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Theorem bnj1373 29376
Description: Technical lemma for bnj60 29408. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1373.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1373.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1373.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1373.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1373.5  |-  ( ta'  <->  [. y  /  x ]. ta )
Assertion
Ref Expression
bnj1373  |-  ( ta'  <->  (
f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
Distinct variable groups:    x, A    B, f    x, R    f,
d, x    x, y
Allowed substitution hints:    ta( x, y, f, d)    A( y, f, d)    B( x, y, d)    C( x, y, f, d)    R( y, f, d)    G( x, y, f, d)    Y( x, y, f, d)    ta'( x, y, f, d)

Proof of Theorem bnj1373
StepHypRef Expression
1 bnj1373.5 . 2  |-  ( ta'  <->  [. y  /  x ]. ta )
2 vex 2804 . . 3  |-  y  e. 
_V
3 bnj1373.3 . . . . . . 7  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
4 bnj1373.1 . . . . . . . 8  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
54bnj1309 29368 . . . . . . 7  |-  ( f  e.  B  ->  A. x  f  e.  B )
63, 5bnj1307 29369 . . . . . 6  |-  ( f  e.  C  ->  A. x  f  e.  C )
76bnj1351 29175 . . . . 5  |-  ( ( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) )  ->  A. x ( f  e.  C  /\  dom  f  =  ( {
y }  u.  trCl ( y ,  A ,  R ) ) ) )
87nfi 1541 . . . 4  |-  F/ x
( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) )
9 bnj1373.4 . . . . 5  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
10 sneq 3664 . . . . . . . 8  |-  ( x  =  y  ->  { x }  =  { y } )
11 bnj1318 29371 . . . . . . . 8  |-  ( x  =  y  ->  trCl (
x ,  A ,  R )  =  trCl ( y ,  A ,  R ) )
1210, 11uneq12d 3343 . . . . . . 7  |-  ( x  =  y  ->  ( { x }  u.  trCl ( x ,  A ,  R ) )  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) )
1312eqeq2d 2307 . . . . . 6  |-  ( x  =  y  ->  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  <->  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
1413anbi2d 684 . . . . 5  |-  ( x  =  y  ->  (
( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )  <-> 
( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) ) )
159, 14syl5bb 248 . . . 4  |-  ( x  =  y  ->  ( ta 
<->  ( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) ) )
168, 15sbciegf 3035 . . 3  |-  ( y  e.  _V  ->  ( [. y  /  x ]. ta  <->  ( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl (
y ,  A ,  R ) ) ) ) )
172, 16ax-mp 8 . 2  |-  ( [. y  /  x ]. ta  <->  ( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
181, 17bitri 240 1  |-  ( ta'  <->  (
f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   _Vcvv 2801   [.wsbc 3004    u. cun 3163    C_ wss 3165   {csn 3653   <.cop 3656   dom cdm 4705    |` cres 4707    Fn wfn 5266   ` cfv 5271    predc-bnj14 29029    trClc-bnj18 29035
This theorem is referenced by:  bnj1374  29377  bnj1384  29378  bnj1398  29380  bnj1450  29396  bnj1489  29402
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-sbc 3005  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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