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Theorem bnj882 28628
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 28390 . 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 4030 . . 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 4030 . . . 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 679 . . . . . . . . . . . . 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 676 . . . . . . . . . . . 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 940 . . . . . . . . . . . 12  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  <->  ( f  Fn  n  /\  ( ph  /\  ps ) ) )
11 3anass 940 . . . . . . . . . . . 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 269 . . . . . . . . . . 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 2673 . . . . . . . . . 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 2492 . . . . . . . . 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 2400 . . . . . . . 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 2444 . . . . . . 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 677 . . . . . 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 2671 . . . . 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 2492 . . . 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 2403 . . 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 2403 . 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 2403 1  |-  trCl ( X ,  A ,  R )  =  U_ f  e.  B  U_ i  e.  dom  f ( f `
 i )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {cab 2366   A.wral 2642   E.wrex 2643    \ cdif 3253   (/)c0 3564   {csn 3750   U_ciun 4028   suc csuc 4517   omcom 4778   dom cdm 4811    Fn wfn 5382   ` cfv 5387    predc-bnj14 28383    trClc-bnj18 28389
This theorem is referenced by:  bnj893  28630  bnj906  28632  bnj916  28635  bnj983  28653  bnj1014  28662  bnj1145  28693  bnj1318  28725
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-rex 2648  df-iun 4030  df-bnj18 28390
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