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Theorem List for Intuitionistic Logic Explorer - 5801-5900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoffval 5801* Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  S   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( F `  x )  =  C )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  ( G `  x )  =  D )   =>    |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  S  |->  ( C R D ) ) )
 
Theoremofrfval 5802* Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  S   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( F `  x )  =  C )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  ( G `  x )  =  D )   =>    |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  S  C R D ) )
 
Theoremfnofval 5803 Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  S   &    |-  (
 ( ph  /\  X  e.  A )  ->  ( F `
  X )  =  C )   &    |-  ( ( ph  /\  X  e.  B ) 
 ->  ( G `  X )  =  D )   &    |-  ( ph  ->  R  Fn  ( U  X.  V ) )   &    |-  ( ph  ->  C  e.  U )   &    |-  ( ph  ->  D  e.  V )   =>    |-  ( ( ph  /\  X  e.  S ) 
 ->  ( ( F  oF R G ) `  X )  =  ( C R D ) )
 
Theoremofrval 5804 Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  S   &    |-  (
 ( ph  /\  X  e.  A )  ->  ( F `
  X )  =  C )   &    |-  ( ( ph  /\  X  e.  B ) 
 ->  ( G `  X )  =  D )   =>    |-  (
 ( ph  /\  F  oR R G  /\  X  e.  S )  ->  C R D )
 
Theoremofmresval 5805 Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
 |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( F (  oF R  |`  ( A  X.  B ) ) G )  =  ( F  oF R G ) )
 
Theoremoff 5806* The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  T )
 )  ->  ( x R y )  e.  U )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : B
 --> T )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  C   =>    |-  ( ph  ->  ( F  oF R G ) : C --> U )
 
Theoremofres 5807 Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  C   =>    |-  ( ph  ->  ( F  oF R G )  =  ( ( F  |`  C )  oF R ( G  |`  C )
 ) )
 
Theoremoffval2 5808* The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  W )   &    |-  ( ( ph  /\  x  e.  A )  ->  C  e.  X )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   &    |-  ( ph  ->  G  =  ( x  e.  A  |->  C ) )   =>    |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  A  |->  ( B R C ) ) )
 
Theoremofrfval2 5809* The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  W )   &    |-  ( ( ph  /\  x  e.  A )  ->  C  e.  X )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   &    |-  ( ph  ->  G  =  ( x  e.  A  |->  C ) )   =>    |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  A  B R C ) )
 
Theoremsuppssof1 5810* Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  ( ph  ->  ( `' A " ( _V  \  { Y } )
 )  C_  L )   &    |-  (
 ( ph  /\  v  e.  R )  ->  ( Y O v )  =  Z )   &    |-  ( ph  ->  A : D --> V )   &    |-  ( ph  ->  B : D
 --> R )   &    |-  ( ph  ->  D  e.  W )   =>    |-  ( ph  ->  ( `' ( A  oF O B ) " ( _V  \  { Z }
 ) )  C_  L )
 
Theoremofco 5811 The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  H : D --> C )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  D  e.  X )   &    |-  ( A  i^i  B )  =  C   =>    |-  ( ph  ->  ( ( F  oF R G )  o.  H )  =  ( ( F  o.  H )  oF R ( G  o.  H ) ) )
 
Theoremoffveqb 5812* Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  A )   &    |-  ( ph  ->  H  Fn  A )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( F `  x )  =  B )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( G `  x )  =  C )   =>    |-  ( ph  ->  ( H  =  ( F  oF R G )  <->  A. x  e.  A  ( H `  x )  =  ( B R C ) ) )
 
Theoremofc12 5813 Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   =>    |-  ( ph  ->  (
 ( A  X.  { B } )  oF R ( A  X.  { C } ) )  =  ( A  X.  { ( B R C ) } ) )
 
Theoremcaofref 5814* Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ( ph  /\  x  e.  S )  ->  x R x )   =>    |-  ( ph  ->  F  oR R F )
 
Theoremcaofinvl 5815* Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  N : S --> S )   &    |-  ( ph  ->  G  =  ( v  e.  A  |->  ( N `  ( F `
  v ) ) ) )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  ( ( N `  x ) R x )  =  B )   =>    |-  ( ph  ->  ( G  oF R F )  =  ( A  X.  { B } ) )
 
Theoremcaofcom 5816* Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x R y )  =  ( y R x ) )   =>    |-  ( ph  ->  ( F  oF R G )  =  ( G  oF R F ) )
 
Theoremcaofrss 5817* Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x R y  ->  x T y ) )   =>    |-  ( ph  ->  ( F  oR R G  ->  F  oR T G ) )
 
Theoremcaoftrn 5818* Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ph  ->  H : A --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x R y  /\  y T z )  ->  x U z ) )   =>    |-  ( ph  ->  ( ( F  oR R G  /\  G  oR T H )  ->  F  oR U H ) )
 
2.6.13  Functions (continued)
 
TheoremresfunexgALT 5819 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. This version has a shorter proof than resfunexg 5461 but requires ax-pow 3977 and ax-un 4227. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( Fun  A  /\  B  e.  C ) 
 ->  ( A  |`  B )  e.  _V )
 
Theoremcofunexg 5820 Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
 |-  ( ( Fun  A  /\  B  e.  C ) 
 ->  ( A  o.  B )  e.  _V )
 
Theoremcofunex2g 5821 Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)
 |-  ( ( A  e.  V  /\  Fun  `' B )  ->  ( A  o.  B )  e.  _V )
 
TheoremfnexALT 5822 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 funimaexg 5054. This version of fnex 5462 uses ax-pow 3977 and ax-un 4227, whereas fnex 5462 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )
 
Theoremfunrnex 5823 If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 5463. (Contributed by NM, 11-Nov-1995.)
 |-  ( dom  F  e.  B  ->  ( Fun  F  ->  ran  F  e.  _V ) )
 
Theoremfornex 5824 If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
 |-  ( A  e.  C  ->  ( F : A -onto-> B  ->  B  e.  _V ) )
 
Theoremf1dmex 5825 If the codomain of a one-to-one function exists, so does its domain. This can be thought of as a form of the Axiom of Replacement. (Contributed by NM, 4-Sep-2004.)
 |-  ( ( F : A -1-1-> B  /\  B  e.  C )  ->  A  e.  _V )
 
Theoremabrexex 5826* Existence of a class abstraction of existentially restricted sets.  x is normally a free-variable parameter in the class expression substituted for  B, which can be thought of as  B ( x ). This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path mptexg 5465, funex 5463, fnex 5462, resfunexg 5461, and funimaexg 5054. See also abrexex2 5833. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  _V   =>    |-  { y  | 
 E. x  e.  A  y  =  B }  e.  _V
 
Theoremabrexexg 5827* Existence of a class abstraction of existentially restricted sets.  x is normally a free-variable parameter in  B. The antecedent assures us that  A is a set. (Contributed by NM, 3-Nov-2003.)
 |-  ( A  e.  V  ->  { y  |  E. x  e.  A  y  =  B }  e.  _V )
 
Theoremiunexg 5828* The existence of an indexed union. 
x is normally a free-variable parameter in  B. (Contributed by NM, 23-Mar-2006.)
 |-  ( ( A  e.  V  /\  A. x  e.  A  B  e.  W )  ->  U_ x  e.  A  B  e.  _V )
 
Theoremabrexex2g 5829* Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( A  e.  V  /\  A. x  e.  A  { y  | 
 ph }  e.  W )  ->  { y  | 
 E. x  e.  A  ph
 }  e.  _V )
 
Theoremopabex3d 5830* Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  { y  |  ps }  e.  _V )   =>    |-  ( ph  ->  {
 <. x ,  y >.  |  ( x  e.  A  /\  ps ) }  e.  _V )
 
Theoremopabex3 5831* Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  A  e.  _V   &    |-  ( x  e.  A  ->  { y  |  ph }  e.  _V )   =>    |- 
 { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  e.  _V
 
Theoremiunex 5832* The existence of an indexed union. 
x is normally a free-variable parameter in the class expression substituted for  B, which can be read informally as  B ( x ). (Contributed by NM, 13-Oct-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  U_ x  e.  A  B  e.  _V
 
Theoremabrexex2 5833* Existence of an existentially restricted class abstraction.  ph is normally has free-variable parameters  x and  y. See also abrexex 5826. (Contributed by NM, 12-Sep-2004.)
 |-  A  e.  _V   &    |-  { y  |  ph }  e.  _V   =>    |-  { y  |  E. x  e.  A  ph
 }  e.  _V
 
Theoremabexssex 5834* Existence of a class abstraction with an existentially quantified expression. Both  x and  y can be free in  ph. (Contributed by NM, 29-Jul-2006.)
 |-  A  e.  _V   &    |-  { y  |  ph }  e.  _V   =>    |-  { y  |  E. x ( x 
 C_  A  /\  ph ) }  e.  _V
 
Theoremabexex 5835* A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
 |-  A  e.  _V   &    |-  ( ph  ->  x  e.  A )   &    |- 
 { y  |  ph }  e.  _V   =>    |- 
 { y  |  E. x ph }  e.  _V
 
Theoremoprabexd 5836* Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  E* z ps )   &    |-  ( ph  ->  F  =  { <.
 <. x ,  y >. ,  z >.  |  (
 ( x  e.  A  /\  y  e.  B )  /\  ps ) }
 )   =>    |-  ( ph  ->  F  e.  _V )
 
Theoremoprabex 5837* Existence of an operation class abstraction. (Contributed by NM, 19-Oct-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  e.  A  /\  y  e.  B )  ->  E* z ph )   &    |-  F  =  { <.
 <. x ,  y >. ,  z >.  |  (
 ( x  e.  A  /\  y  e.  B )  /\  ph ) }   =>    |-  F  e.  _V
 
Theoremoprabex3 5838* Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)
 |-  H  e.  _V   &    |-  F  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  ( H  X.  H )  /\  y  e.  ( H  X.  H ) )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f >. ) 
 /\  z  =  R ) ) }   =>    |-  F  e.  _V
 
Theoremoprabrexex2 5839* Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
 |-  A  e.  _V   &    |-  { <. <. x ,  y >. ,  z >.  |  ph }  e.  _V   =>    |-  {
 <. <. x ,  y >. ,  z >.  |  E. w  e.  A  ph }  e.  _V
 
Theoremab2rexex 5840* Existence of a class abstraction of existentially restricted sets. Variables  x and  y are normally free-variable parameters in the class expression substituted for  C, which can be thought of as  C ( x ,  y ). See comments for abrexex 5826. (Contributed by NM, 20-Sep-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 { z  |  E. x  e.  A  E. y  e.  B  z  =  C }  e.  _V
 
Theoremab2rexex2 5841* Existence of an existentially restricted class abstraction.  ph normally has free-variable parameters  x,  y, and  z. Compare abrexex2 5833. (Contributed by NM, 20-Sep-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  { z  | 
 ph }  e.  _V   =>    |-  { z  |  E. x  e.  A  E. y  e.  B  ph
 }  e.  _V
 
TheoremxpexgALT 5842 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4513 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B )  e.  _V )
 
Theoremoffval3 5843* General value of  ( F  oF R G ) with no assumptions on functionality of  F and  G. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( F  oF R G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
  x ) R ( G `  x ) ) ) )
 
Theoremoffres 5844 Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( F  oF R G )  |`  D )  =  ( ( F  |`  D )  oF R ( G  |`  D )
 ) )
 
Theoremofmres 5845* Equivalent expressions for a restriction of the function operation map. Unlike  oF R which is a proper class,  (  oF R  |  `  ( A  X.  B
) ) can be a set by ofmresex 5846, allowing it to be used as a function or structure argument. By ofmresval 5805, the restricted operation map values are the same as the original values, allowing theorems for  oF R to be reused. (Contributed by NM, 20-Oct-2014.)
 |-  (  oF R  |`  ( A  X.  B ) )  =  (
 f  e.  A ,  g  e.  B  |->  ( f  oF R g ) )
 
Theoremofmresex 5846 Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   =>    |-  ( ph  ->  (  oF R  |`  ( A  X.  B ) )  e.  _V )
 
2.6.14  First and second members of an ordered pair
 
Syntaxc1st 5847 Extend the definition of a class to include the first member an ordered pair function.
 class  1st
 
Syntaxc2nd 5848 Extend the definition of a class to include the second member an ordered pair function.
 class  2nd
 
Definitiondf-1st 5849 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 5855 proves that it does this. For example, ( 1st `  <. 3 , 4  >.) = 3 . Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 4869 and op1stb 4266). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
 |- 
 1st  =  ( x  e.  _V  |->  U. dom  { x } )
 
Definitiondf-2nd 5850 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 5856 proves that it does this. For example,  ( 2nd ` 
<. 3 , 4 
>.) = 4 . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 4872 and op2ndb 4871). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
 |- 
 2nd  =  ( x  e.  _V  |->  U. ran  { x } )
 
Theorem1stvalg 5851 The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( A  e.  _V  ->  ( 1st `  A )  =  U. dom  { A } )
 
Theorem2ndvalg 5852 The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( A  e.  _V  ->  ( 2nd `  A )  =  U. ran  { A } )
 
Theorem1st0 5853 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
 |-  ( 1st `  (/) )  =  (/)
 
Theorem2nd0 5854 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
 |-  ( 2nd `  (/) )  =  (/)
 
Theoremop1st 5855 Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( 1st `  <. A ,  B >. )  =  A
 
Theoremop2nd 5856 Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( 2nd `  <. A ,  B >. )  =  B
 
Theoremop1std 5857 Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( C  =  <. A ,  B >.  ->  ( 1st `  C )  =  A )
 
Theoremop2ndd 5858 Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( C  =  <. A ,  B >.  ->  ( 2nd `  C )  =  B )
 
Theoremop1stg 5859 Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 1st `  <. A ,  B >. )  =  A )
 
Theoremop2ndg 5860 Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
 
Theoremot1stg 5861 Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 5861, ot2ndg 5862, ot3rdgg 5863.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( 1st `  ( 1st `  <. A ,  B ,  C >. ) )  =  A )
 
Theoremot2ndg 5862 Extract the second member of an ordered triple. (See ot1stg 5861 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  B )
 
Theoremot3rdgg 5863 Extract the third member of an ordered triple. (See ot1stg 5861 comment.) (Contributed by NM, 3-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( 2nd `  <. A ,  B ,  C >. )  =  C )
 
Theorem1stval2 5864 Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)
 |-  ( A  e.  ( _V  X.  _V )  ->  ( 1st `  A )  =  |^| |^| A )
 
Theorem2ndval2 5865 Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)
 |-  ( A  e.  ( _V  X.  _V )  ->  ( 2nd `  A )  =  |^| |^| |^| `' { A } )
 
Theoremfo1st 5866 The  1st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |- 
 1st : _V -onto-> _V
 
Theoremfo2nd 5867 The  2nd function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |- 
 2nd : _V -onto-> _V
 
Theoremf1stres 5868 Mapping of a restriction of the 
1st (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B ) --> A
 
Theoremf2ndres 5869 Mapping of a restriction of the 
2nd (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B ) --> B
 
Theoremfo1stresm 5870* Onto mapping of a restriction of the  1st (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
 |-  ( E. y  y  e.  B  ->  ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B ) -onto-> A )
 
Theoremfo2ndresm 5871* Onto mapping of a restriction of the  2nd (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
 |-  ( E. x  x  e.  A  ->  ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B ) -onto-> B )
 
Theorem1stcof 5872 Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
 |-  ( F : A --> ( B  X.  C ) 
 ->  ( 1st  o.  F ) : A --> B )
 
Theorem2ndcof 5873 Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)
 |-  ( F : A --> ( B  X.  C ) 
 ->  ( 2nd  o.  F ) : A --> C )
 
Theoremxp1st 5874 Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  e.  ( B  X.  C )  ->  ( 1st `  A )  e.  B )
 
Theoremxp2nd 5875 Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  e.  ( B  X.  C )  ->  ( 2nd `  A )  e.  C )
 
Theorem1stexg 5876 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
 |-  ( A  e.  V  ->  ( 1st `  A )  e.  _V )
 
Theorem2ndexg 5877 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
 |-  ( A  e.  V  ->  ( 2nd `  A )  e.  _V )
 
Theoremelxp6 5878 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4875. (Contributed by NM, 9-Oct-2004.)
 |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A ) >.  /\  (
 ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C ) ) )
 
Theoremelxp7 5879 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4875. (Contributed by NM, 19-Aug-2006.)
 |-  ( A  e.  ( B  X.  C )  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C ) ) )
 
Theoremeqopi 5880 Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( A  e.  ( V  X.  W ) 
 /\  ( ( 1st `  A )  =  B  /\  ( 2nd `  A )  =  C )
 )  ->  A  =  <. B ,  C >. )
 
Theoremxp2 5881* Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)
 |-  ( A  X.  B )  =  { x  e.  ( _V  X.  _V )  |  ( ( 1st `  x )  e.  A  /\  ( 2nd `  x )  e.  B ) }
 
Theoremunielxp 5882 The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)
 |-  ( A  e.  ( B  X.  C )  ->  U. A  e.  U. ( B  X.  C ) )
 
Theorem1st2nd2 5883 Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)
 |-  ( A  e.  ( B  X.  C )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A ) >. )
 
Theoremxpopth 5884 An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)
 |-  ( ( A  e.  ( C  X.  D ) 
 /\  B  e.  ( R  X.  S ) ) 
 ->  ( ( ( 1st `  A )  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B ) )  <->  A  =  B ) )
 
Theoremeqop 5885 Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  e.  ( V  X.  W )  ->  ( A  =  <. B ,  C >.  <->  ( ( 1st `  A )  =  B  /\  ( 2nd `  A )  =  C )
 ) )
 
Theoremeqop2 5886 Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)
 |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( A  =  <. B ,  C >.  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  =  B  /\  ( 2nd `  A )  =  C ) ) )
 
Theoremop1steq 5887* Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.)
 |-  ( A  e.  ( V  X.  W )  ->  ( ( 1st `  A )  =  B  <->  E. x  A  =  <. B ,  x >. ) )
 
Theorem2nd1st 5888 Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
 |-  ( A  e.  ( B  X.  C )  ->  U. `' { A }  =  <. ( 2nd `  A ) ,  ( 1st `  A ) >. )
 
Theorem1st2nd 5889 Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)
 |-  ( ( Rel  B  /\  A  e.  B ) 
 ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A ) >. )
 
Theorem1stdm 5890 The first ordered pair component of a member of a relation belongs to the domain of the relation. (Contributed by NM, 17-Sep-2006.)
 |-  ( ( Rel  R  /\  A  e.  R ) 
 ->  ( 1st `  A )  e.  dom  R )
 
Theorem2ndrn 5891 The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.)
 |-  ( ( Rel  R  /\  A  e.  R ) 
 ->  ( 2nd `  A )  e.  ran  R )
 
Theorem1st2ndbr 5892 Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)
 |-  ( ( Rel  B  /\  A  e.  B ) 
 ->  ( 1st `  A ) B ( 2nd `  A ) )
 
Theoremreleldm2 5893* Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)
 |-  ( Rel  A  ->  ( B  e.  dom  A  <->  E. x  e.  A  ( 1st `  x )  =  B ) )
 
Theoremreldm 5894* An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)
 |-  ( Rel  A  ->  dom 
 A  =  ran  ( x  e.  A  |->  ( 1st `  x ) ) )
 
Theoremsbcopeq1a 5895 Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2837 that avoids the existential quantifiers of copsexg 4038). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  =  <. x ,  y >.  ->  ( [. ( 1st `  A )  /  x ]. [. ( 2nd `  A )  /  y ]. ph  <->  ph ) )
 
Theoremcsbopeq1a 5896 Equality theorem for substitution of a class  A for an ordered pair  <. x ,  y >. in  B (analog of csbeq1a 2929). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  =  <. x ,  y >.  ->  [_ ( 1st `  A )  /  x ]_ [_ ( 2nd `  A )  /  y ]_ B  =  B )
 
Theoremdfopab2 5897* A way to define an ordered-pair class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |- 
 { <. x ,  y >.  |  ph }  =  { z  e.  ( _V  X.  _V )  | 
 [. ( 1st `  z
 )  /  x ]. [. ( 2nd `  z )  /  y ]. ph }
 
Theoremdfoprab3s 5898* A way to define an operation class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph ) }
 
Theoremdfoprab3 5899* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)
 |-  ( w  =  <. x ,  y >.  ->  ( ph 
 <->  ps ) )   =>    |-  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  ph ) }  =  { <. <. x ,  y >. ,  z >.  |  ps }
 
Theoremdfoprab4 5900* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( w  =  <. x ,  y >.  ->  ( ph 
 <->  ps ) )   =>    |-  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  ph ) }  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ps ) }
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