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Theorem bnj1136 29027
Description: Technical lemma for bnj69 29040. 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 197 . . 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 29026 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  e.  _V )
5 bnj893 28960 . . . . . . 7  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  e.  _V )
6 simp1 955 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
) )  ->  R  FrSe  A )
7 bnj1127 29021 . . . . . . . . . . 11  |-  ( y  e.  trCl ( X ,  A ,  R )  ->  y  e.  A )
873ad2ant3 978 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
) )  ->  y  e.  A )
9 bnj893 28960 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  y  e.  A )  ->  trCl ( y ,  A ,  R )  e.  _V )
106, 8, 9syl2anc 642 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
) )  ->  trCl (
y ,  A ,  R )  e.  _V )
11103expia 1153 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( y  e.  trCl ( X ,  A ,  R )  ->  trCl (
y ,  A ,  R )  e.  _V ) )
1211ralrimiv 2625 . . . . . . 7  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  e.  _V )
13 iunexg 5767 . . . . . . 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 642 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  e.  _V )
154, 14bnj1149 28824 . . . . 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 2367 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  B  e.  _V )
173bnj1137 29025 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  TrFo ( B ,  A ,  R )
)
183bnj931 28802 . . . . 5  |-  pred ( X ,  A ,  R )  C_  B
1918a1i 10 . . . 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 1136 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ta )
221, 20bnj1124 29018 . . 3  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
232, 21, 22syl2anc 642 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  C_  B )
24 bnj906 28962 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
25 bnj1125 29022 . . . . . . 7  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
) )  ->  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
26253expia 1153 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( y  e.  trCl ( X ,  A ,  R )  ->  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) ) )
2726ralrimiv 2625 . . . . 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 3920 . . . . . 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 28822 . . . . . 6  |-  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( X ,  A ,  R )  C_  trCl ( X ,  A ,  R )
3028, 29syl6ss 3191 . . . . 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 15 . . . 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 3351 . . 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 3222 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  B  C_  trCl ( X ,  A ,  R
) )
3423, 33eqssd 3196 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    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:  bnj1408  29066
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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