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Theorem bnj1125 29079
Description: Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1125  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  trCl ( Y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )

Proof of Theorem bnj1125
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simp1 957 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  R  FrSe  A )
2 bnj1127 29078 . . 3  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  Y  e.  A )
323ad2ant3 980 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  Y  e.  A )
4 bnj893 29017 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  e.  _V )
543adant3 977 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  trCl ( X ,  A ,  R )  e.  _V )
6 bnj1029 29055 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
763adant3 977 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
8 simp3 959 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  Y  e.  trCl ( X ,  A ,  R )
)
9 elisset 2934 . . . . 5  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  E. y  y  =  Y )
1093ad2ant3 980 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  E. y 
y  =  Y )
11 df-bnj19 28779 . . . . . . . 8  |-  (  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R )  <->  A. y  e.  trCl  ( X ,  A ,  R )  pred (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
12 rsp 2734 . . . . . . . 8  |-  ( A. y  e.  trCl  ( X ,  A ,  R
)  pred ( y ,  A ,  R ) 
C_  trCl ( X ,  A ,  R )  ->  ( y  e.  trCl ( X ,  A ,  R )  ->  pred (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) ) )
1311, 12sylbi 188 . . . . . . 7  |-  (  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R )  ->  ( y  e. 
trCl ( X ,  A ,  R )  ->  pred ( y ,  A ,  R ) 
C_  trCl ( X ,  A ,  R )
) )
147, 13syl 16 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  (
y  e.  trCl ( X ,  A ,  R )  ->  pred (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) ) )
15 eleq1 2472 . . . . . . 7  |-  ( y  =  Y  ->  (
y  e.  trCl ( X ,  A ,  R )  <->  Y  e.  trCl ( X ,  A ,  R ) ) )
16 bnj602 29004 . . . . . . . 8  |-  ( y  =  Y  ->  pred (
y ,  A ,  R )  =  pred ( Y ,  A ,  R ) )
1716sseq1d 3343 . . . . . . 7  |-  ( y  =  Y  ->  (  pred ( y ,  A ,  R )  C_  trCl ( X ,  A ,  R )  <->  pred ( Y ,  A ,  R
)  C_  trCl ( X ,  A ,  R
) ) )
1815, 17imbi12d 312 . . . . . 6  |-  ( y  =  Y  ->  (
( y  e.  trCl ( X ,  A ,  R )  ->  pred (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )  <->  ( Y  e.  trCl ( X ,  A ,  R )  ->  pred ( Y ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
) ) )
1914, 18syl5ib 211 . . . . 5  |-  ( y  =  Y  ->  (
( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R ) )  -> 
( Y  e.  trCl ( X ,  A ,  R )  ->  pred ( Y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) ) ) )
2019exlimiv 1641 . . . 4  |-  ( E. y  y  =  Y  ->  ( ( R 
FrSe  A  /\  X  e.  A  /\  Y  e. 
trCl ( X ,  A ,  R )
)  ->  ( Y  e.  trCl ( X ,  A ,  R )  ->  pred ( Y ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
) ) )
2110, 20mpcom 34 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  ( Y  e.  trCl ( X ,  A ,  R
)  ->  pred ( Y ,  A ,  R
)  C_  trCl ( X ,  A ,  R
) ) )
228, 21mpd 15 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  pred ( Y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
23 biid 228 . . 3  |-  ( ( R  FrSe  A  /\  Y  e.  A )  <->  ( R  FrSe  A  /\  Y  e.  A )
)
24 biid 228 . . 3  |-  ( ( 
trCl ( X ,  A ,  R )  e.  _V  /\  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R )  /\  pred ( Y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )  <->  (  trCl ( X ,  A ,  R )  e.  _V  /\ 
TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R )  /\  pred ( Y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) ) )
2523, 24bnj1124 29075 . 2  |-  ( ( ( R  FrSe  A  /\  Y  e.  A
)  /\  (  trCl ( X ,  A ,  R )  e.  _V  /\ 
TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R )  /\  pred ( Y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) ) )  ->  trCl ( Y ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
261, 3, 5, 7, 22, 25syl23anc 1191 1  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  trCl ( Y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   A.wral 2674   _Vcvv 2924    C_ wss 3288    predc-bnj14 28770    FrSe w-bnj15 28774    trClc-bnj18 28776    TrFow-bnj19 28778
This theorem is referenced by:  bnj1137  29082  bnj1136  29084  bnj1175  29091  bnj1408  29123  bnj1417  29128  bnj1452  29139
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-reg 7524  ax-inf2 7560
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-1o 6691  df-bnj17 28769  df-bnj14 28771  df-bnj13 28773  df-bnj15 28775  df-bnj18 28777  df-bnj19 28779
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