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Theorem bnj1416 29069
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
bnj1416.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1416.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1416.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1416.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1416.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1416.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1416.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1416.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1416.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1416.10  |-  P  = 
U. H
bnj1416.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1416.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
bnj1416.28  |-  ( ch 
->  dom  P  =  trCl ( x ,  A ,  R ) )
Assertion
Ref Expression
bnj1416  |-  ( ch 
->  dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )

Proof of Theorem bnj1416
StepHypRef Expression
1 bnj1416.12 . . . 4  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
21dmeqi 4880 . . 3  |-  dom  Q  =  dom  ( P  u.  {
<. x ,  ( G `
 Z ) >. } )
3 dmun 4885 . . 3  |-  dom  ( P  u.  { <. x ,  ( G `  Z ) >. } )  =  ( dom  P  u.  dom  { <. x ,  ( G `  Z ) >. } )
4 fvex 5539 . . . . 5  |-  ( G `
 Z )  e. 
_V
54dmsnop 5147 . . . 4  |-  dom  { <. x ,  ( G `
 Z ) >. }  =  { x }
65uneq2i 3326 . . 3  |-  ( dom 
P  u.  dom  { <. x ,  ( G `
 Z ) >. } )  =  ( dom  P  u.  {
x } )
72, 3, 63eqtri 2307 . 2  |-  dom  Q  =  ( dom  P  u.  { x } )
8 bnj1416.28 . . . 4  |-  ( ch 
->  dom  P  =  trCl ( x ,  A ,  R ) )
98uneq1d 3328 . . 3  |-  ( ch 
->  ( dom  P  u.  { x } )  =  (  trCl ( x ,  A ,  R )  u.  { x }
) )
10 uncom 3319 . . 3  |-  (  trCl ( x ,  A ,  R )  u.  {
x } )  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
119, 10syl6eq 2331 . 2  |-  ( ch 
->  ( dom  P  u.  { x } )  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
127, 11syl5eq 2327 1  |-  ( ch 
->  dom  Q  =  ( { 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   [.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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-dm 4699  df-iota 5219  df-fv 5263
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