HomeHome Intuitionistic Logic Explorer
Theorem List (p. 58 of 141)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 5701-5800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfvtp1g 5701 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  ( ( ( A  e.  V  /\  D  e.  W )  /\  ( A  =/=  B  /\  A  =/=  C ) )  ->  ( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  A )  =  D )
 
Theoremfvtp2g 5702 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  ( ( ( B  e.  V  /\  E  e.  W )  /\  ( A  =/=  B  /\  B  =/=  C ) )  ->  ( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  B )  =  E )
 
Theoremfvtp3g 5703 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  ( ( ( C  e.  V  /\  F  e.  W )  /\  ( A  =/=  C  /\  B  =/=  C ) )  ->  ( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  C )  =  F )
 
Theoremfvtp1 5704 The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
 |-  A  e.  _V   &    |-  D  e.  _V   =>    |-  ( ( A  =/=  B 
 /\  A  =/=  C )  ->  ( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  A )  =  D )
 
Theoremfvtp2 5705 The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
 |-  B  e.  _V   &    |-  E  e.  _V   =>    |-  ( ( A  =/=  B 
 /\  B  =/=  C )  ->  ( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  B )  =  E )
 
Theoremfvtp3 5706 The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
 |-  C  e.  _V   &    |-  F  e.  _V   =>    |-  ( ( A  =/=  C 
 /\  B  =/=  C )  ->  ( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  C )  =  F )
 
Theoremfvconst2g 5707 The value of a constant function. (Contributed by NM, 20-Aug-2005.)
 |-  ( ( B  e.  D  /\  C  e.  A )  ->  ( ( A  X.  { B }
 ) `  C )  =  B )
 
Theoremfconst2g 5708 A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.)
 |-  ( B  e.  C  ->  ( F : A --> { B }  <->  F  =  ( A  X.  { B }
 ) ) )
 
Theoremfvconst2 5709 The value of a constant function. (Contributed by NM, 16-Apr-2005.)
 |-  B  e.  _V   =>    |-  ( C  e.  A  ->  ( ( A  X.  { B }
 ) `  C )  =  B )
 
Theoremfconst2 5710 A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.)
 |-  B  e.  _V   =>    |-  ( F : A
 --> { B }  <->  F  =  ( A  X.  { B }
 ) )
 
Theoremfconstfvm 5711* A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5710. (Contributed by Jim Kingdon, 8-Jan-2019.)
 |-  ( E. y  y  e.  A  ->  ( F : A --> { B } 
 <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) ) )
 
Theoremfconst3m 5712* Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)
 |-  ( E. x  x  e.  A  ->  ( F : A --> { B } 
 <->  ( F  Fn  A  /\  A  C_  ( `' F " { B }
 ) ) ) )
 
Theoremfconst4m 5713* Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
 |-  ( E. x  x  e.  A  ->  ( F : A --> { B } 
 <->  ( F  Fn  A  /\  ( `' F " { B } )  =  A ) ) )
 
Theoremresfunexg 5714 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.)
 |-  ( ( Fun  A  /\  B  e.  C ) 
 ->  ( A  |`  B )  e.  _V )
 
Theoremfnex 5715 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 5714. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )
 
Theoremfunex 5716 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 5715. (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 5717* 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 5718* 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 5719* 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
 
Theoremmptexd 5720* If the domain of a function given by maps-to notation is a set, the function is a set. Deduction version of mptexg 5718. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
 |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  _V )
 
Theoremmptrabex 5721* If the domain of a function given by maps-to notation is a class abstraction based on a set, the function is a set. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
 |-  A  e.  _V   =>    |-  ( x  e. 
 { y  e.  A  |  ph }  |->  B )  e.  _V
 
Theoremfex 5722 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 5723* 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 5724 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 5725 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 5726 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 5727 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 5728* 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 5423). (Contributed by BJ, 6-Jul-2022.)
 |-  ( F : A -onto-> B  ->  ( Y  e.  B 
 <-> 
 E. x  e.  A  Y  =  ( F `  x ) ) )
 
Theoremfoelrn 5729* 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 ) )
 
Theoremfoco2 5730 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 5731* 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 5732* 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 5733* TODO: This is the same as issref 4991 (which has a much longer proof). Should we replace issref 4991 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 5734* 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 5735* 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 5736* 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 5737* 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 5738* 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 5739* 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 5738. (Contributed by Jim Kingdon, 10-Jan-2019.)
 |-  ( F  Fn  A  -> 
 U_ x  e.  A  ( F `  x )  =  U. ( F
 " A ) )
 
Theoremfuniunfvdmf 5740* The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5739 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 5741* 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 5742* 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 5743* 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 5744* 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 5745 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 5746* 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 5747* 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 5748 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 5749 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 5750 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 5751 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 5752* 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 5753 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 5754 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 5755 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 5756 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 5757 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 5758 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 5759 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 5760 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 5761* 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 5762* 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 5763 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 5764 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 5765 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 5766 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 5767 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 5768*  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 5769* 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 5770* 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 5771* 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 5772* 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 5773* 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 5774* 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 5775* 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 5776* 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 5777 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 5778 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 5779 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 5780 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 5781 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 5782 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 5783 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 5784 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 5785* 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 5786 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 5787 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 5788 Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
 |-  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  `' R ,  `' S ( A ,  B ) )
 
Theoremisores2 5789 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 5790 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 5791 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 5792 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 5793 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 5794 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 5795 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 5796* Lemma for isose 5797. (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 5797 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 5798 Lemma for isopo 5799. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  ( S  Po  B  ->  R  Po  A ) )
 
Theoremisopo 5799 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 5800 Lemma for isoso 5801. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  ( S  Or  B  ->  R  Or  A ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14073
  Copyright terms: Public domain < Previous  Next >