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Theorem bnj1373 29060
Description: Technical lemma for bnj60 29092. 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 2791 . . 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 29052 . . . . . . 7  |-  ( f  e.  B  ->  A. x  f  e.  B )
63, 5bnj1307 29053 . . . . . 6  |-  ( f  e.  C  ->  A. x  f  e.  C )
76bnj1351 28859 . . . . 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 1538 . . . 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 3651 . . . . . . . 8  |-  ( x  =  y  ->  { x }  =  { y } )
11 bnj1318 29055 . . . . . . . 8  |-  ( x  =  y  ->  trCl (
x ,  A ,  R )  =  trCl ( y ,  A ,  R ) )
1210, 11uneq12d 3330 . . . . . . 7  |-  ( x  =  y  ->  ( { x }  u.  trCl ( x ,  A ,  R ) )  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) )
1312eqeq2d 2294 . . . . . 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 3022 . . 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 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788   [.wsbc 2991    u. cun 3150    C_ wss 3152   {csn 3640   <.cop 3643   dom cdm 4689    |` cres 4691    Fn wfn 5250   ` cfv 5255    predc-bnj14 28713    trClc-bnj18 28719
This theorem is referenced by:  bnj1374  29061  bnj1384  29062  bnj1398  29064  bnj1450  29080  bnj1489  29086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-iun 3907  df-br 4024  df-bnj14 28714  df-bnj18 28720
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