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Theorem bnj18eq1 29275
Description: Equality theorem for transitive closure. (Contributed by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Assertion
Ref Expression
bnj18eq1  |-  ( X  =  Y  ->  trCl ( X ,  A ,  R )  =  trCl ( Y ,  A ,  R ) )

Proof of Theorem bnj18eq1
Dummy variables  f 
i  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj602 29263 . . . . . . . . . . 11  |-  ( X  =  Y  ->  pred ( X ,  A ,  R )  =  pred ( Y ,  A ,  R ) )
21eqeq2d 2307 . . . . . . . . . 10  |-  ( X  =  Y  ->  (
( f `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( f `  (/) )  = 
pred ( Y ,  A ,  R )
) )
323anbi2d 1257 . . . . . . . . 9  |-  ( X  =  Y  ->  (
( 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 ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) ) )
43rexbidv 2577 . . . . . . . 8  |-  ( X  =  Y  ->  ( 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 ) ) )  <->  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `  (/) )  = 
pred ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) ) )
54abbidv 2410 . . . . . . 7  |-  ( X  =  Y  ->  { 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 ) ) ) }  =  { f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } )
65eleq2d 2363 . . . . . 6  |-  ( X  =  Y  ->  (
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 ) ) ) }  <->  f  e.  {
f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } ) )
76anbi1d 685 . . . . 5  |-  ( X  =  Y  ->  (
( 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 ) ) ) }  /\  x  e. 
U_ i  e.  dom  f ( f `  i ) )  <->  ( f  e.  { f  |  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `  (/) )  = 
pred ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) }  /\  x  e. 
U_ i  e.  dom  f ( f `  i ) ) ) )
87rexbidv2 2579 . . . 4  |-  ( X  =  Y  ->  ( 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 ) ) ) } x  e.  U_ i  e.  dom  f ( f `  i )  <->  E. f  e.  { f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } x  e.  U_ i  e.  dom  f ( f `  i ) ) )
98abbidv 2410 . . 3  |-  ( X  =  Y  ->  { x  |  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 ) ) ) } x  e.  U_ i  e.  dom  f ( f `  i ) }  =  { x  |  E. f  e.  {
f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } x  e.  U_ i  e.  dom  f ( f `  i ) } )
10 df-iun 3923 . . 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 )  =  { x  |  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 ) ) ) } x  e.  U_ i  e.  dom  f ( f `  i ) }
11 df-iun 3923 . . 3  |-  U_ f  e.  { f  |  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `  (/) )  = 
pred ( Y ,  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 )  =  { x  |  E. f  e.  { f  |  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( Y ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) } x  e.  U_ i  e.  dom  f ( f `  i ) }
129, 10, 113eqtr4g 2353 . 2  |-  ( X  =  Y  ->  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 )  = 
U_ f  e.  {
f  |  E. n  e.  ( om  \  { (/)
} ) ( f  Fn  n  /\  (
f `  (/) )  = 
pred ( Y ,  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 ) )
13 df-bnj18 29036 . 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 )
14 df-bnj18 29036 . 2  |-  trCl ( Y ,  A ,  R )  =  U_ f  e.  { f  |  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( Y ,  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 )
1512, 13, 143eqtr4g 2353 1  |-  ( X  =  Y  ->  trCl ( X ,  A ,  R )  =  trCl ( Y ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557    \ cdif 3162   (/)c0 3468   {csn 3653   U_ciun 3921   suc csuc 4410   omcom 4672   dom cdm 4705    Fn wfn 5266   ` cfv 5271    predc-bnj14 29029    trClc-bnj18 29035
This theorem is referenced by:  bnj1137  29341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-iun 3923  df-br 4040  df-bnj14 29030  df-bnj18 29036
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