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Theorem List for Metamath Proof Explorer - 23901-24000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxfree2 23901 A partial converse to 19.9t 1789. (Contributed by Stefan Allan, 21-Dec-2008.)
 |-  ( A. x ( ph  ->  A. x ph )  <->  A. x ( -.  ph  ->  A. x  -.  ph ) )
 
TheoremaddltmulALT 23902 A proof readability experiment for addltmul 10159. (Contributed by Stefan Allan, 30-Oct-2010.) (Proof modification is discouraged.)
 |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 2  <  A  /\  2  <  B ) )  ->  ( A  +  B )  < 
 ( A  x.  B ) )
 
19.3  Mathbox for Thierry Arnoux
 
19.3.1  Propositional Calculus - misc additions
 
Theorembian1d 23903 Adding a superfluous conjunct in a biconditionnal. (Contributed by Thierry Arnoux, 26-Feb-2017.)
 |-  ( ph  ->  ( ps  <->  ( ch  /\  th ) ) )   =>    |-  ( ph  ->  ( ( ch  /\  ps ) 
 <->  ( ch  /\  th ) ) )
 
Theoremor3di 23904 Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
 |-  (
 ( ph  \/  ( ps  /\  ch  /\  ta ) )  <->  ( ( ph  \/  ps )  /\  ( ph  \/  ch )  /\  ( ph  \/  ta )
 ) )
 
Theoremor3dir 23905 Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
 |-  (
 ( ( ph  /\  ps  /\ 
 ch )  \/  ta ) 
 <->  ( ( ph  \/  ta )  /\  ( ps 
 \/  ta )  /\  ( ch  \/  ta ) ) )
 
Theorem3o1cs 23906 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
 |-  (
 ( ph  \/  ps  \/  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theorem3o2cs 23907 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
 |-  (
 ( ph  \/  ps  \/  ch )  ->  th )   =>    |-  ( ps  ->  th )
 
Theorem3o3cs 23908 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
 |-  (
 ( ph  \/  ps  \/  ch )  ->  th )   =>    |-  ( ch  ->  th )
 
Theoremadantl3r 23909 Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)
 |-  (
 ( ( ( ph  /\ 
 rh )  /\  mu )  /\  la )  ->  ka )   =>    |-  ( ( ( ( ( ph  /\  si )  /\  rh )  /\  mu )  /\  la )  ->  ka )
 
Theoremadantl4r 23910 Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)
 |-  (
 ( ( ( (
 ph  /\  si )  /\  rh )  /\  mu )  /\  la )  ->  ka )   =>    |-  (
 ( ( ( ( ( ph  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  ->  ka )
 
Theoremadantl5r 23911 Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)
 |-  (
 ( ( ( ( ( ph  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  ->  ka )   =>    |-  (
 ( ( ( ( ( ( ph  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  ->  ka )
 
Theoremadantl6r 23912 Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)
 |-  (
 ( ( ( ( ( ( ph  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  ->  ka )   =>    |-  ( ( ( ( ( ( ( (
 ph  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  ->  ka )
 
19.3.2  Predicate Calculus
 
19.3.2.1  Predicate Calculus - misc additions
 
Theoremabeq2f 23913 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 ) )
 
Theoremeqvincg 23914* A variable introduction law for class equality, deduction version. (Contributed by Thierry Arnoux, 2-Mar-2017.)
 |-  ( A  e.  V  ->  ( A  =  B  <->  E. x ( x  =  A  /\  x  =  B ) ) )
 
19.3.2.2  Restricted quantification - misc additions
 
Theoremreximddv 23915* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 7-Dec-2016.)
 |-  (
 ( ph  /\  ( x  e.  A  /\  ps ) )  ->  ch )   &    |-  ( ph  ->  E. x  e.  A  ps )   =>    |-  ( ph  ->  E. x  e.  A  ch )
 
Theoremraleqbid 23916 Equality deduction for restricted universal quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. x  e.  B  ch ) )
 
Theoremrexeqbid 23917 Equality deduction for restricted existential quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  B  ch ) )
 
Theoremralcom4f 23918* Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ y A   =>    |-  ( A. x  e.  A  A. y ph  <->  A. y A. x  e.  A  ph )
 
Theoremrexcom4f 23919* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ y A   =>    |-  ( E. x  e.  A  E. y ph  <->  E. y E. x  e.  A  ph )
 
Theoremrabid2f 23920 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 )
 
Theorem19.9d2rf 23921 A deduction version of one direction of 19.9 1793 with two variables (Contributed by Thierry Arnoux, 20-Mar-2017.)
 |-  F/ y ph   &    |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )   =>    |-  ( ph  ->  ps )
 
Theorem19.9d2r 23922* A deduction version of one direction of 19.9 1793 with two variables (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )   =>    |-  ( ph  ->  ps )
 
Theoremr19.41vv 23923* Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. Version with two quantifiers (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( E. x  e.  A  E. y  e.  B  ( ph  /\  ps )  <->  ( E. x  e.  A  E. y  e.  B  ph 
 /\  ps ) )
 
19.3.2.3  Substitution (without distinct variables) - misc additions
 
Theoremclelsb3f 23924 Substitution applied to an atomic wff (class version of elsb3 2152). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
 |-  F/_ y A   =>    |-  ( [ x  /  y ] y  e.  A  <->  x  e.  A )
 
19.3.2.4  Existential "at most one" - misc additions
 
Theoremmo5f 23925* Alternate definition of "at most one." (Contributed by Thierry Arnoux, 1-Mar-2017.)
 |-  F/ i ph   &    |-  F/ j ph   =>    |-  ( E* x ph  <->  A. i A. j
 ( ( [ i  /  x ] ph  /\  [
 j  /  x ] ph )  ->  i  =  j ) )
 
Theoremnmo 23926* 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 23927* "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 ) )
 
19.3.2.5  Existential uniqueness - misc additions
 
Theorem2reuswap2 23928* 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 23929* Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. Cf. reuxfr2d 4705 (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 23930* Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. Cf. reuxfrd 4707 (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 23931* Restricted existential quantification over the union of the range of a function. Cf. rexrn 5831 and eluni2 3979. (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 )
 
19.3.2.6  Restricted "at most one" - misc additions
 
TheoremrmoxfrdOLD 23932* 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 23933* 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 23934 "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 23935* 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 23936* 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 23937* 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 ) )
 
19.3.3  General Set Theory
 
19.3.3.1  Class abstractions (a.k.a. class builders)
 
Theoremceqsexv2d 23938* 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
 
Theoremrabbidva2 23939* Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.)
 |-  ( ph  ->  ( ( x  e.  A  /\  ps ) 
 <->  ( x  e.  B  /\  ch ) ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  =  { x  e.  B  |  ch } )
 
TheoremrabexgfGS 23940 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 )
 
19.3.3.2  Image Sets
 
Theoremabrexdomjm 23941* 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 23942* 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 23943* 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 23944* 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 23945* 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 23946* 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 )
 
19.3.3.3  Set relations and operations - misc additions
 
Theoremeqri 23947 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 23948* 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 23949 Intersection with an intersection (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |-  ( A  i^i  ( A  i^i  B ) )  =  ( A  i^i  B )
 
Theoremdifneqnul 23950 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 23951 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 ) ) )
 
19.3.3.4  Unordered pairs
 
Theoremelpreq 23952 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 23953 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 ) ) )
 
19.3.3.5  Conditional operator - misc additions
 
Theoremifeqeqx 23954* An equality theorem tailored for ballotlemsf1o 24724 (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 23955 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 ) )
 
Theoremovif 23956 Move a conditional outside of an operation (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( if ( ph ,  A ,  B ) F C )  =  if ( ph ,  ( A F C ) ,  ( B F C ) )
 
Theoremelimifd 23957 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 23958 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 23959 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
 
19.3.3.6  Indexed union - misc additions
 
Theoremiuneq12daf 23960 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 )
 
Theoremiuneq12df 23961 Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 31-Dec-2016.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  D )
 
Theoremssiun3 23962* 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 23963 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 )
 
Theoremiuninc 23964* 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 23965* 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 23966* The intersection of a set is the complement of the union of the complements. TODO shorten using iundifdifd 23965 (Contributed by Thierry Arnoux, 4-Sep-2016.)
 |-  O  e.  _V   &    |-  A  C_  ~P O   =>    |-  ( A  =/=  (/)  ->  |^| A  =  ( O  \  U_ x  e.  A  ( O  \  x ) ) )
 
Theoremiunrdx 23967* 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 )
 
19.3.3.7  Disjointness - misc additions
 
Theoremcbvdisjf 23968* 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 23969 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 23970* 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 23971* 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 23972* 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 23973* 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 23974* 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 23975* 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 23976* 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 23977* 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 23978* 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 23979* 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 ) )
 
Theoremdisjin 23980 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 23981* Derive a disjunction over a cross 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 23982* Rewrite a countable union as a disjoint union. Cf. iundisj 19395 (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 23983* A disjoint union is disjoint. Cf. iundisj2 19396 (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 23984* 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 23985* 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 23986* A variant of disjex 23985, 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 )  =  (/) ) )
 
19.3.4  Relations and Functions
 
19.3.4.1  Relations - misc additions
 
Theoremdfrel4 23987* A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5731 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 }
 )
 
Theoremrnpropg 23988 The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ran  { <. A ,  C >. ,  <. B ,  D >. }  =  { C ,  D }
 )
 
Theoremxpdisjres 23989 Restriction of a constant function (or other cross product) outside of its domain (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  (
 ( A  i^i  C )  =  (/)  ->  (
 ( A  X.  B )  |`  C )  =  (/) )
 
Theoremcsbcnvg 23990 Move class substitution in and out of the converse of a function. (Contributed by Thierry Arnoux, 8-Feb-2017.)
 |-  ( A  e.  V  ->  `'
 [_ A  /  x ]_ F  =  [_ A  /  x ]_ `' F )
 
Theoremimadifxp 23991 Image of the difference with a cross product (Contributed by Thierry Arnoux, 13-Dec-2017.)
 |-  ( C  C_  A  ->  (
 ( R  \  ( A  X.  B ) )
 " C )  =  ( ( R " C )  \  B ) )
 
19.3.4.2  Functions - misc additions
 
Theoremfdmrn 23992 A different way to write  F is a function. (Contributed by Thierry Arnoux, 7-Dec-2016.)
 |-  ( Fun  F  <->  F : dom  F --> ran  F )
 
Theoremnvof1o 23993 An involution is a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)
 |-  (
 ( F  Fn  A  /\  `' F  =  F )  ->  F : A -1-1-onto-> A )
 
Theoremf1o3d 23994* Describe an implicit one-to-one onto function. (Contributed by Thierry Arnoux, 23-Apr-2017.)
 |-  ( ph  ->  F  =  ( x  e.  A  |->  C ) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  B )   &    |-  ( ( ph  /\  y  e.  B )  ->  D  e.  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  ( x  =  D  <->  y  =  C ) )   =>    |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  ( y  e.  B  |->  D ) ) )
 
Theoremrinvf1o 23995 Sufficient conditions for the restriction of an involution to be a bijection (Contributed by Thierry Arnoux, 7-Dec-2016.)
 |-  Fun  F   &    |-  `' F  =  F   &    |-  ( F " A )  C_  B   &    |-  ( F " B )  C_  A   &    |-  A  C_  dom  F   &    |-  B  C_ 
 dom  F   =>    |-  ( F  |`  A ) : A -1-1-onto-> B
 
Theoremdfimafnf 23996* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Thierry Arnoux, 24-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x F   =>    |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( F " A )  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
 
Theoremfunimass4f 23997 Membership relation for the values of a function whose image is a subclass. (Contributed by Thierry Arnoux, 24-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x F   =>    |-  ( ( Fun 
 F  /\  A  C_  dom  F )  ->  ( ( F
 " A )  C_  B 
 <-> 
 A. x  e.  A  ( F `  x )  e.  B ) )
 
Theoremmptcnv 23998* The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017.)
 |-  ( ph  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( y  e.  C  /\  x  =  D ) ) )   =>    |-  ( ph  ->  `' ( x  e.  A  |->  B )  =  ( y  e.  C  |->  D ) )
 
Theoremfneq12 23999 Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  (
 ( F  =  G  /\  A  =  B ) 
 ->  ( F  Fn  A  <->  G  Fn  B ) )
 
Theoremfimacnvinrn 24000 Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( Fun  F  ->  ( `' F " A )  =  ( `' F "
 ( A  i^i  ran  F ) ) )
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