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Theorem bnj1137 34452
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 3631 . . . . 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 34341 . . . . . . . . 9  |-  pred ( X ,  A ,  R )  C_  A
65sseli 3485 . . . . . . . 8  |-  ( v  e.  pred ( X ,  A ,  R )  ->  v  e.  A )
7 bnj906 34389 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  v  e.  A )  ->  pred ( v ,  A ,  R ) 
C_  trCl ( v ,  A ,  R ) )
87adantlr 712 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  A )  ->  pred (
v ,  A ,  R )  C_  trCl (
v ,  A ,  R ) )
96, 8sylan2 472 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  pred ( X ,  A ,  R ) )  ->  pred ( v ,  A ,  R )  C_  trCl (
v ,  A ,  R ) )
10 bnj906 34389 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
1110sselda 3489 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  pred ( X ,  A ,  R ) )  -> 
v  e.  trCl ( X ,  A ,  R ) )
12 bnj18eq1 34386 . . . . . . . . 9  |-  ( y  =  v  ->  trCl (
y ,  A ,  R )  =  trCl ( v ,  A ,  R ) )
1312ssiun2s 4359 . . . . . . . 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 3499 . . . . . 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 34451 . . . . . . . . . . 11  |-  trCl (
y ,  A ,  R )  C_  A
1716rgenw 2815 . . . . . . . . . 10  |-  A. y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  C_  A
18 iunss 4356 . . . . . . . . . 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 3485 . . . . . . . 8  |-  ( v  e.  U_ y  e. 
trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  ->  v  e.  A )
2120, 8sylan2 472 . . . . . . 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 34449 . . . . . . . . . . . 12  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
) )  ->  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
23223expia 1196 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( y  e.  trCl ( X ,  A ,  R )  ->  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) ) )
2423ralrimiv 2866 . . . . . . . . . 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 4356 . . . . . . . . . 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 3489 . . . . . . . 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 3499 . . . . . 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 3654 . . . . . 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 3522 . . . . 5  |-  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  C_  B
3330, 32syl6ss 3501 . . . 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 473 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  v  e.  B )  ->  pred (
v ,  A ,  R )  C_  B
)
3534ralrimiva 2868 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. v  e.  B  pred ( v ,  A ,  R )  C_  B
)
36 df-bnj19 34150 . 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 366    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804    u. cun 3459    C_ wss 3461   U_ciun 4315    predc-bnj14 34141    FrSe w-bnj15 34145    trClc-bnj18 34147    TrFow-bnj19 34149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-reg 8010  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-1o 7122  df-bnj17 34140  df-bnj14 34142  df-bnj13 34144  df-bnj15 34146  df-bnj18 34148  df-bnj19 34150
This theorem is referenced by:  bnj1136  34454
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