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Theorem bnj1373 32026
Description: Technical lemma for bnj60 32058. 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 2980 . . 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 32018 . . . . . . 7  |-  ( f  e.  B  ->  A. x  f  e.  B )
63, 5bnj1307 32019 . . . . . 6  |-  ( f  e.  C  ->  A. x  f  e.  C )
76bnj1351 31825 . . . . 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 1596 . . . 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 3892 . . . . . . . 8  |-  ( x  =  y  ->  { x }  =  { y } )
11 bnj1318 32021 . . . . . . . 8  |-  ( x  =  y  ->  trCl (
x ,  A ,  R )  =  trCl ( y ,  A ,  R ) )
1210, 11uneq12d 3516 . . . . . . 7  |-  ( x  =  y  ->  ( { x }  u.  trCl ( x ,  A ,  R ) )  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) )
1312eqeq2d 2454 . . . . . 6  |-  ( x  =  y  ->  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  <->  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
1413anbi2d 703 . . . . 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 257 . . . 4  |-  ( x  =  y  ->  ( ta 
<->  ( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) ) )
168, 15sbciegf 3223 . . 3  |-  ( y  e.  _V  ->  ( [. y  /  x ]. ta  <->  ( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl (
y ,  A ,  R ) ) ) ) )
172, 16ax-mp 5 . 2  |-  ( [. y  /  x ]. ta  <->  ( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
181, 17bitri 249 1  |-  ( ta'  <->  (
f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2720   E.wrex 2721   _Vcvv 2977   [.wsbc 3191    u. cun 3331    C_ wss 3333   {csn 3882   <.cop 3888   dom cdm 4845    |` cres 4847    Fn wfn 5418   ` cfv 5423    predc-bnj14 31681    trClc-bnj18 31687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-iun 4178  df-br 4298  df-bnj14 31682  df-bnj18 31688
This theorem is referenced by:  bnj1374  32027  bnj1384  32028  bnj1398  32030  bnj1450  32046  bnj1489  32052
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