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Theorem bnj1137 29025
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 2347 . . . . 5  |-  ( v  e.  B  <->  v  e.  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) ) )
3 elun 3316 . . . . 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 240 . . . 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 28914 . . . . . . . . 9  |-  pred ( X ,  A ,  R )  C_  A
65sseli 3176 . . . . . . . 8  |-  ( v  e.  pred ( X ,  A ,  R )  ->  v  e.  A )
7 bnj906 28962 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  v  e.  A )  ->  pred ( v ,  A ,  R ) 
C_  trCl ( v ,  A ,  R ) )
87adantlr 695 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  A )  ->  pred (
v ,  A ,  R )  C_  trCl (
v ,  A ,  R ) )
96, 8sylan2 460 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  pred ( X ,  A ,  R ) )  ->  pred ( v ,  A ,  R )  C_  trCl (
v ,  A ,  R ) )
10 bnj906 28962 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
1110sselda 3180 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  pred ( X ,  A ,  R ) )  -> 
v  e.  trCl ( X ,  A ,  R ) )
12 bnj18eq1 28959 . . . . . . . . 9  |-  ( y  =  v  ->  trCl (
y ,  A ,  R )  =  trCl ( v ,  A ,  R ) )
1312ssiun2s 3946 . . . . . . . 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 15 . . . . . . 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 3189 . . . . . 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 29024 . . . . . . . . . . 11  |-  trCl (
y ,  A ,  R )  C_  A
1716rgenw 2610 . . . . . . . . . 10  |-  A. y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  C_  A
18 iunss 3943 . . . . . . . . . 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 200 . . . . . . . . 9  |-  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  C_  A
2019sseli 3176 . . . . . . . 8  |-  ( v  e.  U_ y  e. 
trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  ->  v  e.  A )
2120, 8sylan2 460 . . . . . . 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 29022 . . . . . . . . . . . 12  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
) )  ->  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
23223expia 1153 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( y  e.  trCl ( X ,  A ,  R )  ->  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) ) )
2423ralrimiv 2625 . . . . . . . . . 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 3943 . . . . . . . . . 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 203 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
2726sselda 3180 . . . . . . . 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 15 . . . . . . 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 3189 . . . . . 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 760 . . . . 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 3339 . . . . . 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 3211 . . . . 5  |-  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  C_  B
3330, 32syl6ss 3191 . . . 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 461 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  B )  ->  pred (
v ,  A ,  R )  C_  B
)
3534ralrimiva 2626 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. v  e.  B  pred ( v ,  A ,  R )  C_  B
)
36 df-bnj19 28722 . 2  |-  (  TrFo ( B ,  A ,  R )  <->  A. v  e.  B  pred ( v ,  A ,  R
)  C_  B )
3735, 36sylibr 203 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  TrFo ( B ,  A ,  R )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    u. cun 3150    C_ wss 3152   U_ciun 3905    predc-bnj14 28713    FrSe w-bnj15 28717    trClc-bnj18 28719    TrFow-bnj19 28721
This theorem is referenced by:  bnj1136  29027
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-bnj17 28712  df-bnj14 28714  df-bnj13 28716  df-bnj15 28718  df-bnj18 28720  df-bnj19 28722
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