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Theorem List for Intuitionistic Logic Explorer - 5501-5600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfnex 5501 If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 5500. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )
 
Theoremfunex 5502 If the domain of a function exists, so does the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 5501. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995.)
 |-  ( ( Fun  F  /\  dom  F  e.  B )  ->  F  e.  _V )
 
Theoremopabex 5503* Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)
 |-  A  e.  _V   &    |-  ( x  e.  A  ->  E* y ph )   =>    |-  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  e.  _V
 
Theoremmptexg 5504* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  e.  V  ->  ( x  e.  A  |->  B )  e.  _V )
 
Theoremmptex 5505* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.)
 |-  A  e.  _V   =>    |-  ( x  e.  A  |->  B )  e. 
 _V
 
Theoremfex 5506 If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)
 |-  ( ( F : A
 --> B  /\  A  e.  C )  ->  F  e.  _V )
 
Theoremeufnfv 5507* A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 E! f ( f  Fn  A  /\  A. x  e.  A  (
 f `  x )  =  B )
 
Theoremfunfvima 5508 A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)
 |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) )
 
Theoremfunfvima2 5509 A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)
 |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) )
 
Theoremfunfvima3 5510 A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.)
 |-  ( ( Fun  F  /\  F  C_  G )  ->  ( A  e.  dom  F 
 ->  ( F `  A )  e.  ( G " { A } )
 ) )
 
Theoremfnfvima 5511 The function value of an operand in a set is contained in the image of that set, using the  Fn abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.)
 |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S ) 
 ->  ( F `  X )  e.  ( F " S ) )
 
Theoremfoima2 5512* Given an onto function, an element is in its codomain if and only if it is the image of an element of its domain (see foima 5222). (Contributed by BJ, 6-Jul-2022.)
 |-  ( F : A -onto-> B  ->  ( Y  e.  B 
 <-> 
 E. x  e.  A  Y  =  ( F `  x ) ) )
 
Theoremfoelrn 5513* Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.) (Proof shortened by BJ, 6-Jul-2022.)
 |-  ( ( F : A -onto-> B  /\  C  e.  B )  ->  E. x  e.  A  C  =  ( F `  x ) )
 
TheoremfoelrnOLD 5514* Obsolete proof of foelrn 5513 as of 6-Jul-2022. (Contributed by Jeff Madsen, 4-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( F : A -onto-> B  /\  C  e.  B )  ->  E. x  e.  A  C  =  ( F `  x ) )
 
Theoremfoco2 5515 If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  ( ( F : B
 --> C  /\  G : A
 --> B  /\  ( F  o.  G ) : A -onto-> C )  ->  F : B -onto-> C )
 
Theoremrexima 5516* Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |-  ( x  =  ( F `  y ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( ( F  Fn  A  /\  B  C_  A )  ->  ( E. x  e.  ( F " B ) ph  <->  E. y  e.  B  ps ) )
 
Theoremralima 5517* Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |-  ( x  =  ( F `  y ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( ( F  Fn  A  /\  B  C_  A )  ->  ( A. x  e.  ( F " B ) ph  <->  A. y  e.  B  ps ) )
 
Theoremidref 5518* TODO: This is the same as issref 4801 (which has a much longer proof). Should we replace issref 4801 with this one? - NM 9-May-2016.

Two ways to state a relation is reflexive. (Adapted from Tarski.) (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Proof modification is discouraged.)

 |-  ( (  _I  |`  A ) 
 C_  R  <->  A. x  e.  A  x R x )
 
Theoremelabrex 5519* Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)
 |-  B  e.  _V   =>    |-  ( x  e.  A  ->  B  e.  { y  |  E. x  e.  A  y  =  B } )
 
Theoremabrexco 5520* Composition of two image maps  C ( y ) and 
B ( w ). (Contributed by NM, 27-May-2013.)
 |-  B  e.  _V   &    |-  (
 y  =  B  ->  C  =  D )   =>    |-  { x  |  E. y  e.  { z  |  E. w  e.  A  z  =  B } x  =  C }  =  { x  |  E. w  e.  A  x  =  D }
 
Theoremimaiun 5521* The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( A " U_ x  e.  B  C )  = 
 U_ x  e.  B  ( A " C )
 
Theoremimauni 5522* 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.)
 |-  ( A " U. B )  =  U_ x  e.  B  ( A " x )
 
Theoremfniunfv 5523* 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.)
 |-  ( F  Fn  A  -> 
 U_ x  e.  A  ( F `  x )  =  U. ran  F )
 
Theoremfuniunfvdm 5524* The indexed union of a function's values is the union of its image under the index class. This theorem is a slight variation of fniunfv 5523. (Contributed by Jim Kingdon, 10-Jan-2019.)
 |-  ( F  Fn  A  -> 
 U_ x  e.  A  ( F `  x )  =  U. ( F
 " A ) )
 
Theoremfuniunfvdmf 5525* The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5524 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.)
 |-  F/_ x F   =>    |-  ( F  Fn  A  -> 
 U_ x  e.  A  ( F `  x )  =  U. ( F
 " A ) )
 
Theoremeluniimadm 5526* Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.)
 |-  ( F  Fn  A  ->  ( B  e.  U. ( F " A )  <->  E. x  e.  A  B  e.  ( F `  x ) ) )
 
Theoremelunirn 5527* Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.)
 |-  ( Fun  F  ->  ( A  e.  U. ran  F  <->  E. x  e.  dom  F  A  e.  ( F `
  x ) ) )
 
Theoremfnunirn 5528* Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( F  Fn  I  ->  ( A  e.  U. ran  F  <->  E. x  e.  I  A  e.  ( F `  x ) ) )
 
Theoremdff13 5529* 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.)
 |-  ( F : A -1-1-> B  <-> 
 ( F : A --> B  /\  A. x  e.  A  A. y  e.  A  ( ( F `
  x )  =  ( F `  y
 )  ->  x  =  y ) ) )
 
Theoremf1veqaeq 5530 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.)
 |-  ( ( F : A -1-1-> B  /\  ( C  e.  A  /\  D  e.  A ) )  ->  ( ( F `  C )  =  ( F `  D )  ->  C  =  D )
 )
 
Theoremdff13f 5531* 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.)
 |-  F/_ x F   &    |-  F/_ y F   =>    |-  ( F : A -1-1-> B  <->  ( F : A
 --> B  /\  A. x  e.  A  A. y  e.  A  ( ( F `
  x )  =  ( F `  y
 )  ->  x  =  y ) ) )
 
Theoremf1mpt 5532* Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  F  =  ( x  e.  A  |->  C )   &    |-  ( x  =  y  ->  C  =  D )   =>    |-  ( F : A -1-1-> B  <->  (
 A. x  e.  A  C  e.  B  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) ) )
 
Theoremf1fveq 5533 Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.)
 |-  ( ( F : A -1-1-> B  /\  ( C  e.  A  /\  D  e.  A ) )  ->  ( ( F `  C )  =  ( F `  D )  <->  C  =  D ) )
 
Theoremf1elima 5534 Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( F : A -1-1-> B  /\  X  e.  A  /\  Y  C_  A )  ->  ( ( F `
  X )  e.  ( F " Y ) 
 <->  X  e.  Y ) )
 
Theoremf1imass 5535 Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 |-  ( ( F : A -1-1-> B  /\  ( C 
 C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  C_  ( F
 " D )  <->  C  C_  D ) )
 
Theoremf1imaeq 5536 Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 |-  ( ( F : A -1-1-> B  /\  ( C 
 C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  =  ( F " D )  <->  C  =  D ) )
 
Theoremdff1o6 5537* A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)
 |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  ran 
 F  =  B  /\  A. x  e.  A  A. y  e.  A  (
 ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) )
 
Theoremf1ocnvfv1 5538 The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)
 |-  ( ( F : A
 -1-1-onto-> B  /\  C  e.  A )  ->  ( `' F `  ( F `  C ) )  =  C )
 
Theoremf1ocnvfv2 5539 The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)
 |-  ( ( F : A
 -1-1-onto-> B  /\  C  e.  B )  ->  ( F `  ( `' F `  C ) )  =  C )
 
Theoremf1ocnvfv 5540 Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004.)
 |-  ( ( F : A
 -1-1-onto-> B  /\  C  e.  A )  ->  ( ( F `
  C )  =  D  ->  ( `' F `  D )  =  C ) )
 
Theoremf1ocnvfvb 5541 Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)
 |-  ( ( F : A
 -1-1-onto-> B  /\  C  e.  A  /\  D  e.  B ) 
 ->  ( ( F `  C )  =  D  <->  ( `' F `  D )  =  C ) )
 
Theoremf1ocnvdm 5542 The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.)
 |-  ( ( F : A
 -1-1-onto-> B  /\  C  e.  B )  ->  ( `' F `  C )  e.  A )
 
Theoremf1ocnvfvrneq 5543 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.)
 |-  ( ( F : A -1-1-> B  /\  ( C  e.  ran  F  /\  D  e.  ran  F ) )  ->  ( ( `' F `  C )  =  ( `' F `  D )  ->  C  =  D ) )
 
Theoremfcof1 5544 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.)
 |-  ( ( F : A
 --> B  /\  ( R  o.  F )  =  (  _I  |`  A ) )  ->  F : A -1-1-> B )
 
Theoremfcofo 5545 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.)
 |-  ( ( F : A
 --> B  /\  S : B
 --> A  /\  ( F  o.  S )  =  (  _I  |`  B ) )  ->  F : A -onto-> B )
 
Theoremcbvfo 5546* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( F `  x )  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( F : A -onto-> B  ->  ( A. x  e.  A  ph  <->  A. y  e.  B  ps ) )
 
Theoremcbvexfo 5547* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)
 |-  ( ( F `  x )  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( F : A -onto-> B  ->  ( E. x  e.  A  ph  <->  E. y  e.  B  ps ) )
 
Theoremcocan1 5548 An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( F : B -1-1-> C  /\  H : A
 --> B  /\  K : A
 --> B )  ->  (
 ( F  o.  H )  =  ( F  o.  K )  <->  H  =  K ) )
 
Theoremcocan2 5549 A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B )  ->  ( ( H  o.  F )  =  ( K  o.  F ) 
 <->  H  =  K ) )
 
Theoremfcof1o 5550 Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( ( F : A --> B  /\  G : B --> A ) 
 /\  ( ( F  o.  G )  =  (  _I  |`  B ) 
 /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( F : A -1-1-onto-> B  /\  `' F  =  G ) )
 
Theoremfoeqcnvco 5551 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.)
 |-  ( ( F : A -onto-> B  /\  G : A -onto-> B )  ->  ( F  =  G  <->  ( F  o.  `' G )  =  (  _I  |`  B )
 ) )
 
Theoremf1eqcocnv 5552 Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
 |-  ( ( F : A -1-1-> B  /\  G : A -1-1-> B )  ->  ( F  =  G  <->  ( `' F  o.  G )  =  (  _I  |`  A )
 ) )
 
Theoremfliftrel 5553*  F, a function lift, is a subset of  R  X.  S. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   =>    |-  ( ph  ->  F  C_  ( R  X.  S ) )
 
Theoremfliftel 5554* Elementhood in the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   =>    |-  ( ph  ->  ( C F D  <->  E. x  e.  X  ( C  =  A  /\  D  =  B ) ) )
 
Theoremfliftel1 5555* Elementhood in the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   =>    |-  ( ( ph  /\  x  e.  X )  ->  A F B )
 
Theoremfliftcnv 5556* Converse of the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   =>    |-  ( ph  ->  `' F  =  ran  ( x  e.  X  |->  <. B ,  A >. ) )
 
Theoremfliftfun 5557* The function  F is the unique function defined by  F `  A  =  B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   &    |-  ( x  =  y  ->  A  =  C )   &    |-  ( x  =  y  ->  B  =  D )   =>    |-  ( ph  ->  ( Fun  F  <->  A. x  e.  X  A. y  e.  X  ( A  =  C  ->  B  =  D ) ) )
 
Theoremfliftfund 5558* The function  F is the unique function defined by  F `  A  =  B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   &    |-  ( x  =  y  ->  A  =  C )   &    |-  ( x  =  y  ->  B  =  D )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  A  =  C ) )  ->  B  =  D )   =>    |-  ( ph  ->  Fun  F )
 
Theoremfliftfuns 5559* The function  F is the unique function defined by  F `  A  =  B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   =>    |-  ( ph  ->  ( Fun  F  <->  A. y  e.  X  A. z  e.  X  (
 [_ y  /  x ]_ A  =  [_ z  /  x ]_ A  ->  [_ y  /  x ]_ B  =  [_ z  /  x ]_ B ) ) )
 
Theoremfliftf 5560* The domain and range of the function  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   =>    |-  ( ph  ->  ( Fun  F  <->  F : ran  ( x  e.  X  |->  A ) --> S ) )
 
Theoremfliftval 5561* The value of the function  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   &    |-  ( x  =  Y  ->  A  =  C )   &    |-  ( x  =  Y  ->  B  =  D )   &    |-  ( ph  ->  Fun 
 F )   =>    |-  ( ( ph  /\  Y  e.  X )  ->  ( F `  C )  =  D )
 
Theoremisoeq1 5562 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 |-  ( H  =  G  ->  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  G  Isom  R ,  S  ( A ,  B ) ) )
 
Theoremisoeq2 5563 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 |-  ( R  =  T  ->  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  T ,  S  ( A ,  B ) ) )
 
Theoremisoeq3 5564 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 |-  ( S  =  T  ->  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  R ,  T  ( A ,  B ) ) )
 
Theoremisoeq4 5565 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 |-  ( A  =  C  ->  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  R ,  S  ( C ,  B ) ) )
 
Theoremisoeq5 5566 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 |-  ( B  =  C  ->  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  R ,  S  ( A ,  C ) ) )
 
Theoremnfiso 5567 Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  F/_ x H   &    |-  F/_ x R   &    |-  F/_ x S   &    |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/ x  H  Isom  R ,  S  ( A ,  B )
 
Theoremisof1o 5568 An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B )
 
Theoremisorel 5569 An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)
 |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  e.  A  /\  D  e.  A ) )  ->  ( C R D  <->  ( H `  C ) S ( H `  D ) ) )
 
Theoremisoresbr 5570* A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.)
 |-  ( ( F  |`  A ) 
 Isom  R ,  S  ( A ,  ( F
 " A ) ) 
 ->  A. x  e.  A  A. y  e.  A  ( x R y  ->  ( F `  x ) S ( F `  y ) ) )
 
Theoremisoid 5571 Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
 |-  (  _I  |`  A ) 
 Isom  R ,  R  ( A ,  A )
 
Theoremisocnv 5572 Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  R  ( B ,  A ) )
 
Theoremisocnv2 5573 Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
 |-  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  `' R ,  `' S ( A ,  B ) )
 
Theoremisores2 5574 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
 |-  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  R ,  ( S  i^i  ( B  X.  B ) ) ( A ,  B ) )
 
Theoremisores1 5575 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
 |-  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  ( R  i^i  ( A  X.  A ) ) ,  S ( A ,  B ) )
 
Theoremisores3 5576 Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)
 |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  K  C_  A  /\  X  =  ( H " K ) )  ->  ( H  |`  K )  Isom  R ,  S  ( K ,  X ) )
 
Theoremisotr 5577 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.)
 |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  G  Isom  S ,  T  ( B ,  C ) )  ->  ( G  o.  H )  Isom  R ,  T  ( A ,  C ) )
 
Theoremiso0 5578 The empty set is an  R ,  S isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.)
 |-  (/)  Isom  R ,  S  ( (/) ,  (/) )
 
Theoremisoini 5579 Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)
 |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  D  e.  A )  ->  ( H " ( A  i^i  ( `' R " { D } )
 ) )  =  ( B  i^i  ( `' S " { ( H `  D ) }
 ) ) )
 
Theoremisoini2 5580 Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)
 |-  C  =  ( A  i^i  ( `' R " { X } )
 )   &    |-  D  =  ( B  i^i  ( `' S " { ( H `  X ) } )
 )   =>    |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  X  e.  A )  ->  ( H  |`  C ) 
 Isom  R ,  S  ( C ,  D ) )
 
Theoremisoselem 5581* Lemma for isose 5582. (Contributed by Mario Carneiro, 23-Jun-2015.)
 |-  ( ph  ->  H  Isom  R ,  S  ( A ,  B ) )   &    |-  ( ph  ->  ( H " x )  e.  _V )   =>    |-  ( ph  ->  ( R Se  A  ->  S Se  B ) )
 
Theoremisose 5582 An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  ( R Se  A  <->  S Se 
 B ) )
 
Theoremisopolem 5583 Lemma for isopo 5584. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  ( S  Po  B  ->  R  Po  A ) )
 
Theoremisopo 5584 An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  ( R  Po  A 
 <->  S  Po  B ) )
 
Theoremisosolem 5585 Lemma for isoso 5586. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  ( S  Or  B  ->  R  Or  A ) )
 
Theoremisoso 5586 An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  ( R  Or  A 
 <->  S  Or  B ) )
 
Theoremf1oiso 5587* Any one-to-one onto function determines an isomorphism with an induced relation  S. Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by NM, 30-Apr-2004.)
 |-  ( ( H : A
 -1-1-onto-> B  /\  S  =  { <. z ,  w >.  | 
 E. x  e.  A  E. y  e.  A  ( ( z  =  ( H `  x )  /\  w  =  ( H `  y ) )  /\  x R y ) } )  ->  H  Isom  R ,  S  ( A ,  B ) )
 
Theoremf1oiso2 5588* Any one-to-one onto function determines an isomorphism with an induced relation  S. (Contributed by Mario Carneiro, 9-Mar-2013.)
 |-  S  =  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B )  /\  ( `' H `  x ) R ( `' H `  y ) ) }   =>    |-  ( H : A -1-1-onto-> B  ->  H  Isom  R ,  S  ( A ,  B ) )
 
2.6.9  Restricted iota (description binder)
 
Syntaxcrio 5589 Extend class notation with restricted description binder.
 class  ( iota_ x  e.  A  ph )
 
Definitiondf-riota 5590 Define restricted description binder. In case there is no unique  x such that  ( x  e.  A  /\  ph ) holds, it evaluates to the empty set. See also comments for df-iota 4967. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 2-Sep-2018.)
 |-  ( iota_ x  e.  A  ph )  =  ( iota
 x ( x  e.  A  /\  ph )
 )
 
Theoremriotaeqdv 5591* Formula-building deduction for iota. (Contributed by NM, 15-Sep-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( iota_ x  e.  A  ps )  =  ( iota_ x  e.  B  ps ) )
 
Theoremriotabidv 5592* Formula-building deduction for restricted iota. (Contributed by NM, 15-Sep-2011.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 iota_ x  e.  A  ps )  =  ( iota_ x  e.  A  ch ) )
 
Theoremriotaeqbidv 5593* 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 ) )
 
Theoremriotaexg 5594* Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.)
 |-  ( A  e.  V  ->  ( iota_ x  e.  A  ps )  e.  _V )
 
Theoremriotav 5595 An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
 |-  ( iota_ x  e.  _V  ph )  =  ( iota
 x ph )
 
Theoremriotauni 5596 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 5597* 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 )
 
Theoremnfriotadxy 5598* Deduction version of nfriota 5599. (Contributed by Jim Kingdon, 12-Jan-2019.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/_ x (
 iota_ y  e.  A  ps ) )
 
Theoremnfriota 5599* 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 5600* 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 )
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