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Theorem bnj1125 29435
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 958 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  R  FrSe  A )
2 bnj1127 29434 . . 3  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  Y  e.  A )
323ad2ant3 981 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  Y  e.  A )
4 bnj893 29373 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  e.  _V )
543adant3 978 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  trCl ( X ,  A ,  R )  e.  _V )
6 bnj1029 29411 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
763adant3 978 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
8 simp3 960 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  Y  e.  trCl ( X ,  A ,  R )
)
9 elisset 2968 . . . . 5  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  E. y  y  =  Y )
1093ad2ant3 981 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  E. y 
y  =  Y )
11 df-bnj19 29135 . . . . . . . 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 2768 . . . . . . . 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 189 . . . . . . 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 2498 . . . . . . 7  |-  ( y  =  Y  ->  (
y  e.  trCl ( X ,  A ,  R )  <->  Y  e.  trCl ( X ,  A ,  R ) ) )
16 bnj602 29360 . . . . . . . 8  |-  ( y  =  Y  ->  pred (
y ,  A ,  R )  =  pred ( Y ,  A ,  R ) )
1716sseq1d 3377 . . . . . . 7  |-  ( y  =  Y  ->  (  pred ( y ,  A ,  R )  C_  trCl ( X ,  A ,  R )  <->  pred ( Y ,  A ,  R
)  C_  trCl ( X ,  A ,  R
) ) )
1815, 17imbi12d 313 . . . . . 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 212 . . . . 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 1645 . . . 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 35 . . 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 229 . . 3  |-  ( ( R  FrSe  A  /\  Y  e.  A )  <->  ( R  FrSe  A  /\  Y  e.  A )
)
24 biid 229 . . 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 29431 . 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 1192 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 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    C_ wss 3322    predc-bnj14 29126    FrSe w-bnj15 29130    trClc-bnj18 29132    TrFow-bnj19 29134
This theorem is referenced by:  bnj1137  29438  bnj1136  29440  bnj1175  29447  bnj1408  29479  bnj1417  29484  bnj1452  29495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-reg 7563  ax-inf2 7599
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-1o 6727  df-bnj17 29125  df-bnj14 29127  df-bnj13 29129  df-bnj15 29131  df-bnj18 29133  df-bnj19 29135
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