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Theorem bnj882 32274
Description: Definition (using hypotheses for readability) of the function giving the transitive closure of  X in  A by  R. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj882.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj882.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj882.3  |-  D  =  ( om  \  { (/)
} )
bnj882.4  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
Assertion
Ref Expression
bnj882  |-  trCl ( X ,  A ,  R )  =  U_ f  e.  B  U_ i  e.  dom  f ( f `
 i )
Distinct variable groups:    A, f,
i, n, y    R, f, i, n, y    f, X, i, n, y
Allowed substitution hints:    ph( y, f, i, n)    ps( y,
f, i, n)    B( y, f, i, n)    D( y, f, i, n)

Proof of Theorem bnj882
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 df-bnj18 32038 . 2  |-  trCl ( X ,  A ,  R )  =  U_ f  e.  { f  |  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } U_ i  e. 
dom  f ( f `
 i )
2 df-iun 4284 . . 3  |-  U_ f  e.  B  U_ i  e. 
dom  f ( f `
 i )  =  { w  |  E. f  e.  B  w  e.  U_ i  e.  dom  f ( f `  i ) }
3 df-iun 4284 . . . 4  |-  U_ f  e.  { f  |  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } U_ i  e. 
dom  f ( f `
 i )  =  { w  |  E. f  e.  { f  |  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } w  e.  U_ i  e.  dom  f ( f `  i ) }
4 bnj882.4 . . . . . . . . 9  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
5 bnj882.3 . . . . . . . . . . 11  |-  D  =  ( om  \  { (/)
} )
6 bnj882.1 . . . . . . . . . . . . . 14  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
7 bnj882.2 . . . . . . . . . . . . . 14  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
86, 7anbi12i 697 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ps )  <->  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) )
98anbi2i 694 . . . . . . . . . . . 12  |-  ( ( f  Fn  n  /\  ( ph  /\  ps )
)  <->  ( f  Fn  n  /\  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) ) )
10 3anass 969 . . . . . . . . . . . 12  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  <->  ( f  Fn  n  /\  ( ph  /\  ps ) ) )
11 3anass 969 . . . . . . . . . . . 12  |-  ( ( f  Fn  n  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )  <-> 
( f  Fn  n  /\  ( ( f `  (/) )  =  pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) ) )
129, 10, 113bitr4i 277 . . . . . . . . . . 11  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  <->  ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) )
135, 12rexeqbii 2872 . . . . . . . . . 10  |-  ( E. n  e.  D  ( f  Fn  n  /\  ph 
/\  ps )  <->  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) )
1413abbii 2588 . . . . . . . . 9  |-  { f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }  =  {
f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) }
154, 14eqtri 2483 . . . . . . . 8  |-  B  =  { f  |  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) }
1615eleq2i 2532 . . . . . . 7  |-  ( f  e.  B  <->  f  e.  { f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } )
1716anbi1i 695 . . . . . 6  |-  ( ( f  e.  B  /\  w  e.  U_ i  e. 
dom  f ( f `
 i ) )  <-> 
( f  e.  {
f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) }  /\  w  e. 
U_ i  e.  dom  f ( f `  i ) ) )
1817rexbii2 2859 . . . . 5  |-  ( E. f  e.  B  w  e.  U_ i  e. 
dom  f ( f `
 i )  <->  E. f  e.  { f  |  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } w  e.  U_ i  e.  dom  f ( f `  i ) )
1918abbii 2588 . . . 4  |-  { w  |  E. f  e.  B  w  e.  U_ i  e. 
dom  f ( f `
 i ) }  =  { w  |  E. f  e.  {
f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } w  e.  U_ i  e.  dom  f ( f `  i ) }
203, 19eqtr4i 2486 . . 3  |-  U_ f  e.  { f  |  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } U_ i  e. 
dom  f ( f `
 i )  =  { w  |  E. f  e.  B  w  e.  U_ i  e.  dom  f ( f `  i ) }
212, 20eqtr4i 2486 . 2  |-  U_ f  e.  B  U_ i  e. 
dom  f ( f `
 i )  = 
U_ f  e.  {
f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } U_ i  e. 
dom  f ( f `
 i )
221, 21eqtr4i 2486 1  |-  trCl ( X ,  A ,  R )  =  U_ f  e.  B  U_ i  e.  dom  f ( f `
 i )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {cab 2439   A.wral 2799   E.wrex 2800    \ cdif 3436   (/)c0 3748   {csn 3988   U_ciun 4282   suc csuc 4832   dom cdm 4951    Fn wfn 5524   ` cfv 5529   omcom 6589    predc-bnj14 32031    trClc-bnj18 32037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-rex 2805  df-iun 4284  df-bnj18 32038
This theorem is referenced by:  bnj893  32276  bnj906  32278  bnj916  32281  bnj983  32299  bnj1014  32308  bnj1145  32339  bnj1318  32371
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