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Theorem bnj1137 33486
Description: Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1137.1  |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
Assertion
Ref Expression
bnj1137  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  TrFo ( B ,  A ,  R )
)
Distinct variable groups:    y, A    y, R    y, X
Allowed substitution hint:    B( y)

Proof of Theorem bnj1137
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 bnj1137.1 . . . . . 6  |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
21eleq2i 2545 . . . . 5  |-  ( v  e.  B  <->  v  e.  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) ) )
3 elun 3650 . . . . 5  |-  ( v  e.  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R
)  trCl ( y ,  A ,  R ) )  <->  ( v  e. 
pred ( X ,  A ,  R )  \/  v  e.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) ) )
42, 3bitri 249 . . . 4  |-  ( v  e.  B  <->  ( v  e.  pred ( X ,  A ,  R )  \/  v  e.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) ) )
5 bnj213 33375 . . . . . . . . 9  |-  pred ( X ,  A ,  R )  C_  A
65sseli 3505 . . . . . . . 8  |-  ( v  e.  pred ( X ,  A ,  R )  ->  v  e.  A )
7 bnj906 33423 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  v  e.  A )  ->  pred ( v ,  A ,  R ) 
C_  trCl ( v ,  A ,  R ) )
87adantlr 714 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  A )  ->  pred (
v ,  A ,  R )  C_  trCl (
v ,  A ,  R ) )
96, 8sylan2 474 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  pred ( X ,  A ,  R ) )  ->  pred ( v ,  A ,  R )  C_  trCl (
v ,  A ,  R ) )
10 bnj906 33423 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
1110sselda 3509 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  pred ( X ,  A ,  R ) )  -> 
v  e.  trCl ( X ,  A ,  R ) )
12 bnj18eq1 33420 . . . . . . . . 9  |-  ( y  =  v  ->  trCl (
y ,  A ,  R )  =  trCl ( v ,  A ,  R ) )
1312ssiun2s 4375 . . . . . . . 8  |-  ( v  e.  trCl ( X ,  A ,  R )  ->  trCl ( v ,  A ,  R ) 
C_  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
1411, 13syl 16 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  pred ( X ,  A ,  R ) )  ->  trCl ( v ,  A ,  R )  C_  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )
159, 14sstrd 3519 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  pred ( X ,  A ,  R ) )  ->  pred ( v ,  A ,  R )  C_  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )
16 bnj1147 33485 . . . . . . . . . . 11  |-  trCl (
y ,  A ,  R )  C_  A
1716rgenw 2828 . . . . . . . . . 10  |-  A. y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  C_  A
18 iunss 4372 . . . . . . . . . 10  |-  ( U_ y  e.  trCl  ( X ,  A ,  R
)  trCl ( y ,  A ,  R ) 
C_  A  <->  A. y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  C_  A
)
1917, 18mpbir 209 . . . . . . . . 9  |-  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  C_  A
2019sseli 3505 . . . . . . . 8  |-  ( v  e.  U_ y  e. 
trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  ->  v  e.  A )
2120, 8sylan2 474 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  ->  pred ( v ,  A ,  R )  C_  trCl (
v ,  A ,  R ) )
22 bnj1125 33483 . . . . . . . . . . . 12  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
) )  ->  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
23223expia 1198 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( y  e.  trCl ( X ,  A ,  R )  ->  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) ) )
2423ralrimiv 2879 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
25 iunss 4372 . . . . . . . . . 10  |-  ( U_ y  e.  trCl  ( X ,  A ,  R
)  trCl ( y ,  A ,  R ) 
C_  trCl ( X ,  A ,  R )  <->  A. y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
2624, 25sylibr 212 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
2726sselda 3509 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  -> 
v  e.  trCl ( X ,  A ,  R ) )
2827, 13syl 16 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  ->  trCl ( v ,  A ,  R )  C_  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )
2921, 28sstrd 3519 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  ->  pred ( v ,  A ,  R )  C_  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )
3015, 29jaodan 783 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( v  e.  pred ( X ,  A ,  R )  \/  v  e.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) ) )  ->  pred ( v ,  A ,  R ) 
C_  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
31 ssun2 3673 . . . . . 6  |-  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  C_  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R
)  trCl ( y ,  A ,  R ) )
3231, 1sseqtr4i 3542 . . . . 5  |-  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  C_  B
3330, 32syl6ss 3521 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( v  e.  pred ( X ,  A ,  R )  \/  v  e.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) ) )  ->  pred ( v ,  A ,  R ) 
C_  B )
344, 33sylan2b 475 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  B )  ->  pred (
v ,  A ,  R )  C_  B
)
3534ralrimiva 2881 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. v  e.  B  pred ( v ,  A ,  R )  C_  B
)
36 df-bnj19 33185 . 2  |-  (  TrFo ( B ,  A ,  R )  <->  A. v  e.  B  pred ( v ,  A ,  R
)  C_  B )
3735, 36sylibr 212 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  TrFo ( B ,  A ,  R )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817    u. cun 3479    C_ wss 3481   U_ciun 4331    predc-bnj14 33176    FrSe w-bnj15 33180    trClc-bnj18 33182    TrFow-bnj19 33184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-reg 8030  ax-inf2 8070
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6696  df-1o 7142  df-bnj17 33175  df-bnj14 33177  df-bnj13 33179  df-bnj15 33181  df-bnj18 33183  df-bnj19 33185
This theorem is referenced by:  bnj1136  33488
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