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Theorem bnj1125 34191
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 996 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  R  FrSe  A )
2 bnj1127 34190 . . 3  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  Y  e.  A )
323ad2ant3 1019 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  Y  e.  A )
4 bnj893 34129 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  e.  _V )
543adant3 1016 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  trCl ( X ,  A ,  R )  e.  _V )
6 bnj1029 34167 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
763adant3 1016 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
8 simp3 998 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  Y  e.  trCl ( X ,  A ,  R )
)
9 elisset 3120 . . . . 5  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  E. y  y  =  Y )
1093ad2ant3 1019 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R
) )  ->  E. y 
y  =  Y )
11 df-bnj19 33892 . . . . . . . 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 2823 . . . . . . . 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 2529 . . . . . . 7  |-  ( y  =  Y  ->  (
y  e.  trCl ( X ,  A ,  R )  <->  Y  e.  trCl ( X ,  A ,  R ) ) )
16 bnj602 34116 . . . . . . . 8  |-  ( y  =  Y  ->  pred (
y ,  A ,  R )  =  pred ( Y ,  A ,  R ) )
1716sseq1d 3526 . . . . . . 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 1723 . . . 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 34187 . 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 1235 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 973    = wceq 1395   E.wex 1613    e. wcel 1819   A.wral 2807   _Vcvv 3109    C_ wss 3471    predc-bnj14 33883    FrSe w-bnj15 33887    trClc-bnj18 33889    TrFow-bnj19 33891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-reg 8036  ax-inf2 8075
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-1o 7148  df-bnj17 33882  df-bnj14 33884  df-bnj13 33886  df-bnj15 33888  df-bnj18 33890  df-bnj19 33892
This theorem is referenced by:  bnj1137  34194  bnj1136  34196  bnj1175  34203  bnj1408  34235  bnj1417  34240  bnj1452  34251
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