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Theorem bnj1491 34533
Description: Technical lemma for bnj60 34538. 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
bnj1491.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1491.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1491.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1491.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1491.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1491.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1491.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1491.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1491.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1491.10  |-  P  = 
U. H
bnj1491.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1491.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
bnj1491.13  |-  ( ch 
->  ( Q  e.  C  /\  dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
Assertion
Ref Expression
bnj1491  |-  ( ( ch  /\  Q  e. 
_V )  ->  E. f
( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
Distinct variable groups:    A, f    f, G    R, f    x, f
Allowed substitution hints:    ps( x, y, f, d)    ch( x, y, f, d)    ta( x, y, f, d)    A( x, y, d)    B( x, y, f, d)    C( x, y, f, d)    D( x, y, f, d)    P( x, y, f, d)    Q( x, y, f, d)    R( x, y, d)    G( x, y, d)    H( x, y, f, d)    Y( x, y, f, d)    Z( x, y, f, d)    ta'( x, y, f, d)

Proof of Theorem bnj1491
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 bnj1491.13 . 2  |-  ( ch 
->  ( Q  e.  C  /\  dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
2 bnj1491.1 . . . . 5  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
3 bnj1491.2 . . . . 5  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
4 bnj1491.3 . . . . 5  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
5 bnj1491.4 . . . . 5  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
6 bnj1491.5 . . . . 5  |-  D  =  { x  e.  A  |  -.  E. f ta }
7 bnj1491.6 . . . . 5  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
8 bnj1491.7 . . . . 5  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
9 bnj1491.8 . . . . 5  |-  ( ta'  <->  [. y  /  x ]. ta )
10 bnj1491.9 . . . . 5  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
11 bnj1491.10 . . . . 5  |-  P  = 
U. H
12 bnj1491.11 . . . . 5  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
13 bnj1491.12 . . . . 5  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
142, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13bnj1466 34529 . . . 4  |-  ( w  e.  Q  ->  A. f  w  e.  Q )
1514nfcii 2606 . . 3  |-  F/_ f Q
164bnj1317 34300 . . . . . 6  |-  ( w  e.  C  ->  A. f  w  e.  C )
1716nfcii 2606 . . . . 5  |-  F/_ f C
1815, 17nfel 2629 . . . 4  |-  F/ f  Q  e.  C
1915nfdm 5233 . . . . 5  |-  F/_ f dom  Q
2019nfeq1 2631 . . . 4  |-  F/ f dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
2118, 20nfan 1933 . . 3  |-  F/ f ( Q  e.  C  /\  dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
22 eleq1 2526 . . . 4  |-  ( f  =  Q  ->  (
f  e.  C  <->  Q  e.  C ) )
23 dmeq 5192 . . . . 5  |-  ( f  =  Q  ->  dom  f  =  dom  Q )
2423eqeq1d 2456 . . . 4  |-  ( f  =  Q  ->  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  <->  dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
2522, 24anbi12d 708 . . 3  |-  ( f  =  Q  ->  (
( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )  <-> 
( Q  e.  C  /\  dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) ) )
2615, 21, 25spcegf 3187 . 2  |-  ( Q  e.  _V  ->  (
( Q  e.  C  /\  dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )  ->  E. f ( f  e.  C  /\  dom  f  =  ( {
x }  u.  trCl ( x ,  A ,  R ) ) ) ) )
271, 26mpan9 467 1  |-  ( ( ch  /\  Q  e. 
_V )  ->  E. f
( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398   E.wex 1617    e. wcel 1823   {cab 2439    =/= wne 2649   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106   [.wsbc 3324    u. cun 3459    C_ wss 3461   (/)c0 3783   {csn 4016   <.cop 4022   U.cuni 4235   class class class wbr 4439   dom cdm 4988    |` cres 4990    Fn wfn 5565   ` cfv 5570    predc-bnj14 34160    FrSe w-bnj15 34164    trClc-bnj18 34166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-dm 4998  df-res 5000  df-iota 5534  df-fv 5578
This theorem is referenced by:  bnj1312  34534
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