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Theorem bnj1125 28784
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 955 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  R  FrSe  A )
2 bnj1127 28783 . . 3  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  Y  e.  A )
323ad2ant3 978 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  Y  e.  A )
4 bnj893 28722 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  e.  _V )
543adant3 975 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  trCl ( X ,  A ,  R )  e.  _V )
6 bnj1029 28760 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
763adant3 975 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
8 simp3 957 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  Y  e.  trCl ( X ,  A ,  R )
)
9 elisset 2874 . . . . 5  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  E. y  y  =  Y )
1093ad2ant3 978 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  E. y 
y  =  Y )
11 df-bnj19 28484 . . . . . . . 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 2679 . . . . . . . 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 187 . . . . . . 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 15 . . . . . 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 2418 . . . . . . 7  |-  ( y  =  Y  ->  (
y  e.  trCl ( X ,  A ,  R )  <->  Y  e.  trCl ( X ,  A ,  R ) ) )
16 bnj602 28709 . . . . . . . 8  |-  ( y  =  Y  ->  pred (
y ,  A ,  R )  =  pred ( Y ,  A ,  R ) )
1716sseq1d 3281 . . . . . . 7  |-  ( y  =  Y  ->  (  pred ( y ,  A ,  R )  C_  trCl ( X ,  A ,  R )  <->  pred ( Y ,  A ,  R
)  C_  trCl ( X ,  A ,  R
) ) )
1815, 17imbi12d 311 . . . . . 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 210 . . . . 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 1634 . . . 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 32 . . 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 14 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  pred ( Y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
23 biid 227 . . 3  |-  ( ( R  FrSe  A  /\  Y  e.  A )  <->  ( R  FrSe  A  /\  Y  e.  A )
)
24 biid 227 . . 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 28780 . 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 1189 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 358    /\ w3a 934   E.wex 1541    = wceq 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864    C_ wss 3228    predc-bnj14 28475    FrSe w-bnj15 28479    trClc-bnj18 28481    TrFow-bnj19 28483
This theorem is referenced by:  bnj1137  28787  bnj1136  28789  bnj1175  28796  bnj1408  28828  bnj1417  28833  bnj1452  28844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-reg 7396  ax-inf2 7432
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-1o 6566  df-bnj17 28474  df-bnj14 28476  df-bnj13 28478  df-bnj15 28480  df-bnj18 28482  df-bnj19 28484
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