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Theorem bnj1137 32338
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 2532 . . . . 5  |-  ( v  e.  B  <->  v  e.  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) ) )
3 elun 3608 . . . . 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 32227 . . . . . . . . 9  |-  pred ( X ,  A ,  R )  C_  A
65sseli 3463 . . . . . . . 8  |-  ( v  e.  pred ( X ,  A ,  R )  ->  v  e.  A )
7 bnj906 32275 . . . . . . . . 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 32275 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
1110sselda 3467 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  pred ( X ,  A ,  R ) )  -> 
v  e.  trCl ( X ,  A ,  R ) )
12 bnj18eq1 32272 . . . . . . . . 9  |-  ( y  =  v  ->  trCl (
y ,  A ,  R )  =  trCl ( v ,  A ,  R ) )
1312ssiun2s 4325 . . . . . . . 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 3477 . . . . . 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 32337 . . . . . . . . . . 11  |-  trCl (
y ,  A ,  R )  C_  A
1716rgenw 2901 . . . . . . . . . 10  |-  A. y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  C_  A
18 iunss 4322 . . . . . . . . . 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 3463 . . . . . . . 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 32335 . . . . . . . . . . . 12  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
) )  ->  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
23223expia 1190 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( y  e.  trCl ( X ,  A ,  R )  ->  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) ) )
2423ralrimiv 2828 . . . . . . . . . 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 4322 . . . . . . . . . 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 3467 . . . . . . . 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 3477 . . . . . 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 3631 . . . . . 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 3500 . . . . 5  |-  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  C_  B
3330, 32syl6ss 3479 . . . 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 2830 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. v  e.  B  pred ( v ,  A ,  R )  C_  B
)
36 df-bnj19 32037 . 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 1370    e. wcel 1758   A.wral 2799    u. cun 3437    C_ wss 3439   U_ciun 4282    predc-bnj14 32028    FrSe w-bnj15 32032    trClc-bnj18 32034    TrFow-bnj19 32036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-reg 7921  ax-inf2 7961
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-om 6590  df-1o 7033  df-bnj17 32027  df-bnj14 32029  df-bnj13 32031  df-bnj15 32033  df-bnj18 32035  df-bnj19 32037
This theorem is referenced by:  bnj1136  32340
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