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Theorem bnj1373 29911
Description: Technical lemma for bnj60 29943. 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 3034 . . 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 29903 . . . . . . 7  |-  ( f  e.  B  ->  A. x  f  e.  B )
63, 5bnj1307 29904 . . . . . 6  |-  ( f  e.  C  ->  A. x  f  e.  C )
76bnj1351 29710 . . . . 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 1682 . . . 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 3969 . . . . . . . 8  |-  ( x  =  y  ->  { x }  =  { y } )
11 bnj1318 29906 . . . . . . . 8  |-  ( x  =  y  ->  trCl (
x ,  A ,  R )  =  trCl ( y ,  A ,  R ) )
1210, 11uneq12d 3580 . . . . . . 7  |-  ( x  =  y  ->  ( { x }  u.  trCl ( x ,  A ,  R ) )  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) )
1312eqeq2d 2481 . . . . . 6  |-  ( x  =  y  ->  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  <->  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
1413anbi2d 718 . . . . 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 265 . . . 4  |-  ( x  =  y  ->  ( ta 
<->  ( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) ) )
168, 15sbciegf 3287 . . 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 257 1  |-  ( ta'  <->  (
f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   {cab 2457   A.wral 2756   E.wrex 2757   _Vcvv 3031   [.wsbc 3255    u. cun 3388    C_ wss 3390   {csn 3959   <.cop 3965   dom cdm 4839    |` cres 4841    Fn wfn 5584   ` cfv 5589    predc-bnj14 29565    trClc-bnj18 29571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-iun 4271  df-br 4396  df-bnj14 29566  df-bnj18 29572
This theorem is referenced by:  bnj1374  29912  bnj1384  29913  bnj1398  29915  bnj1450  29931  bnj1489  29937
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