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Theorem bnj1491 29403
Description: Technical lemma for bnj60 29408. 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 29399 . . . 4  |-  ( w  e.  Q  ->  A. f  w  e.  Q )
1716nfcii 2423 . . 3  |-  F/_ f Q
186bnj1317 29170 . . . . . 6  |-  ( w  e.  C  ->  A. f  w  e.  C )
1918nfcii 2423 . . . . 5  |-  F/_ f C
2017, 19nfel 2440 . . . 4  |-  F/ f  Q  e.  C
2117nfdm 4936 . . . . 5  |-  F/_ f dom  Q
2221nfeq1 2441 . . . 4  |-  F/ f dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
2320, 22nfan 1783 . . 3  |-  F/ f ( Q  e.  C  /\  dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
24 eleq1 2356 . . . 4  |-  ( f  =  Q  ->  (
f  e.  C  <->  Q  e.  C ) )
25 dmeq 4895 . . . . 5  |-  ( f  =  Q  ->  dom  f  =  dom  Q )
2625eqeq1d 2304 . . . 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 2877 . 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 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801   [.wsbc 3004    u. cun 3163    C_ wss 3165   (/)c0 3468   {csn 3653   <.cop 3656   U.cuni 3843   class class class wbr 4039   dom cdm 4705    |` cres 4707    Fn wfn 5266   ` cfv 5271    predc-bnj14 29029    FrSe w-bnj15 29033    trClc-bnj18 29035
This theorem is referenced by:  bnj1312  29404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-dm 4715  df-res 4717  df-iota 5235  df-fv 5279
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