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Theorem bnj1491 29087
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
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 simpr 447 . 2  |-  ( ( ch  /\  Q  e. 
_V )  ->  Q  e.  _V )
2 bnj1491.13 . . 3  |-  ( ch 
->  ( Q  e.  C  /\  dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
32adantr 451 . 2  |-  ( ( ch  /\  Q  e. 
_V )  ->  ( Q  e.  C  /\  dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
4 bnj1491.1 . . . . 5  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
5 bnj1491.2 . . . . 5  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
6 bnj1491.3 . . . . 5  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
7 bnj1491.4 . . . . 5  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
8 bnj1491.5 . . . . 5  |-  D  =  { x  e.  A  |  -.  E. f ta }
9 bnj1491.6 . . . . 5  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
10 bnj1491.7 . . . . 5  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
11 bnj1491.8 . . . . 5  |-  ( ta'  <->  [. y  /  x ]. ta )
12 bnj1491.9 . . . . 5  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
13 bnj1491.10 . . . . 5  |-  P  = 
U. H
14 bnj1491.11 . . . . 5  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
15 bnj1491.12 . . . . 5  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
164, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15bnj1466 29083 . . . 4  |-  ( w  e.  Q  ->  A. f  w  e.  Q )
1716nfcii 2410 . . 3  |-  F/_ f Q
186bnj1317 28854 . . . . . 6  |-  ( w  e.  C  ->  A. f  w  e.  C )
1918nfcii 2410 . . . . 5  |-  F/_ f C
2017, 19nfel 2427 . . . 4  |-  F/ f  Q  e.  C
2117nfdm 4920 . . . . 5  |-  F/_ f dom  Q
2221nfeq1 2428 . . . 4  |-  F/ f dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
2320, 22nfan 1771 . . 3  |-  F/ f ( Q  e.  C  /\  dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
24 eleq1 2343 . . . 4  |-  ( f  =  Q  ->  (
f  e.  C  <->  Q  e.  C ) )
25 dmeq 4879 . . . . 5  |-  ( f  =  Q  ->  dom  f  =  dom  Q )
2625eqeq1d 2291 . . . 4  |-  ( f  =  Q  ->  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  <->  dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
2724, 26anbi12d 691 . . 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 ) ) ) ) )
2817, 23, 27spcegf 2864 . 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 ) ) ) ) )
291, 3, 28sylc 56 1  |-  ( ( ch  /\  Q  e. 
_V )  ->  E. f
( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788   [.wsbc 2991    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643   U.cuni 3827   class class class wbr 4023   dom cdm 4689    |` cres 4691    Fn wfn 5250   ` cfv 5255    predc-bnj14 28713    FrSe w-bnj15 28717    trClc-bnj18 28719
This theorem is referenced by:  bnj1312  29088
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-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-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-dm 4699  df-res 4701  df-iota 5219  df-fv 5263
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