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Theorem bnj1416 34442
Description: Technical lemma for bnj60 34465. 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 5117 . . 3  |-  dom  Q  =  dom  ( P  u.  {
<. x ,  ( G `
 Z ) >. } )
3 dmun 5122 . . 3  |-  dom  ( P  u.  { <. x ,  ( G `  Z ) >. } )  =  ( dom  P  u.  dom  { <. x ,  ( G `  Z ) >. } )
4 fvex 5784 . . . . 5  |-  ( G `
 Z )  e. 
_V
54dmsnop 5390 . . . 4  |-  dom  { <. x ,  ( G `
 Z ) >. }  =  { x }
65uneq2i 3569 . . 3  |-  ( dom 
P  u.  dom  { <. x ,  ( G `
 Z ) >. } )  =  ( dom  P  u.  {
x } )
72, 3, 63eqtri 2415 . 2  |-  dom  Q  =  ( dom  P  u.  { x } )
8 bnj1416.28 . . . 4  |-  ( ch 
->  dom  P  =  trCl ( x ,  A ,  R ) )
98uneq1d 3571 . . 3  |-  ( ch 
->  ( dom  P  u.  { x } )  =  (  trCl ( x ,  A ,  R )  u.  { x }
) )
10 uncom 3562 . . 3  |-  (  trCl ( x ,  A ,  R )  u.  {
x } )  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
119, 10syl6eq 2439 . 2  |-  ( ch 
->  ( dom  P  u.  { x } )  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
127, 11syl5eq 2435 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 367    /\ w3a 971    = wceq 1399   E.wex 1620    e. wcel 1826   {cab 2367    =/= wne 2577   A.wral 2732   E.wrex 2733   {crab 2736   [.wsbc 3252    u. cun 3387    C_ wss 3389   (/)c0 3711   {csn 3944   <.cop 3950   U.cuni 4163   class class class wbr 4367   dom cdm 4913    |` cres 4915    Fn wfn 5491   ` cfv 5496    predc-bnj14 34087    FrSe w-bnj15 34091    trClc-bnj18 34093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-dm 4923  df-iota 5460  df-fv 5504
This theorem is referenced by:  bnj1312  34461
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