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Theorem bnj1416 29345
Description: Technical lemma for bnj60 29368. 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 5063 . . 3  |-  dom  Q  =  dom  ( P  u.  {
<. x ,  ( G `
 Z ) >. } )
3 dmun 5068 . . 3  |-  dom  ( P  u.  { <. x ,  ( G `  Z ) >. } )  =  ( dom  P  u.  dom  { <. x ,  ( G `  Z ) >. } )
4 fvex 5734 . . . . 5  |-  ( G `
 Z )  e. 
_V
54dmsnop 5336 . . . 4  |-  dom  { <. x ,  ( G `
 Z ) >. }  =  { x }
65uneq2i 3490 . . 3  |-  ( dom 
P  u.  dom  { <. x ,  ( G `
 Z ) >. } )  =  ( dom  P  u.  {
x } )
72, 3, 63eqtri 2459 . 2  |-  dom  Q  =  ( dom  P  u.  { x } )
8 bnj1416.28 . . . 4  |-  ( ch 
->  dom  P  =  trCl ( x ,  A ,  R ) )
98uneq1d 3492 . . 3  |-  ( ch 
->  ( dom  P  u.  { x } )  =  (  trCl ( x ,  A ,  R )  u.  { x }
) )
10 uncom 3483 . . 3  |-  (  trCl ( x ,  A ,  R )  u.  {
x } )  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
119, 10syl6eq 2483 . 2  |-  ( ch 
->  ( dom  P  u.  { x } )  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
127, 11syl5eq 2479 1  |-  ( ch 
->  dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701   [.wsbc 3153    u. cun 3310    C_ wss 3312   (/)c0 3620   {csn 3806   <.cop 3809   U.cuni 4007   class class class wbr 4204   dom cdm 4870    |` cres 4872    Fn wfn 5441   ` cfv 5446    predc-bnj14 28989    FrSe w-bnj15 28993    trClc-bnj18 28995
This theorem is referenced by:  bnj1312  29364
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-dm 4880  df-iota 5410  df-fv 5454
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