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Theorem List for Metamath Proof Explorer - 27501-27600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnmo 27501* Negation of "at most one". (Contributed by Thierry Arnoux, 26-Feb-2017.)
 |-  F/ y ph   =>    |-  ( -.  E* x ph  <->  A. y E. x (
 ph  /\  x  =/=  y ) )
 
Theoremmoimd 27502* "At most one" is preserved through implication (notice wff reversal). (Contributed by Thierry Arnoux, 25-Feb-2017.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E* x ch  ->  E* x ps ) )
 
Theoremrmoeq 27503* Equality's restricted existential "at most one" property. (Contributed by Thierry Arnoux, 30-Mar-2018.)
 |-  ( A  e.  V  ->  E* x  e.  B  x  =  A )
 
21.4.2.5  Existential uniqueness - misc additions
 
Theorem2reuswap2 27504* A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.)
 |-  ( A. x  e.  A  E* y ( y  e.  B  /\  ph )  ->  ( E! x  e.  A  E. y  e.  B  ph  ->  E! y  e.  B  E. x  e.  A  ph ) )
 
Theoremreuxfr3d 27505* Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. Cf. reuxfr2d 4585 (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
 |-  (
 ( ph  /\  y  e.  C )  ->  A  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E* y  e.  C  x  =  A )   =>    |-  ( ph  ->  ( E! x  e.  B  E. y  e.  C  ( x  =  A  /\  ps )  <->  E! y  e.  C  ps ) )
 
Theoremreuxfr4d 27506* Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. Cf. reuxfrd 4587 (Contributed by Thierry Arnoux, 7-Apr-2017.)
 |-  (
 ( ph  /\  y  e.  C )  ->  A  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E! y  e.  C  x  =  A )   &    |-  (
 ( ph  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E! x  e.  B  ps 
 <->  E! y  e.  C  ch ) )
 
Theoremrexunirn 27507* Restricted existential quantification over the union of the range of a function. Cf. rexrn 5935 and eluni2 4167. (Contributed by Thierry Arnoux, 19-Sep-2017.)
 |-  F  =  ( x  e.  A  |->  B )   &    |-  ( x  e.  A  ->  B  e.  V )   =>    |-  ( E. x  e.  A  E. y  e.  B  ph  ->  E. y  e.  U. ran  F ph )
 
21.4.2.6  Restricted "at most one" - misc additions
 
TheoremrmoxfrdOLD 27508* Transfer "at most one" restricted quantification from a variable  x to another variable  y contained in expression 
A. (Contributed by Thierry Arnoux, 7-Apr-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (
 ( ph  /\  y  e.  C )  ->  A  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E! y  e.  C  x  =  A )   &    |-  (
 ( ph  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E* x ( x  e.  B  /\  ps ) 
 <->  E* y ( y  e.  C  /\  ch ) ) )
 
Theoremrmoxfrd 27509* Transfer "at most one" restricted quantification from a variable  x to another variable  y contained in expression 
A. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
 |-  (
 ( ph  /\  y  e.  C )  ->  A  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E! y  e.  C  x  =  A )   &    |-  (
 ( ph  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E* x  e.  B  ps 
 <->  E* y  e.  C  ch ) )
 
Theoremssrmo 27510 "At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  C_  B  ->  ( E* x  e.  B  ph  ->  E* x  e.  A  ph ) )
 
Theoremrmo3f 27511* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/ y ph   =>    |-  ( E* x  e.  A  ph  <->  A. x  e.  A  A. y  e.  A  (
 ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
 
Theoremrmo4fOLD 27512* Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E* x ( x  e.  A  /\  ph )  <->  A. x  e.  A  A. y  e.  A  ( ( ph  /\  ps )  ->  x  =  y ) )
 
Theoremrmo4f 27513* Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E* x  e.  A  ph  <->  A. x  e.  A  A. y  e.  A  ( ( ph  /\  ps )  ->  x  =  y ) )
 
21.4.3  General Set Theory
 
21.4.3.1  Class abstractions (a.k.a. class builders)
 
Theoremceqsexv2d 27514* Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ps   =>    |- 
 E. x ph
 
Theoremabeq2f 27515 Equality of a class variable and a class abstraction. In this version, the fact that  x is a non-free variable in  A is explicitely stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.)
 |-  F/_ x A   =>    |-  ( A  =  { x  |  ph }  <->  A. x ( x  e.  A  <->  ph ) )
 
Theoremrabrab 27516 Abstract builder restricted to another restricted abstract builder (Contributed by Thierry Arnoux, 30-Aug-2017.)
 |-  { x  e.  { x  e.  A  |  ph }  |  ps }  =  { x  e.  A  |  ( ph  /\ 
 ps ) }
 
Theoremrabtru 27517 Abtract builder using the constant wff T. (Contributed by Thierry Arnoux, 4-May-2020.)
 |-  F/_ x A   =>    |- 
 { x  e.  A  | T.  }  =  A
 
Theoremrabid2f 27518 An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
 |-  F/_ x A   =>    |-  ( A  =  { x  e.  A  |  ph
 } 
 <-> 
 A. x  e.  A  ph )
 
TheoremrabexgfGS 27519 Separation Scheme in terms of a restricted class abstraction. To be removed in profit of Glauco's equivalent version. (Contributed by Thierry Arnoux, 11-May-2017.)
 |-  F/_ x A   =>    |-  ( A  e.  V  ->  { x  e.  A  |  ph }  e.  _V )
 
Theoremrabsnel 27520* Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by Thierry Arnoux, 15-Sep-2018.)
 |-  B  e.  _V   =>    |-  ( { x  e.  A  |  ph }  =  { B }  ->  B  e.  A )
 
Theoremforesf1o 27521* From a surjective function, *choose* a subset of the domain, such that the restricted function is bijective. (Contributed by Thierry Arnoux, 27-Jan-2020.)
 |-  (
 ( A  e.  V  /\  F : A -onto-> B )  ->  E. x  e.  ~P  A ( F  |`  x ) : x -1-1-onto-> B )
 
Theoremrabfodom 27522* Domination relation for restricted abstract class builders, based on a surjective function. (Contributed by Thierry Arnoux, 27-Jan-2020.)
 |-  (
 ( ph  /\  x  e.  A  /\  y  =  ( F `  x ) )  ->  ( ch  <->  ps ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A -onto-> B )   =>    |-  ( ph  ->  { y  e.  B  |  ch }  ~<_  { x  e.  A  |  ps } )
 
21.4.3.2  Image Sets
 
Theoremabrexdomjm 27523* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 y  e.  A  ->  E* x ph )   =>    |-  ( A  e.  V  ->  { x  |  E. y  e.  A  ph
 }  ~<_  A )
 
Theoremabrexdom2jm 27524* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  e.  V  ->  { x  |  E. y  e.  A  x  =  B } 
 ~<_  A )
 
Theoremabrexexd 27525* Existence of a class abstraction of existentially restricted sets. (Contributed by Thierry Arnoux, 10-May-2017.)
 |-  F/_ x A   &    |-  ( ph  ->  A  e.  _V )   =>    |-  ( ph  ->  { y  |  E. x  e.  A  y  =  B }  e.  _V )
 
Theoremelabreximd 27526* Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
 |-  F/ x ph   &    |-  F/ x ch   &    |-  ( A  =  B  ->  ( ch  <->  ps ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  C )  ->  ps )   =>    |-  (
 ( ph  /\  A  e.  { y  |  E. x  e.  C  y  =  B } )  ->  ch )
 
Theoremelabreximdv 27527* Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
 |-  ( A  =  B  ->  ( ch  <->  ps ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  C )  ->  ps )   =>    |-  (
 ( ph  /\  A  e.  { y  |  E. x  e.  C  y  =  B } )  ->  ch )
 
Theoremabrexss 27528* A necessary condition for an image set to be a subset. (Contributed by Thierry Arnoux, 6-Feb-2017.)
 |-  F/_ x C   =>    |-  ( A. x  e.  A  B  e.  C  ->  { y  |  E. x  e.  A  y  =  B }  C_  C )
 
21.4.3.3  Set relations and operations - misc additions
 
Theoremeqri 27529 Infer equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 7-Oct-2017.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  ( x  e.  A  <->  x  e.  B )   =>    |-  A  =  B
 
Theoremrabss3d 27530* Subclass law for restricted abstraction. (Contributed by Thierry Arnoux, 25-Sep-2017.)
 |-  (
 ( ph  /\  ( x  e.  A  /\  ps ) )  ->  x  e.  B )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  C_ 
 { x  e.  B  |  ps } )
 
Theoreminin 27531 Intersection with an intersection (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |-  ( A  i^i  ( A  i^i  B ) )  =  ( A  i^i  B )
 
Theoreminindif 27532 See inundif 3822 (Contributed by Thierry Arnoux, 13-Sep-2017.)
 |-  (
 ( A  i^i  C )  i^i  ( A  \  C ) )  =  (/)
 
Theoremdifneqnul 27533 If the difference of two sets is not empty, then the sets are not equal. (Contributed by Thierry Arnoux, 28-Feb-2017.)
 |-  (
 ( A  \  B )  =/=  (/)  ->  A  =/=  B )
 
Theoremdifeq 27534 Rewriting an equation with set difference, without using quantifiers. (Contributed by Thierry Arnoux, 24-Sep-2017.)
 |-  (
 ( A  \  B )  =  C  <->  ( ( C  i^i  B )  =  (/)  /\  ( C  u.  B )  =  ( A  u.  B ) ) )
 
Theoremindifundif 27535 A remarkable equation with sets. (Contributed by Thierry Arnoux, 18-May-2020.)
 |-  (
 ( ( A  i^i  B )  \  C )  u.  ( A  \  B ) )  =  ( A  \  ( B  i^i  C ) )
 
Theoremelin1d 27536 Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
 |-  ( ph  ->  X  e.  ( A  i^i  B ) )   =>    |-  ( ph  ->  X  e.  A )
 
Theoremelin2d 27537 Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
 |-  ( ph  ->  X  e.  ( A  i^i  B ) )   =>    |-  ( ph  ->  X  e.  B )
 
Theoremsselpwd 27538 Elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
 |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  A  e.  ~P B )
 
Theoremelpwincl1 27539 Closure of intersection with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
 |-  ( ph  ->  A  e.  ~P C )   =>    |-  ( ph  ->  ( A  i^i  B )  e. 
 ~P C )
 
Theoremelpwdifcl 27540 Closure of set difference with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
 |-  ( ph  ->  A  e.  ~P C )   =>    |-  ( ph  ->  ( A  \  B )  e. 
 ~P C )
 
Theoremelpwiuncl 27541* Closure of indexed union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 27-May-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ~P C )   =>    |-  ( ph  ->  U_ k  e.  A  B  e.  ~P C )
 
21.4.3.4  Unordered pairs
 
Theoremelpreq 27542 Equality wihin a pair (Contributed by Thierry Arnoux, 23-Aug-2017.)
 |-  ( ph  ->  X  e.  { A ,  B }
 )   &    |-  ( ph  ->  Y  e.  { A ,  B } )   &    |-  ( ph  ->  ( X  =  A  <->  Y  =  A ) )   =>    |-  ( ph  ->  X  =  Y )
 
Theorempreqsnd 27543 Equivalence for a pair equal to a singleton, deduction form. (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  _V )   =>    |-  ( ph  ->  ( { A ,  B }  =  { C } 
 <->  ( A  =  C  /\  B  =  C ) ) )
 
21.4.3.5  Conditional operator - misc additions
 
Theoremifeqeqx 27544* An equality theorem tailored for ballotlemsf1o 28635 (Contributed by Thierry Arnoux, 14-Apr-2017.)
 |-  ( x  =  X  ->  A  =  C )   &    |-  ( x  =  Y  ->  B  =  a )   &    |-  ( x  =  X  ->  ( ch  <->  th ) )   &    |-  ( x  =  Y  ->  ( ch  <->  ps ) )   &    |-  ( ph  ->  a  =  C )   &    |-  ( ( ph  /\  ps )  ->  th )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  e.  W )   =>    |-  ( ( ph  /\  x  =  if ( ps ,  X ,  Y )
 )  ->  a  =  if ( ch ,  A ,  B ) )
 
Theoremifbieq12d2 27545 Equivalence deduction for conditional operators. (Contributed by Thierry Arnoux, 14-Feb-2017.)
 |-  ( ph  ->  ( ps  <->  ch ) )   &    |-  (
 ( ph  /\  ps )  ->  A  =  C )   &    |-  ( ( ph  /\  -.  ps )  ->  B  =  D )   =>    |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  C ,  D ) )
 
Theoremelimifd 27546 Elimination of a conditional operator contained in a wff  ch. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  ( if ( ps ,  A ,  B )  =  A  ->  ( ch  <->  th ) ) )   &    |-  ( ph  ->  ( if ( ps ,  A ,  B )  =  B  ->  ( ch  <->  ta ) ) )   =>    |-  ( ph  ->  ( ch  <->  (
 ( ps  /\  th )  \/  ( -.  ps  /\ 
 ta ) ) ) )
 
Theoremelim2if 27547 Elimination of two conditional operators contained in a wff  ch. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
 )  =  A  ->  ( ch  <->  th ) )   &    |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
 )  =  B  ->  ( ch  <->  ta ) )   &    |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
 )  =  C  ->  ( ch  <->  et ) )   =>    |-  ( ch  <->  ( ( ph  /\ 
 th )  \/  ( -.  ph  /\  ( ( ps  /\  ta )  \/  ( -.  ps  /\  et ) ) ) ) )
 
Theoremelim2ifim 27548 Elimination of two conditional operators for an implication. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
 )  =  A  ->  ( ch  <->  th ) )   &    |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
 )  =  B  ->  ( ch  <->  ta ) )   &    |-  ( if ( ph ,  A ,  if ( ps ,  B ,  C )
 )  =  C  ->  ( ch  <->  et ) )   &    |-  ( ph  ->  th )   &    |-  ( ( -.  ph  /\  ps )  ->  ta )   &    |-  ( ( -.  ph  /\  -.  ps )  ->  et )   =>    |- 
 ch
 
21.4.3.6  Set union
 
Theoremuniinn0 27549* Sufficient and necessary condition for a union to intersect with a given set. (Contributed by Thierry Arnoux, 27-Jan-2020.)
 |-  (
 ( U. A  i^i  B )  =/=  (/)  <->  E. x  e.  A  ( x  i^i  B )  =/=  (/) )
 
21.4.3.7  Indexed union - misc additions
 
Theoremcbviunf 27550* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/_ y B   &    |-  F/_ x C   &    |-  ( x  =  y 
 ->  B  =  C )   =>    |-  U_ x  e.  A  B  =  U_ y  e.  A  C
 
Theoremiuneq12daf 27551 Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 13-Mar-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   &    |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  =  D )   =>    |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  D )
 
Theoremiunin1f 27552 Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4296 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) (Revised by Thierry Arnoux, 2-May-2020.)
 |-  F/_ x C   =>    |-  U_ x  e.  A  ( B  i^i  C )  =  ( U_ x  e.  A  B  i^i  C )
 
Theoremiunxsngf 27553* A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.) (Revised by Thierry Arnoux, 2-May-2020.)
 |-  F/_ x C   &    |-  ( x  =  A  ->  B  =  C )   =>    |-  ( A  e.  V  -> 
 U_ x  e.  { A } B  =  C )
 
Theoremssiun3 27554* Subset equivalence for an indexed union. (Contributed by Thierry Arnoux, 17-Oct-2016.)
 |-  ( A. y  e.  C  E. x  e.  A  y  e.  B  <->  C  C_  U_ x  e.  A  B )
 
Theoremssiun2sf 27555 Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.)
 |-  F/_ x A   &    |-  F/_ x C   &    |-  F/_ x D   &    |-  ( x  =  C  ->  B  =  D )   =>    |-  ( C  e.  A  ->  D  C_  U_ x  e.  A  B )
 
Theoremiunxdif3 27556* An indexed union where some terms are the empty set. See iunxdif2 4291 (Contributed by Thierry Arnoux, 4-May-2020.)
 |-  F/_ x E   =>    |-  ( A. x  e.  E  B  =  (/)  ->  U_ x  e.  ( A 
 \  E ) B  =  U_ x  e.  A  B )
 
Theoremiuninc 27557* The union of an increasing collection of sets is its last element. (Contributed by Thierry Arnoux, 22-Jan-2017.)
 |-  ( ph  ->  F  Fn  NN )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( F `  n )  C_  ( F `  ( n  +  1 ) ) )   =>    |-  ( ( ph  /\  i  e.  NN )  ->  U_ n  e.  ( 1 ... i
 ) ( F `  n )  =  ( F `  i ) )
 
Theoremiundifdifd 27558* The intersection of a set is the complement of the union of the complements. (Contributed by Thierry Arnoux, 19-Dec-2016.)
 |-  ( A  C_  ~P O  ->  ( A  =/=  (/)  ->  |^| A  =  ( O  \  U_ x  e.  A  ( O  \  x ) ) ) )
 
Theoremiundifdif 27559* The intersection of a set is the complement of the union of the complements. TODO shorten using iundifdifd 27558 (Contributed by Thierry Arnoux, 4-Sep-2016.)
 |-  O  e.  _V   &    |-  A  C_  ~P O   =>    |-  ( A  =/=  (/)  ->  |^| A  =  ( O  \  U_ x  e.  A  ( O  \  x ) ) )
 
Theoremiunrdx 27560* Re-index an indexed union. (Contributed by Thierry Arnoux, 6-Apr-2017.)
 |-  ( ph  ->  F : A -onto-> C )   &    |-  ( ( ph  /\  y  =  ( F `
  x ) ) 
 ->  D  =  B )   =>    |-  ( ph  ->  U_ x  e.  A  B  =  U_ y  e.  C  D )
 
Theoremiunpreima 27561* Preimage of an indexed union. (Contributed by Thierry Arnoux, 27-Mar-2018.)
 |-  ( Fun  F  ->  ( `' F " U_ x  e.  A  B )  = 
 U_ x  e.  A  ( `' F " B ) )
 
21.4.3.8  Disjointness - misc additions
 
Theoremdisjnf 27562* In case  x is not free in  B, disjointness is not so interesting since it reduces to cases where  A is a singleton. (Google Groups discussion with Peter Masza) (Contributed by Thierry Arnoux, 26-Jul-2018.)
 |-  (Disj  x  e.  A  B  <->  ( B  =  (/) 
 \/  E* x  x  e.  A ) )
 
Theoremcbvdisjf 27563* Change bound variables in a disjoint collection. (Contributed by Thierry Arnoux, 6-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ y B   &    |-  F/_ x C   &    |-  ( x  =  y  ->  B  =  C )   =>    |-  (Disj  x  e.  A  B  <-> Disj  y  e.  A  C )
 
Theoremdisjss1f 27564 A subset of a disjoint collection is disjoint. (Contributed by Thierry Arnoux, 6-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  C_  B  ->  (Disj  x  e.  B  C  -> Disj  x  e.  A  C ) )
 
Theoremdisjdifprg 27565* A trivial partition into a subset and its complement. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 -> Disj 
 x  e.  { ( B  \  A ) ,  A } x )
 
Theoremdisjdifprg2 27566* A trivial partition of a set into its difference and intersection with another set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  ( A  e.  V  -> Disj  x  e.  { ( A  \  B ) ,  ( A  i^i  B ) } x )
 
Theoremdisji2f 27567* Property of a disjoint collection: if  B ( x )  =  C and  B ( Y )  =  D, and  x  =/=  Y, then  B and  C are disjoint. (Contributed by Thierry Arnoux, 30-Dec-2016.)
 |-  F/_ x C   &    |-  ( x  =  Y  ->  B  =  C )   =>    |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A )  /\  x  =/= 
 Y )  ->  ( B  i^i  C )  =  (/) )
 
Theoremdisjif 27568* Property of a disjoint collection: if  B ( x ) and 
B ( Y )  =  D have a common element  Z, then  x  =  Y. (Contributed by Thierry Arnoux, 30-Dec-2016.)
 |-  F/_ x C   &    |-  ( x  =  Y  ->  B  =  C )   =>    |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A )  /\  ( Z  e.  B  /\  Z  e.  C ) )  ->  x  =  Y )
 
Theoremdisjorf 27569* Two ways to say that a collection 
B ( i ) for  i  e.  A is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ i A   &    |-  F/_ j A   &    |-  ( i  =  j  ->  B  =  C )   =>    |-  (Disj  i  e.  A  B 
 <-> 
 A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( B  i^i  C )  =  (/) ) )
 
Theoremdisjorsf 27570* Two ways to say that a collection 
B ( i ) for  i  e.  A is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ x A   =>    |-  (Disj  x  e.  A  B 
 <-> 
 A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
 
Theoremdisjif2 27571* Property of a disjoint collection: if  B ( x ) and 
B ( Y )  =  D have a common element  Z, then  x  =  Y. (Contributed by Thierry Arnoux, 6-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x C   &    |-  ( x  =  Y  ->  B  =  C )   =>    |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A )  /\  ( Z  e.  B  /\  Z  e.  C )
 )  ->  x  =  Y )
 
Theoremdisjabrex 27572* Rewriting a disjoint collection into a partition of its image set (Contributed by Thierry Arnoux, 30-Dec-2016.)
 |-  (Disj  x  e.  A  B  -> Disj  y  e.  { z  |  E. x  e.  A  z  =  B } y )
 
Theoremdisjabrexf 27573* Rewriting a disjoint collection into a partition of its image set (Contributed by Thierry Arnoux, 30-Dec-2016.) (Revised by Thierry Arnoux, 9-Mar-2017.)
 |-  F/_ x A   =>    |-  (Disj  x  e.  A  B  -> Disj  y  e.  { z  |  E. x  e.  A  z  =  B }
 y )
 
Theoremdisjpreima 27574* A preimage of a disjoint set is disjoint. (Contributed by Thierry Arnoux, 7-Feb-2017.)
 |-  (
 ( Fun  F  /\ Disj  x  e.  A  B )  -> Disj  x  e.  A  ( `' F " B ) )
 
Theoremdisjrnmpt 27575* Rewriting a disjoint collection using the range of a mapping. (Contributed by Thierry Arnoux, 27-May-2020.)
 |-  (Disj  x  e.  A  B  -> Disj  y  e.  ran  ( x  e.  A  |->  B ) y )
 
Theoremdisjin 27576 If a collection is disjoint, so is the collection of the intersections with a given set. (Contributed by Thierry Arnoux, 14-Feb-2017.)
 |-  (Disj  x  e.  B  C  -> Disj  x  e.  B  ( C  i^i  A ) )
 
Theoremdisjxpin 27577* Derive a disjunction over a Cartesian product from the disjunctions over its first and second elements. (Contributed by Thierry Arnoux, 9-Mar-2018.)
 |-  ( x  =  ( 1st `  p )  ->  C  =  E )   &    |-  ( y  =  ( 2nd `  p )  ->  D  =  F )   &    |-  ( ph  -> Disj  x  e.  A  C )   &    |-  ( ph  -> Disj  y  e.  B  D )   =>    |-  ( ph  -> Disj  p  e.  ( A  X.  B ) ( E  i^i  F ) )
 
Theoremiundisjf 27578* Rewrite a countable union as a disjoint union. Cf. iundisj 22043 (Contributed by Thierry Arnoux, 31-Dec-2016.)
 |-  F/_ k A   &    |-  F/_ n B   &    |-  ( n  =  k  ->  A  =  B )   =>    |-  U_ n  e.  NN  A  =  U_ n  e. 
 NN  ( A  \  U_ k  e.  ( 1..^ n ) B )
 
Theoremiundisj2f 27579* A disjoint union is disjoint. Cf. iundisj2 22044 (Contributed by Thierry Arnoux, 30-Dec-2016.)
 |-  F/_ k A   &    |-  F/_ n B   &    |-  ( n  =  k  ->  A  =  B )   =>    |- Disj  n  e.  NN  ( A 
 \  U_ k  e.  (
 1..^ n ) B )
 
Theoremdisjrdx 27580* Re-index a disjunct collection statement. (Contributed by Thierry Arnoux, 7-Apr-2017.)
 |-  ( ph  ->  F : A -1-1-onto-> C )   &    |-  ( ( ph  /\  y  =  ( F `  x ) )  ->  D  =  B )   =>    |-  ( ph  ->  (Disj  x  e.  A  B  <-> Disj  y  e.  C  D ) )
 
Theoremdisjex 27581* Two ways to say that two classes are disjoint (or equal). (Contributed by Thierry Arnoux, 4-Oct-2016.)
 |-  (
 ( E. z ( z  e.  A  /\  z  e.  B )  ->  A  =  B )  <-> 
 ( A  =  B  \/  ( A  i^i  B )  =  (/) ) )
 
Theoremdisjexc 27582* A variant of disjex 27581, applicable for more generic families. (Contributed by Thierry Arnoux, 4-Oct-2016.)
 |-  ( x  =  y  ->  A  =  B )   =>    |-  ( ( E. z ( z  e.  A  /\  z  e.  B )  ->  x  =  y )  ->  ( A  =  B  \/  ( A  i^i  B )  =  (/) ) )
 
Theoremdisjunsn 27583* Append an element to a disjoint collection. Similar to ralunsn 4151, gsumunsn 17100, etc. (Contributed by Thierry Arnoux, 28-Mar-2018.)
 |-  ( x  =  M  ->  B  =  C )   =>    |-  ( ( M  e.  V  /\  -.  M  e.  A )  ->  (Disj  x  e.  ( A  u.  { M }
 ) B  <->  (Disj  x  e.  A  B  /\  ( U_ x  e.  A  B  i^i  C )  =  (/) ) ) )
 
Theoremdisjun0 27584* Adding the empty element preserves disjointness. (Contributed by Thierry Arnoux, 30-May-2020.)
 |-  (Disj  x  e.  A  x  -> Disj  x  e.  ( A  u.  { (/) } ) x )
 
Theoremdisjiunel 27585* A set of elements B of a disjoint set A is disjoint with another element of that set. (Contributed by Thierry Arnoux, 24-May-2020.)
 |-  ( ph  -> Disj  x  e.  A  B )   &    |-  ( x  =  Y  ->  B  =  D )   &    |-  ( ph  ->  E  C_  A )   &    |-  ( ph  ->  Y  e.  ( A  \  E ) )   =>    |-  ( ph  ->  ( U_ x  e.  E  B  i^i  D )  =  (/) )
 
Theoremdisjuniel 27586* A set of elements B of a disjoint set A is disjoint with another element of that set. (Contributed by Thierry Arnoux, 24-May-2020.)
 |-  ( ph  -> Disj  x  e.  A  x )   &    |-  ( ph  ->  B  C_  A )   &    |-  ( ph  ->  C  e.  ( A  \  B ) )   =>    |-  ( ph  ->  (
 U. B  i^i  C )  =  (/) )
 
21.4.4  Relations and Functions
 
21.4.4.1  Relations - misc additions
 
Theoremdfrel4 27587* A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5819 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) (Revised by Thierry Arnoux, 11-May-2017.)
 |-  F/_ x R   &    |-  F/_ y R   =>    |-  ( Rel  R  <->  R  =  { <. x ,  y >.  |  x R y }
 )
 
Theoremxpdisjres 27588 Restriction of a constant function (or other Cartesian product) outside of its domain (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  (
 ( A  i^i  C )  =  (/)  ->  (
 ( A  X.  B )  |`  C )  =  (/) )
 
Theoremopeldifid 27589 Ordered pair elementhood outside of the diagonal. (Contributed by Thierry Arnoux, 1-Jan-2020.)
 |-  ( Rel  A  ->  ( <. X ,  Y >.  e.  ( A  \  _I  )  <->  ( <. X ,  Y >.  e.  A  /\  X  =/=  Y ) ) )
 
Theoremdifres 27590 Case when set difference in unaffected by restriction. (Contributed by Thierry Arnoux, 1-Jan-2020.)
 |-  ( A  C_  ( B  X.  _V )  ->  ( A 
 \  ( C  |`  B ) )  =  ( A 
 \  C ) )
 
Theoremimadifxp 27591 Image of the difference with a Cartesian product (Contributed by Thierry Arnoux, 13-Dec-2017.)
 |-  ( C  C_  A  ->  (
 ( R  \  ( A  X.  B ) )
 " C )  =  ( ( R " C )  \  B ) )
 
Theoremrelfi 27592 A relation (set) is finite if and only if both its domain and range are finite. (Contributed by Thierry Arnoux, 27-Aug-2017.)
 |-  ( Rel  A  ->  ( A  e.  Fin  <->  ( dom  A  e.  Fin  /\  ran  A  e.  Fin ) ) )
 
Theoremfcoinver 27593 Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 27594 (Contributed by Thierry Arnoux, 3-Jan-2020.)
 |-  ( F  Fn  X  ->  ( `' F  o.  F )  Er  X )
 
Theoremfcoinvbr 27594 Binary relation for the equivalence relation from fcoinver 27593. (Contributed by Thierry Arnoux, 3-Jan-2020.)
 |-  .~  =  ( `' F  o.  F )   =>    |-  ( ( F  Fn  A  /\  X  e.  A  /\  Y  e.  A ) 
 ->  ( X  .~  Y  <->  ( F `  X )  =  ( F `  Y ) ) )
 
Theorembrabgaf 27595* The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.) (Revised by Thierry Arnoux, 17-May-2020.)
 |-  F/ x ps   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   &    |-  R  =  { <. x ,  y >.  |  ph }   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A R B  <->  ps ) )
 
Theorembrelg 27596 Two things in a binary relation belong to the relation's domain. (Contributed by Thierry Arnoux, 29-Aug-2017.)
 |-  (
 ( R  C_  ( C  X.  D )  /\  A R B )  ->  ( A  e.  C  /\  B  e.  D ) )
 
Theorembr8d 27597* Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by Thierry Arnoux, 21-Mar-2019.)
 |-  (
 a  =  A  ->  ( ps  <->  ch ) )   &    |-  (
 b  =  B  ->  ( ch  <->  th ) )   &    |-  (
 c  =  C  ->  ( th  <->  ta ) )   &    |-  (
 d  =  D  ->  ( ta  <->  et ) )   &    |-  (
 e  =  E  ->  ( et  <->  ze ) )   &    |-  (
 f  =  F  ->  ( ze  <->  si ) )   &    |-  (
 g  =  G  ->  (
 si 
 <->  rh ) )   &    |-  ( h  =  H  ->  ( rh  <->  mu ) )   &    |-  ( ph  ->  R  =  { <. p ,  q >.  | 
 E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  E. e  e.  P  E. f  e.  P  E. g  e.  P  E. h  e.  P  ( p  =  <. <.
 a ,  b >. , 
 <. c ,  d >. >.  /\  q  =  <. <.
 e ,  f >. , 
 <. g ,  h >. >.  /\  ps ) } )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  P )   &    |-  ( ph  ->  D  e.  P )   &    |-  ( ph  ->  E  e.  P )   &    |-  ( ph  ->  F  e.  P )   &    |-  ( ph  ->  G  e.  P )   &    |-  ( ph  ->  H  e.  P )   =>    |-  ( ph  ->  ( <.
 <. A ,  B >. , 
 <. C ,  D >. >. R <. <. E ,  F >. ,  <. G ,  H >.
 >. 
 <->  mu ) )
 
Theoremopabdm 27598* Domain of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.)
 |-  ( R  =  { <. x ,  y >.  |  ph }  ->  dom 
 R  =  { x  |  E. y ph } )
 
Theoremopabrn 27599* Range of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.)
 |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ran 
 R  =  { y  |  E. x ph } )
 
Theoremssrelf 27600* A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Thierry Arnoux, 6-Nov-2017.)
 |-  F/ x ph   &    |-  F/ y ph   &    |-  F/_ x A   &    |-  F/_ y A   &    |-  F/_ x B   &    |-  F/_ y B   =>    |-  ( Rel  A  ->  ( A  C_  B  <->  A. x A. y
 ( <. x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B ) ) )
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