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Theorem bnj1137 29800
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 2500 . . . . 5  |-  ( v  e.  B  <->  v  e.  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) ) )
3 elun 3606 . . . . 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 252 . . . 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 29689 . . . . . . . . 9  |-  pred ( X ,  A ,  R )  C_  A
65sseli 3460 . . . . . . . 8  |-  ( v  e.  pred ( X ,  A ,  R )  ->  v  e.  A )
7 bnj906 29737 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  v  e.  A )  ->  pred ( v ,  A ,  R ) 
C_  trCl ( v ,  A ,  R ) )
87adantlr 719 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  A )  ->  pred (
v ,  A ,  R )  C_  trCl (
v ,  A ,  R ) )
96, 8sylan2 476 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  pred ( X ,  A ,  R ) )  ->  pred ( v ,  A ,  R )  C_  trCl (
v ,  A ,  R ) )
10 bnj906 29737 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
1110sselda 3464 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  pred ( X ,  A ,  R ) )  -> 
v  e.  trCl ( X ,  A ,  R ) )
12 bnj18eq1 29734 . . . . . . . . 9  |-  ( y  =  v  ->  trCl (
y ,  A ,  R )  =  trCl ( v ,  A ,  R ) )
1312ssiun2s 4340 . . . . . . . 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 17 . . . . . . 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 3474 . . . . . 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 29799 . . . . . . . . . . 11  |-  trCl (
y ,  A ,  R )  C_  A
1716rgenw 2786 . . . . . . . . . 10  |-  A. y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  C_  A
18 iunss 4337 . . . . . . . . . 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 212 . . . . . . . . 9  |-  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  C_  A
2019sseli 3460 . . . . . . . 8  |-  ( v  e.  U_ y  e. 
trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  ->  v  e.  A )
2120, 8sylan2 476 . . . . . . 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 29797 . . . . . . . . . . . 12  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
) )  ->  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
23223expia 1207 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( y  e.  trCl ( X ,  A ,  R )  ->  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) ) )
2423ralrimiv 2837 . . . . . . . . . 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 4337 . . . . . . . . . 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 215 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
2726sselda 3464 . . . . . . . 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 17 . . . . . . 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 3474 . . . . . 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 792 . . . . 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 3630 . . . . . 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 3497 . . . . 5  |-  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  C_  B
3330, 32syl6ss 3476 . . . 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 477 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  B )  ->  pred (
v ,  A ,  R )  C_  B
)
3534ralrimiva 2839 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. v  e.  B  pred ( v ,  A ,  R )  C_  B
)
36 df-bnj19 29498 . 2  |-  (  TrFo ( B ,  A ,  R )  <->  A. v  e.  B  pred ( v ,  A ,  R
)  C_  B )
3735, 36sylibr 215 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  TrFo ( B ,  A ,  R )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1868   A.wral 2775    u. cun 3434    C_ wss 3436   U_ciun 4296    predc-bnj14 29489    FrSe w-bnj15 29493    trClc-bnj18 29495    TrFow-bnj19 29497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-reg 8110  ax-inf2 8149
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-om 6704  df-1o 7187  df-bnj17 29488  df-bnj14 29490  df-bnj13 29492  df-bnj15 29494  df-bnj18 29496  df-bnj19 29498
This theorem is referenced by:  bnj1136  29802
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