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Theorem bnj1136 29293
Description: Technical lemma for bnj69 29306. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1136.1  |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
bnj1136.2  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
bnj1136.3  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
Assertion
Ref Expression
bnj1136  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  B )
Distinct variable groups:    y, A    y, R    y, X
Allowed substitution hints:    th( y)    ta( y)    B( y)

Proof of Theorem bnj1136
StepHypRef Expression
1 bnj1136.2 . . . 4  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
21biimpri 198 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  th )
3 bnj1136.1 . . . . 5  |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
4 bnj1148 29292 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  e.  _V )
5 bnj893 29226 . . . . . . 7  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  e.  _V )
6 simp1 957 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
) )  ->  R  FrSe  A )
7 bnj1127 29287 . . . . . . . . . . 11  |-  ( y  e.  trCl ( X ,  A ,  R )  ->  y  e.  A )
873ad2ant3 980 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
) )  ->  y  e.  A )
9 bnj893 29226 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  y  e.  A )  ->  trCl ( y ,  A ,  R )  e.  _V )
106, 8, 9syl2anc 643 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
) )  ->  trCl (
y ,  A ,  R )  e.  _V )
11103expia 1155 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( y  e.  trCl ( X ,  A ,  R )  ->  trCl (
y ,  A ,  R )  e.  _V ) )
1211ralrimiv 2780 . . . . . . 7  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  e.  _V )
13 iunexg 5979 . . . . . . 7  |-  ( ( 
trCl ( X ,  A ,  R )  e.  _V  /\  A. y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  e.  _V )  ->  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  e.  _V )
145, 12, 13syl2anc 643 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  e.  _V )
154, 14bnj1149 29090 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  e. 
_V )
163, 15syl5eqel 2519 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  B  e.  _V )
173bnj1137 29291 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  TrFo ( B ,  A ,  R )
)
183bnj931 29068 . . . . 5  |-  pred ( X ,  A ,  R )  C_  B
1918a1i 11 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_  B )
20 bnj1136.3 . . . 4  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
2116, 17, 19, 20syl3anbrc 1138 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ta )
221, 20bnj1124 29284 . . 3  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
232, 21, 22syl2anc 643 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  C_  B )
24 bnj906 29228 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
25 bnj1125 29288 . . . . . . 7  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
) )  ->  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
26253expia 1155 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( y  e.  trCl ( X ,  A ,  R )  ->  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) ) )
2726ralrimiv 2780 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
28 ss2iun 4100 . . . . . 6  |-  ( A. y  e.  trCl  ( X ,  A ,  R
)  trCl ( y ,  A ,  R ) 
C_  trCl ( X ,  A ,  R )  ->  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  C_  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( X ,  A ,  R ) )
29 bnj1143 29088 . . . . . 6  |-  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( X ,  A ,  R )  C_  trCl ( X ,  A ,  R )
3028, 29syl6ss 3352 . . . . 5  |-  ( A. y  e.  trCl  ( X ,  A ,  R
)  trCl ( y ,  A ,  R ) 
C_  trCl ( X ,  A ,  R )  ->  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
3127, 30syl 16 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
3224, 31unssd 3515 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  C_  trCl ( X ,  A ,  R ) )
333, 32syl5eqss 3384 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  B  C_  trCl ( X ,  A ,  R
) )
3423, 33eqssd 3357 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    u. cun 3310    C_ wss 3312   U_ciun 4085    predc-bnj14 28979    FrSe w-bnj15 28983    trClc-bnj18 28985    TrFow-bnj19 28987
This theorem is referenced by:  bnj1408  29332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-reg 7552  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-1o 6716  df-bnj17 28978  df-bnj14 28980  df-bnj13 28982  df-bnj15 28984  df-bnj18 28986  df-bnj19 28988
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