B  =  C )   &    |-  ( ( ph  /\  x  e.  A )  ->  E! y  e.  A  x  =  B )   =>    |-  ( ( ph  /\  E! x  e.  A  ps )  ->  ( iota_ x  e.  A ps )  =  C )
 
Theoremeusvobj2 6332* 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  e.  _V   =>    |-  ( E! x E. y  e.  A  x  =  B  ->  ( E. y  e.  A  x  =  B  <->  A. y  e.  A  x  =  B )
 )
 
Theoremeusvobj1 6333* 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  e.  _V   =>    |-  ( E! x E. y  e.  A  x  =  B  ->  (
 iota x E. y  e.  A  x  =  B )  =  ( iota x
 A. y  e.  A  x  =  B )
 )
 
Theoremf1ofveu 6334* There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.)
 |-  ( ( F : A
 -1-1-onto-> B  /\  C  e.  B )  ->  E! x  e.  A  ( F `  x )  =  C )
 
Theoremf1ocnvfv3 6335* 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 : A
 -1-1-onto-> B  /\  C  e.  B )  ->  ( `' F `  C )  =  (
 iota_ x  e.  A ( F `  x )  =  C ) )
 
Theoremriotaund 6336* Restricted iota equals the undefined value of its domain of discourse  A when not meaningful. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( -.  E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( Undef `  A )
 )
 
Theoremriotaprc 6337* For proper classes, restricted and unrestricted iota are the same. (Contributed by NM, 15-Sep-2011.)
 |-  ( -.  A  e.  _V 
 ->  ( iota_ x  e.  A ph )  =  ( iota
 x ( x  e.  A  /\  ph )
 ) )
 
Theoremriotassuni 6338* The restricted iota class is limited in size by the base set. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( iota_ x  e.  A ph )  C_  ( ~P U. A  u.  U. A )
 
Theoremriotaclbg 6339* Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( A  e.  V  ->  ( E! x  e.  A  ph  <->  ( iota_ x  e.  A ph )  e.  A ) )
 
Theoremriotaclb 6340* Closure of restricted iota. (Contributed by NM, 15-Sep-2011.)
 |-  A  e.  _V   =>    |-  ( E! x  e.  A  ph  <->  ( iota_ x  e.  A ph )  e.  A )
 
Theoremriotaundb 6341* Restricted iota equals the undefined value of its domain of discourse  A when not meaningful. (Contributed by NM, 26-Sep-2011.)
 |-  A  e.  _V   =>    |-  ( -.  E! x  e.  A  ph  <->  ( iota_ x  e.  A ph )  =  ( Undef `  A )
 )
 
Theoremriotasvd 6342* Deduction version of riotasv 6347. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( ph  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )   &    |-  ( ph  ->  D  e.  A )   =>    |-  ( ( ph  /\  A  e.  V )  ->  (
 ( y  e.  B  /\  ps )  ->  D  =  C ) )
 
TheoremriotasvdOLD 6343* Deduction version of riotasv 6347. (Contributed by NM, 1-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  A. y ph )   &    |-  ( ph  ->  D  =  (
 iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )   =>    |-  ( ( ( ph  /\  A  e.  V ) 
 /\  D  e.  A  /\  ( y  e.  B  /\  ps ) )  ->  D  =  C )
 
Theoremriotasv2d 6344* Value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4539). Special case of riota2f 6321. (Contributed by NM, 2-Mar-2013.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ y F )   &    |-  ( ph  ->  F/ y ch )   &    |-  ( ph  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )   &    |-  ( ( ph  /\  y  =  E ) 
 ->  ( ps  <->  ch ) )   &    |-  (
 ( ph  /\  y  =  E )  ->  C  =  F )   &    |-  ( ph  ->  D  e.  A )   &    |-  ( ph  ->  E  e.  B )   &    |-  ( ph  ->  ch )   =>    |-  (
 ( ph  /\  A  e.  V )  ->  D  =  F )
 
Theoremriotasv2dOLD 6345* Value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4539). Special case of riota2f 6321. (Contributed by NM, 1-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  A. y ph )   &    |-  ( ph  ->  ( z  e.  F  ->  A. y  z  e.  F ) )   &    |-  ( ph  ->  ( ch  ->  A. y ch )
 )   &    |-  ( ph  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )   &    |-  ( ph  ->  ( y  =  E  ->  ( ps  <->  ch ) ) )   &    |-  ( ph  ->  ( y  =  E  ->  C  =  F ) )   =>    |-  ( ( (
 ph  /\  A  e.  V )  /\  ( D  e.  A  /\  E  e.  B  /\  ch )
 )  ->  D  =  F )
 
Theoremriotasv2s 6346* The value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4539) in the form of a substitution instance. Special case of riota2f 6321. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
 |-  D  =  ( iota_ x  e.  A A. y  e.  B  ( ph  ->  x  =  C ) )   =>    |-  ( ( A  e.  V  /\  D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  D  =  [_ E  /  y ]_ C )
 
Theoremriotasv 6347* Value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4539). Special case of riota2f 6321. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
 |-  A  e.  _V   &    |-  D  =  ( iota_ x  e.  A A. y  e.  B  ( ph  ->  x  =  C ) )   =>    |-  ( ( D  e.  A  /\  y  e.  B  /\  ph )  ->  D  =  C )
 
Theoremriotasv3d 6348* A property  ch holding for a representative of a single-valued class expression  C ( y ) (see e.g. reusv2 4539) also holds for its description binder  D (in the form of property  th). (Contributed by NM, 5-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ y th )   &    |-  ( ph  ->  D  =  (
 iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )   &    |-  ( ( ph  /\  C  =  D )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( ph  ->  ( ( y  e.  B  /\  ps )  ->  ch ) )   &    |-  ( ph  ->  D  e.  A )   &    |-  ( ph  ->  E. y  e.  B  ps )   =>    |-  ( ( ph  /\  A  e.  V ) 
 ->  th )
 
Theoremriotasv3dOLD 6349* A property  ch holding for a representative of a single-valued class expression  C ( y ) (see e.g. reusv2 4539) also holds for its description binder  D (in the form of property  th). (Contributed by NM, 1-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  A. y ph )   &    |-  ( ph  ->  ( th  ->  A. y th ) )   &    |-  ( ph  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )   &    |-  ( ph  ->  ( C  =  D  ->  ( ch  <->  th ) ) )   &    |-  ( ph  ->  ( ( y  e.  B  /\  ps )  ->  ch ) )   =>    |-  ( ( ph  /\  ( A  e.  V  /\  D  e.  A  /\  E. y  e.  B  ps ) )  ->  th )
 
2.4.19  Functions on ordinals; strictly monotone ordinal functions
 
Theoremiunon 6350* The indexed union of a set of ordinal numbers  B ( x ) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.)
 |-  ( ( A  e.  V  /\  A. x  e.  A  B  e.  On )  ->  U_ x  e.  A  B  e.  On )
 
TheoremiunonOLD 6351* The indexed union of a set of ordinal numbers  B ( x ) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A. x  e.  A  B  e.  On  -> 
 U_ x  e.  A  B  e.  On )
 
Theoremiinon 6352* The nonempty indexed intersection of a class of ordinal numbers  B ( x ) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
 |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  |^|_
 x  e.  A  B  e.  On )
 
Theoremonfununi 6353* A property of functions on ordinal numbers. Generalization of Theorem Schema 8E of [Enderton] p. 218. (Contributed by Eric Schmidt, 26-May-2009.)
 |-  ( Lim  y  ->  ( F `  y )  =  U_ x  e.  y  ( F `  x ) )   &    |-  (
 ( x  e.  On  /\  y  e.  On  /\  x  C_  y )  ->  ( F `  x ) 
 C_  ( F `  y ) )   =>    |-  ( ( S  e.  T  /\  S  C_ 
 On  /\  S  =/=  (/) )  ->  ( F ` 
 U. S )  = 
 U_ x  e.  S  ( F `  x ) )
 
Theoremonovuni 6354* A variant of onfununi 6353 for operations. (Contributed by Eric Schmidt, 26-May-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  ( Lim  y  ->  ( A F y )  =  U_ x  e.  y  ( A F x ) )   &    |-  (
 ( x  e.  On  /\  y  e.  On  /\  x  C_  y )  ->  ( A F x ) 
 C_  ( A F y ) )   =>    |-  ( ( S  e.  T  /\  S  C_ 
 On  /\  S  =/=  (/) )  ->  ( A F U. S )  = 
 U_ x  e.  S  ( A F x ) )
 
Theoremonoviun 6355* A variant of onovuni 6354 with indexed unions. (Contributed by Eric Schmidt, 26-May-2009.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
 |-  ( Lim  y  ->  ( A F y )  =  U_ x  e.  y  ( A F x ) )   &    |-  (
 ( x  e.  On  /\  y  e.  On  /\  x  C_  y )  ->  ( A F x ) 
 C_  ( A F y ) )   =>    |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  ( A F U_ z  e.  K  L )  =  U_ z  e.  K  ( A F L ) )
 
Theoremonnseq 6356* There are no length  om decreasing sequences in the ordinals. See also noinfep 7355 for a stronger version assuming Regularity. (Contributed by Mario Carneiro, 19-May-2015.)
 |-  ( ( F `  (/) )  e.  On  ->  E. x  e.  om  -.  ( F `  suc  x )  e.  ( F `  x ) )
 
Syntaxwsmo 6357 Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals.
 wff  Smo  A
 
Definitiondf-smo 6358* Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.)
 |-  ( Smo  A  <->  ( A : dom  A --> On  /\  Ord  dom  A 
 /\  A. x  e.  dom  A
 A. y  e.  dom  A ( x  e.  y  ->  ( A `  x )  e.  ( A `  y ) ) ) )
 
Theoremdfsmo2 6359* Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.)
 |-  ( Smo  F  <->  ( F : dom  F --> On  /\  Ord  dom  F 
 /\  A. x  e.  dom  F
 A. y  e.  x  ( F `  y )  e.  ( F `  x ) ) )
 
Theoremissmo 6360* Conditions for which  A is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.)
 |-  A : B --> On   &    |-  Ord  B   &    |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  e.  y  ->  ( A `  x )  e.  ( A `  y ) ) )   &    |-  dom 
 A  =  B   =>    |-  Smo  A
 
Theoremissmo2 6361* Alternative definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)
 |-  ( F : A --> B  ->  ( ( B 
 C_  On  /\  Ord  A  /\  A. x  e.  A  A. y  e.  x  ( F `  y )  e.  ( F `  x ) )  ->  Smo  F ) )
 
Theoremsmoeq 6362 Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |-  ( A  =  B  ->  ( Smo  A  <->  Smo  B ) )
 
Theoremsmodm 6363 The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |-  ( Smo  A  ->  Ord 
 dom  A )
 
Theoremsmores 6364 A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 16-Nov-2011.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
 |-  ( ( Smo  A  /\  B  e.  dom  A )  ->  Smo  ( A  |`  B ) )
 
Theoremsmores3 6365 A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B )  /\  Ord 
 B )  ->  Smo  ( A  |`  C ) )
 
Theoremsmores2 6366 A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)
 |-  ( ( Smo  F  /\  Ord  A )  ->  Smo  ( F  |`  A ) )
 
Theoremsmodm2 6367 The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.)
 |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )
 
Theoremsmofvon2 6368 The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)
 |-  ( Smo  F  ->  ( F `  B )  e.  On )
 
Theoremiordsmo 6369 The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |- 
 Ord  A   =>    |- 
 Smo  (  _I  |`  A )
 
Theoremsmo0 6370 The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)
 |- 
 Smo  (/)
 
Theoremsmofvon 6371 If  B is a strictly monotone ordinal function, and  A is in the domain of  B, then the value of the function at 
A is an ordinal. (Contributed by Andrew Salmon, 20-Nov-2011.)
 |-  ( ( Smo  B  /\  A  e.  dom  B )  ->  ( B `  A )  e.  On )
 
Theoremsmoel 6372 If  x is less than  y then a strictly monotone function's value will be strictly less at  x than at  y. (Contributed by Andrew Salmon, 22-Nov-2011.)
 |-  ( ( Smo  B  /\  A  e.  dom  B  /\  C  e.  A ) 
 ->  ( B `  C )  e.  ( B `  A ) )
 
Theoremsmoiun 6373* The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.)
 |-  ( ( Smo  B  /\  A  e.  dom  B )  ->  U_ x  e.  A  ( B `  x ) 
 C_  ( B `  A ) )
 
Theoremsmoiso 6374 If  F is an isomorphism from an ordinal  A onto  B, which is a subset of the ordinals, then 
F is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.)
 |-  ( ( F  Isom  _E 
 ,  _E  ( A ,  B )  /\  Ord 
 A  /\  B  C_  On )  ->  Smo  F )
 
Theoremsmoel2 6375 A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.)
 |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( B  e.  A  /\  C  e.  B ) )  ->  ( F `  C )  e.  ( F `  B ) )
 
Theoremsmo11 6376 A strictly monotone ordinal function is one-to-one. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( ( F : A
 --> B  /\  Smo  F )  ->  F : A -1-1-> B )
 
Theoremsmoord 6377 A strictly monotone ordinal function preserves strict ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)
 |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A ) )  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) )
 
Theoremsmoword 6378 A strictly monotone ordinal function preserves weak ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)
 |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A ) )  ->  ( C  C_  D  <->  ( F `  C )  C_  ( F `
  D ) ) )
 
Theoremsmogt 6379 A strictly monotone ordinal function is greater than or equal to its argument. Exercise 1 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 23-Nov-2011.) (Revised by Mario Carneiro, 28-Feb-2013.)
 |-  ( ( F  Fn  A  /\  Smo  F  /\  C  e.  A )  ->  C  C_  ( F `  C ) )
 
Theoremsmorndom 6380 The range of a strictly monotone ordinal function dominates the domain. (Contributed by Mario Carneiro, 13-Mar-2013.)
 |-  ( ( F : A
 --> B  /\  Smo  F  /\  Ord  B )  ->  A  C_  B )
 
Theoremsmoiso2 6381 The strictly monotone ordinal functions are also epsilon isomorphisms of subclasses of  On. (Contributed by Mario Carneiro, 20-Mar-2013.)
 |-  ( ( Ord  A  /\  B  C_  On )  ->  ( ( F : A -onto-> B  /\  Smo  F ) 
 <->  F  Isom  _E  ,  _E  ( A ,  B ) ) )
 
2.4.20  "Strong" transfinite recursion
 
Syntaxcrecs 6382 Notation for a function defined by strong transfinite recursion.
 class recs ( F )
 
Definitiondf-recs 6383* Define a function recs ( F ) on  On, the class of ordinal numbers, by transfinite recursion given a rule  F which sets the next value given all values so far. See df-rdg 6418 for more details on why this definition is desirable. Unlike df-rdg 6418 which restricts the update rule to use only the previous value, this version allows the update rule to use all previous values, which is why it is described as "strong", although it is actually more primitive. See recsfnon 6411 and recsval 6412 for the primary contract of this definition.

EDITORIAL: there are several existing versions of this construction without the definition, notably in ordtype 7242, zorn2 8128, and dfac8alem 7651. (Contributed by Stefan O'Rear, 18-Jan-2015.) (New usage is discouraged.)

 |- recs
 ( F )  = 
 U. { f  | 
 E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }
 
Theoremrecseq 6384 Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  ( F  =  G  -> recs ( F )  = recs ( G ) )
 
Theoremnfrecs 6385 Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F/_ x F   =>    |-  F/_ xrecs ( F )
 
Theoremtfrlem1 6386* A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47. (Contributed by NM, 23-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  ( A  e.  On  ->  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( A. x  e.  A  ( ( F `
  x )  =  ( B `  ( F  |`  x ) ) 
 /\  ( G `  x )  =  ( B `  ( G  |`  x ) ) )  ->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) ) )
 
Theoremtfrlem2 6387* Lemma for transfinite recursion. This provides some messy details needed to link tfrlem1 6386 into the main proof. (Contributed by NM, 23-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( ( <. x ,  y >.  e.  F  /\  <. x ,  z >.  e.  G )  ->  ( A  e.  On  ->  ( A. w ( A  e.  On  ->  ( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `
  w )  =  ( B `  ( G  |`  w ) ) ) ) )  ->  y  =  z )
 ) ) )
 
Theoremtfrlem3 6388* Lemma for transfinite recursion. Let  A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in  A for later use. (Contributed by NM, 9-Apr-1995.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |-  A  =  { g  |  E. z  e.  On  ( g  Fn  z  /\  A. y  e.  z  ( g `  y
 )  =  ( F `
  ( g  |`  y ) ) ) }
 
Theoremtfrlem3a 6389* Lemma for transfinite recursion. Let  A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in  A for later use. (Contributed by NM, 22-Jul-2012.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |-  A  =  { g  |  E. x  e.  On  ( g  Fn  x  /\  A. y  e.  x  ( g `  y
 )  =  ( F `
  ( g  |`  y ) ) ) }
 
Theoremtfrlem4 6390* Lemma for transfinite recursion.  A is the class of all "acceptable" functions, and  F is their union. First we show that an acceptable function is in fact a function. (Contributed by NM, 9-Apr-1995.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |-  ( g  e.  A  ->  Fun  g )
 
Theoremtfrlem5 6391* Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |-  ( ( g  e.  A  /\  h  e.  A )  ->  (
 ( <. x ,  u >.  e.  g  /\  <. x ,  v >.  e.  h )  ->  u  =  v ) )
 
Theoremrecsfval 6392* Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |- recs
 ( F )  = 
 U. A
 
Theoremtfrlem6 6393* Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |- 
 Rel recs ( F )
 
Theoremtfrlem7 6394* Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |- 
 Fun recs ( F )
 
Theoremtfrlem8 6395* Lemma for transfinite recursion. The domain of recs is ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |- 
 Ord  dom recs ( F )
 
Theoremtfrlem9 6396* Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |-  ( B  e.  dom recs ( F )  ->  (recs ( F ) `  B )  =  ( F `  (recs ( F )  |`  B ) ) )
 
Theoremtfrlem9a 6397* Lemma for transfinite recursion. Without using ax-rep 4132, show that all the restrictions of recs are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |-  ( B  e.  dom recs ( F )  ->  (recs ( F )  |`  B )  e.  _V )
 
Theoremtfrlem10 6398* Lemma for transfinite recursion. We define class  C by extending recs with one ordered pair. We will assume, falsely, that domain of recs is a member of, and thus not equal to,  On. Using this assumption we will prove facts about  C that will lead to a contradiction in tfrlem14 6402, thus showing the domain of recs does in fact equal  On. Here we show (under the false assumption) that  C is a function extending the domain of recs
( F ) by one. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   &    |-  C  =  (recs ( F )  u. 
 { <. dom recs ( F ) ,  ( F ` recs
 ( F ) )
 >. } )   =>    |-  ( dom recs ( F )  e.  On  ->  C  Fn  suc  dom recs ( F ) )
 
Theoremtfrlem11 6399* Lemma for transfinite recursion. Compute the value of  C. (Contributed by NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   &    |-  C  =  (recs ( F )  u. 
 { <. dom recs ( F ) ,  ( F ` recs
 ( F ) )
 >. } )   =>    |-  ( dom recs ( F )  e.  On  ->  ( B  e.  suc  dom recs ( F )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
 
Theoremtfrlem12 6400* Lemma for transfinite recursion. Show  C is an acceptable function. (Contributed by NM, 15-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   &    |-  C  =  (recs ( F )  u. 
 { <. dom recs ( F ) ,  ( F ` recs
 ( F ) )
 >. } )   =>    |-  (recs ( F )  e.  _V  ->  C  e.  A )
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