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Theorem bnj1125 31983
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 988 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  R  FrSe  A )
2 bnj1127 31982 . . 3  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  Y  e.  A )
323ad2ant3 1011 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  Y  e.  A )
4 bnj893 31921 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  e.  _V )
543adant3 1008 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  trCl ( X ,  A ,  R )  e.  _V )
6 bnj1029 31959 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
763adant3 1008 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
8 simp3 990 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  Y  e.  trCl ( X ,  A ,  R )
)
9 elisset 2983 . . . . 5  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  E. y  y  =  Y )
1093ad2ant3 1011 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  E. y 
y  =  Y )
11 df-bnj19 31685 . . . . . . . 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 2776 . . . . . . . 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 195 . . . . . . 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 2503 . . . . . . 7  |-  ( y  =  Y  ->  (
y  e.  trCl ( X ,  A ,  R )  <->  Y  e.  trCl ( X ,  A ,  R ) ) )
16 bnj602 31908 . . . . . . . 8  |-  ( y  =  Y  ->  pred (
y ,  A ,  R )  =  pred ( Y ,  A ,  R ) )
1716sseq1d 3383 . . . . . . 7  |-  ( y  =  Y  ->  (  pred ( y ,  A ,  R )  C_  trCl ( X ,  A ,  R )  <->  pred ( Y ,  A ,  R
)  C_  trCl ( X ,  A ,  R
) ) )
1815, 17imbi12d 320 . . . . . 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 219 . . . . 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 1688 . . . 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 36 . . 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 236 . . 3  |-  ( ( R  FrSe  A  /\  Y  e.  A )  <->  ( R  FrSe  A  /\  Y  e.  A )
)
24 biid 236 . . 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 31979 . 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 1225 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   A.wral 2715   _Vcvv 2972    C_ wss 3328    predc-bnj14 31676    FrSe w-bnj15 31680    trClc-bnj18 31682    TrFow-bnj19 31684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-reg 7807  ax-inf2 7847
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-om 6477  df-1o 6920  df-bnj17 31675  df-bnj14 31677  df-bnj13 31679  df-bnj15 31681  df-bnj18 31683  df-bnj19 31685
This theorem is referenced by:  bnj1137  31986  bnj1136  31988  bnj1175  31995  bnj1408  32027  bnj1417  32032  bnj1452  32043
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