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Theorem bnj1416 33174
Description: Technical lemma for bnj60 33197. 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 5202 . . 3  |-  dom  Q  =  dom  ( P  u.  {
<. x ,  ( G `
 Z ) >. } )
3 dmun 5207 . . 3  |-  dom  ( P  u.  { <. x ,  ( G `  Z ) >. } )  =  ( dom  P  u.  dom  { <. x ,  ( G `  Z ) >. } )
4 fvex 5874 . . . . 5  |-  ( G `
 Z )  e. 
_V
54dmsnop 5480 . . . 4  |-  dom  { <. x ,  ( G `
 Z ) >. }  =  { x }
65uneq2i 3655 . . 3  |-  ( dom 
P  u.  dom  { <. x ,  ( G `
 Z ) >. } )  =  ( dom  P  u.  {
x } )
72, 3, 63eqtri 2500 . 2  |-  dom  Q  =  ( dom  P  u.  { x } )
8 bnj1416.28 . . . 4  |-  ( ch 
->  dom  P  =  trCl ( x ,  A ,  R ) )
98uneq1d 3657 . . 3  |-  ( ch 
->  ( dom  P  u.  { x } )  =  (  trCl ( x ,  A ,  R )  u.  { x }
) )
10 uncom 3648 . . 3  |-  (  trCl ( x ,  A ,  R )  u.  {
x } )  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
119, 10syl6eq 2524 . 2  |-  ( ch 
->  ( dom  P  u.  { x } )  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
127, 11syl5eq 2520 1  |-  ( ch 
->  dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   [.wsbc 3331    u. cun 3474    C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   U.cuni 4245   class class class wbr 4447   dom cdm 4999    |` cres 5001    Fn wfn 5581   ` cfv 5586    predc-bnj14 32820    FrSe w-bnj15 32824    trClc-bnj18 32826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-dm 5009  df-iota 5549  df-fv 5594
This theorem is referenced by:  bnj1312  33193
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