HomeHome Metamath Proof Explorer
Theorem List (p. 56 of 322)
< Previous  Next >
Browser slow? Try the
Unicode version.

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

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21498)
  Hilbert Space Explorer  Hilbert Space Explorer
(21499-23021)
  Users' Mathboxes  Users' Mathboxes
(23022-32154)
 

Theorem List for Metamath Proof Explorer - 5501-5600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremf1cocnv2 5501 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
 |-  ( F : A -1-1-> B 
 ->  ( F  o.  `' F )  =  (  _I  |`  ran  F )
 )
 
Theoremf1ococnv1 5502 The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain. (Contributed by NM, 13-Dec-2003.)
 |-  ( F : A -1-1-onto-> B  ->  ( `' F  o.  F )  =  (  _I  |`  A ) )
 
Theoremf1cocnv1 5503 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
 |-  ( F : A -1-1-> B 
 ->  ( `' F  o.  F )  =  (  _I  |`  A ) )
 
Theoremfuncoeqres 5504 Re-express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( ( Fun  G  /\  ( F  o.  G )  =  H )  ->  ( F  |`  ran  G )  =  ( H  o.  `' G ) )
 
Theoremffoss 5505* Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)
 |-  F  e.  _V   =>    |-  ( F : A
 --> B  <->  E. x ( F : A -onto-> x  /\  x  C_  B ) )
 
Theoremf11o 5506* Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)
 |-  F  e.  _V   =>    |-  ( F : A -1-1-> B  <->  E. x ( F : A -1-1-onto-> x  /\  x  C_  B ) )
 
Theoremf10 5507 The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
 |-  (/) : (/) -1-1-> A
 
Theoremf1o00 5508 One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
 |-  ( F : (/) -1-1-onto-> A  <->  ( F  =  (/)  /\  A  =  (/) ) )
 
Theoremfo00 5509 Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
 |-  ( F : (/) -onto-> A  <-> 
 ( F  =  (/)  /\  A  =  (/) ) )
 
Theoremf1o0 5510 One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
 |-  (/) : (/)
 -1-1-onto-> (/)
 
Theoremf1oi 5511 A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  (  _I  |`  A ) : A -1-1-onto-> A
 
Theoremf1ovi 5512 The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.)
 |- 
 _I  : _V -1-1-onto-> _V
 
Theoremf1osn 5513 A singleton of an ordered pair is one-to-one onto function. (Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 { <. A ,  B >. } : { A }
 -1-1-onto-> { B }
 
Theoremf1osng 5514 A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. } : { A } -1-1-onto-> { B } )
 
Theoremf1oprswap 5515 A two-element swap is a bijection on a pair. (Contributed by Mario Carneiro, 23-Jan-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. ,  <. B ,  A >. } : { A ,  B } -1-1-onto-> { A ,  B }
 )
 
Theoremtz6.12-2 5516* Function value when  F is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  ( -.  E! x  A F x  ->  ( F `  A )  =  (/) )
 
Theoremfveu 5517* The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)
 |-  ( E! x  A F x  ->  ( F `
  A )  = 
 U. { x  |  A F x } )
 
Theorembrprcneu 5518* If  A is a proper class, then there is no unique binary relationship with  A as the first element. (Contributed by Scott Fenton, 7-Oct-2017.)
 |-  ( -.  A  e.  _V 
 ->  -.  E! x  A F x )
 
Theoremfvprc 5519 A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)
 |-  ( -.  A  e.  _V 
 ->  ( F `  A )  =  (/) )
 
Theoremfv2 5520* Alternate definition of function value. Definition 10.11 of [Quine] p. 68. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( F `  A )  =  U. { x  |  A. y ( A F y  <->  y  =  x ) }
 
Theoremdffv3 5521* A definition of function value in terms of iota. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  ( F `  A )  =  ( iota x x  e.  ( F
 " { A }
 ) )
 
Theoremdffv4 5522* The previous definition of function value, from before the  iota operator was introduced. Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 5041), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
 |-  ( F `  A )  =  U. { x  |  ( F " { A } )  =  { x } }
 
Theoremelfv 5523* Membership in a function value. (Contributed by NM, 30-Apr-2004.)
 |-  ( A  e.  ( F `  B )  <->  E. x ( A  e.  x  /\  A. y ( B F y 
 <->  y  =  x ) ) )
 
Theoremfveq1 5524 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
 |-  ( F  =  G  ->  ( F `  A )  =  ( G `  A ) )
 
Theoremfveq2 5525 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
 |-  ( A  =  B  ->  ( F `  A )  =  ( F `  B ) )
 
Theoremfveq1i 5526 Equality inference for function value. (Contributed by NM, 2-Sep-2003.)
 |-  F  =  G   =>    |-  ( F `  A )  =  ( G `  A )
 
Theoremfveq1d 5527 Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  ->  ( F `  A )  =  ( G `  A ) )
 
Theoremfveq2i 5528 Equality inference for function value. (Contributed by NM, 28-Jul-1999.)
 |-  A  =  B   =>    |-  ( F `  A )  =  ( F `  B )
 
Theoremfveq2d 5529 Equality deduction for function value. (Contributed by NM, 29-May-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F `  A )  =  ( F `  B ) )
 
Theoremfveq12i 5530 Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)
 |-  F  =  G   &    |-  A  =  B   =>    |-  ( F `  A )  =  ( G `  B )
 
Theoremfveq12d 5531 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F `  A )  =  ( G `  B ) )
 
Theoremnffv 5532 Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x F   &    |-  F/_ x A   =>    |-  F/_ x ( F `
  A )
 
Theoremnffvmpt1 5533* Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  F/_ x ( ( x  e.  A  |->  B ) `
  C )
 
Theoremnffvd 5534 Deduction version of bound-variable hypothesis builder nffv 5532. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( ph  ->  F/_ x F )   &    |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/_ x ( F `  A ) )
 
Theoremcsbfv12g 5535 Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)
 |-  ( A  e.  C  -> 
 [_ A  /  x ]_ ( F `  B )  =  ( [_ A  /  x ]_ F ` 
 [_ A  /  x ]_ B ) )
 
Theoremcsbfv12gALT 5536 Move class substitution in and out of a function value.(This is csbfv12g 5535 with a shortened proof, shortened by Alan Sare, 10-Nov-2012.) The proof is derived from the virtual deduction proof csbfv12gALTVD 28675. Although the proof is shorter, the total number of steps of all theorems used in the proof is probably longer. (Contributed by NM, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  C  -> 
 [_ A  /  x ]_ ( F `  B )  =  ( [_ A  /  x ]_ F ` 
 [_ A  /  x ]_ B ) )
 
Theoremcsbfv2g 5537* Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)
 |-  ( A  e.  C  -> 
 [_ A  /  x ]_ ( F `  B )  =  ( F ` 
 [_ A  /  x ]_ B ) )
 
Theoremcsbfvg 5538* Substitution for a function value. (Contributed by NM, 1-Jan-2006.)
 |-  ( A  e.  C  -> 
 [_ A  /  x ]_ ( F `  x )  =  ( F `  A ) )
 
Theoremfvex 5539 The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by NM, 30-Dec-1996.)
 |-  ( F `  A )  e.  _V
 
Theoremfvif 5540 Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( F `  if ( ph ,  A ,  B ) )  =  if ( ph ,  ( F `  A ) ,  ( F `  B ) )
 
Theoremfv3 5541* Alternate definition of the value of a function. Definition 6.11 of [TakeutiZaring] p. 26. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( F `  A )  =  { x  |  ( E. y ( x  e.  y  /\  A F y )  /\  E! y  A F y ) }
 
Theoremfvres 5542 The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
 |-  ( A  e.  B  ->  ( ( F  |`  B ) `
  A )  =  ( F `  A ) )
 
Theoremfunssfv 5543 The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.)
 |-  ( ( Fun  F  /\  G  C_  F  /\  A  e.  dom  G ) 
 ->  ( F `  A )  =  ( G `  A ) )
 
Theoremtz6.12-1 5544* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
 |-  ( ( A F y  /\  E! y  A F y )  ->  ( F `  A )  =  y )
 
Theoremtz6.12 5545* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.)
 |-  ( ( <. A ,  y >.  e.  F  /\  E! y <. A ,  y >.  e.  F )  ->  ( F `  A )  =  y )
 
Theoremtz6.12f 5546* Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)
 |-  F/_ y F   =>    |-  ( ( <. A ,  y >.  e.  F  /\  E! y <. A ,  y >.  e.  F )  ->  ( F `  A )  =  y )
 
Theoremtz6.12c 5547* Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
 |-  ( E! y  A F y  ->  (
 ( F `  A )  =  y  <->  A F y ) )
 
Theoremtz6.12i 5548 Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( B  =/=  (/)  ->  (
 ( F `  A )  =  B  ->  A F B ) )
 
Theoremfvbr0 5549 Two possibilities for the behavior of a function value. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  ( X F ( F `  X )  \/  ( F `  X )  =  (/) )
 
Theoremfvrn0 5550 A function value is a member of the range plus null. (Contributed by Scott Fenton, 8-Jun-2011.) (Revised by Stefan O'Rear, 3-Jan-2015.)
 |-  ( F `  X )  e.  ( ran  F  u.  { (/) } )
 
Theoremfvssunirn 5551 The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( F `  X )  C_  U. ran  F
 
Theoremndmfv 5552 The value of a class outside its domain is the empty set. (Contributed by NM, 24-Aug-1995.)
 |-  ( -.  A  e.  dom 
 F  ->  ( F `  A )  =  (/) )
 
Theoremndmfvrcl 5553 Reverse closure law for function with the empty set not in its domain. (Contributed by NM, 26-Apr-1996.)
 |- 
 dom  F  =  S   &    |-  -.  (/) 
 e.  S   =>    |-  ( ( F `  A )  e.  S  ->  A  e.  S )
 
Theoremelfvdm 5554 If a function value has a member, the argument belongs to the domain. (Contributed by NM, 12-Feb-2007.)
 |-  ( A  e.  ( F `  B )  ->  B  e.  dom  F )
 
Theoremelfvex 5555 If a function value has a member, the argument is a set. (Contributed by Mario Carneiro, 6-Nov-2015.)
 |-  ( A  e.  ( F `  B )  ->  B  e.  _V )
 
Theoremelfvexd 5556 If a function value is nonempty, its argument is a set. Deduction form of elfvex 5555. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ( B `  C ) )   =>    |-  ( ph  ->  C  e.  _V )
 
Theoremnfvres 5557 The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)
 |-  ( -.  A  e.  B  ->  ( ( F  |`  B ) `  A )  =  (/) )
 
Theoremnfunsn 5558 If the restriction of a class to a singleton is not a function, its value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( -.  Fun  ( F  |`  { A }
 )  ->  ( F `  A )  =  (/) )
 
Theoremfv01 5559 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
 |-  ( (/) `  A )  =  (/)
 
Theoremfveqres 5560 Equal values imply equal values in a restriction. (Contributed by NM, 13-Nov-1995.)
 |-  ( ( F `  A )  =  ( G `  A )  ->  ( ( F  |`  B ) `
  A )  =  ( ( G  |`  B ) `
  A ) )
 
Theoremfunbrfv 5561 The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( Fun  F  ->  ( A F B  ->  ( F `  A )  =  B ) )
 
Theoremfunopfv 5562 The second element in an ordered pair member of a function is the function's value. (Contributed by NM, 19-Jul-1996.)
 |-  ( Fun  F  ->  (
 <. A ,  B >.  e.  F  ->  ( F `  A )  =  B ) )
 
Theoremfnbrfvb 5563 Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `
  B )  =  C  <->  B F C ) )
 
Theoremfnopfvb 5564 Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `
  B )  =  C  <->  <. B ,  C >.  e.  F ) )
 
Theoremfunbrfvb 5565 Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)
 |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  ( ( F `
  A )  =  B  <->  A F B ) )
 
Theoremfunopfvb 5566 Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.)
 |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  ( ( F `
  A )  =  B  <->  <. A ,  B >.  e.  F ) )
 
Theoremfunbrfv2b 5567 Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)
 |-  ( Fun  F  ->  ( A F B  <->  ( A  e.  dom 
 F  /\  ( F `  A )  =  B ) ) )
 
Theoremdffn5 5568* Representation of a function in terms of its values. (Contributed by FL, 14-Sep-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
 
Theoremfnrnfv 5569* The range of a function expressed as a collection of the function's values. (Contributed by NM, 20-Oct-2005.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  ( F  Fn  A  ->  ran  F  =  {
 y  |  E. x  e.  A  y  =  ( F `  x ) } )
 
Theoremfvelrnb 5570* A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)
 |-  ( F  Fn  A  ->  ( B  e.  ran  F  <->  E. x  e.  A  ( F `  x )  =  B ) )
 
Theoremdfimafn 5571* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)
 |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( F " A )  =  { y  |  E. x  e.  A  ( F `  x )  =  y } )
 
Theoremdfimafn2 5572* Alternate definition of the image of a function as an indexed union of singletons of function values. (Contributed by Raph Levien, 20-Nov-2006.)
 |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( F " A )  =  U_ x  e.  A  { ( F `
  x ) }
 )
 
Theoremfunimass4 5573* Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.)
 |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( ( F " A )  C_  B  <->  A. x  e.  A  ( F `  x )  e.  B ) )
 
Theoremfvelima 5574* Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42. (Contributed by NM, 29-Apr-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ( Fun  F  /\  A  e.  ( F
 " B ) ) 
 ->  E. x  e.  B  ( F `  x )  =  A )
 
Theoremfeqmptd 5575* Deduction form of dffn5 5568. (Contributed by Mario Carneiro, 8-Jan-2015.)
 |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
 
Theoremfeqresmpt 5576* Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  C  C_  A )   =>    |-  ( ph  ->  ( F  |`  C )  =  ( x  e.  C  |->  ( F `  x ) ) )
 
Theoremdffn5f 5577* Representation of a function in terms of its values. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  F/_ x F   =>    |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
 
Theoremfvelimab 5578* Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.)
 |-  ( ( F  Fn  A  /\  B  C_  A )  ->  ( C  e.  ( F " B )  <->  E. x  e.  B  ( F `  x )  =  C ) )
 
Theoremfvi 5579 The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( A  e.  V  ->  (  _I  `  A )  =  A )
 
Theoremfviss 5580 The value of the identity function is a subset of the argument. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  (  _I  `  A )  C_  A
 
Theoremfniinfv 5581* The indexed intersection of a function's values is the intersection of its range. (Contributed by NM, 20-Oct-2005.)
 |-  ( F  Fn  A  -> 
 |^|_ x  e.  A  ( F `  x )  =  |^| ran  F )
 
Theoremfnsnfv 5582 Singleton of function value. (Contributed by NM, 22-May-1998.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { ( F `
  B ) }  =  ( F " { B } ) )
 
Theoremfnimapr 5583 The image of a pair under a funtion. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A ) 
 ->  ( F " { B ,  C }
 )  =  { ( F `  B ) ,  ( F `  C ) } )
 
Theoremssimaex 5584* The existence of a subimage. (Contributed by NM, 8-Apr-2007.)
 |-  A  e.  _V   =>    |-  ( ( Fun 
 F  /\  B  C_  ( F " A ) ) 
 ->  E. x ( x 
 C_  A  /\  B  =  ( F " x ) ) )
 
Theoremssimaexg 5585* The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
 |-  ( ( A  e.  C  /\  Fun  F  /\  B  C_  ( F " A ) )  ->  E. x ( x  C_  A  /\  B  =  ( F " x ) ) )
 
Theoremfunfv 5586 A simplified expression for the value of a function when we know it's a function. (Contributed by NM, 22-May-1998.)
 |-  ( Fun  F  ->  ( F `  A )  =  U. ( F
 " { A }
 ) )
 
Theoremfunfv2 5587* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by NM, 22-May-1998.)
 |-  ( Fun  F  ->  ( F `  A )  =  U. { y  |  A F y }
 )
 
Theoremfunfv2f 5588 The value of a function. Version of funfv2 5587 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.)
 |-  F/_ y A   &    |-  F/_ y F   =>    |-  ( Fun  F  ->  ( F `  A )  =  U. { y  |  A F y }
 )
 
Theoremfvun 5589 Value of the union of two functions when the domains are separate. (Contributed by FL, 7-Nov-2011.)
 |-  ( ( ( Fun 
 F  /\  Fun  G ) 
 /\  ( dom  F  i^i  dom  G )  =  (/) )  ->  ( ( F  u.  G ) `
  A )  =  ( ( F `  A )  u.  ( G `  A ) ) )
 
Theoremfvun1 5590 The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  ( ( F  u.  G ) `  X )  =  ( F `  X ) )
 
Theoremfvun2 5591 The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  B ) )  ->  ( ( F  u.  G ) `  X )  =  ( G `  X ) )
 
Theoremdffv2 5592 Alternate definition of function value df-fv 5263 that doesn't require dummy variables. (Contributed by NM, 4-Aug-2010.)
 |-  ( F `  A )  =  U. ( ( F " { A } )  \  U. U. ( ( ( F  |`  { A } )  o.  `' ( F  |`  { A } ) )  \  _I  ) )
 
Theoremdmfco 5593 Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
 |-  ( ( Fun  G  /\  A  e.  dom  G )  ->  ( A  e.  dom  ( F  o.  G ) 
 <->  ( G `  A )  e.  dom  F ) )
 
Theoremfvco2 5594 Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (Contributed by NM, 9-Oct-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 16-Oct-2014.)
 |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) `  X )  =  ( F `  ( G `  X ) ) )
 
Theoremfvco 5595 Value of a function composition. Similar to Exercise 5 of [TakeutiZaring] p. 28. (Contributed by NM, 22-Apr-2006.) (Proof shortened by Mario Carneiro, 26-Dec-2014.)
 |-  ( ( Fun  G  /\  A  e.  dom  G )  ->  ( ( F  o.  G ) `  A )  =  ( F `  ( G `  A ) ) )
 
Theoremfvco3 5596 Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.)
 |-  ( ( G : A
 --> B  /\  C  e.  A )  ->  ( ( F  o.  G ) `
  C )  =  ( F `  ( G `  C ) ) )
 
Theoremfvco4i 5597 Conditions for a composition to be expandable without conditions on the argument. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  (/)  =  ( F `  (/) )   &    |-  Fun  G   =>    |-  ( ( F  o.  G ) `  X )  =  ( F `  ( G `  X ) )
 
Theoremfvopab3g 5598* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  ( x  e.  C  ->  E! y ph )   &    |-  F  =  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) }   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( F `
  A )  =  B  <->  ch ) )
 
Theoremfvopab3ig 5599* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  ( x  e.  C  ->  E* y ph )   &    |-  F  =  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) }   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ch  ->  ( F `  A )  =  B ) )
 
Theoremfvmptg 5600* Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( x  =  A  ->  B  =  C )   &    |-  F  =  ( x  e.  D  |->  B )   =>    |-  ( ( A  e.  D  /\  C  e.  R )  ->  ( F `  A )  =  C )
    < 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 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32154
  Copyright terms: Public domain < Previous  Next >