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Theorem recexprlemelu 6927
Description: Membership in the upper cut of  B. Lemma for recexpr 6942. (Contributed by Jim Kingdon, 27-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
Assertion
Ref Expression
recexprlemelu  |-  ( C  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  C  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
Distinct variable groups:    x, y, A   
x, B, y    x, C, y

Proof of Theorem recexprlemelu
StepHypRef Expression
1 elex 2619 . 2  |-  ( C  e.  ( 2nd `  B
)  ->  C  e.  _V )
2 ltrelnq 6669 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
32brel 4438 . . . . . 6  |-  ( y 
<Q  C  ->  ( y  e.  Q.  /\  C  e.  Q. ) )
43simprd 112 . . . . 5  |-  ( y 
<Q  C  ->  C  e. 
Q. )
5 elex 2619 . . . . 5  |-  ( C  e.  Q.  ->  C  e.  _V )
64, 5syl 14 . . . 4  |-  ( y 
<Q  C  ->  C  e. 
_V )
76adantr 270 . . 3  |-  ( ( y  <Q  C  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  C  e.  _V )
87exlimiv 1530 . 2  |-  ( E. y ( y  <Q  C  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  C  e.  _V )
9 breq2 3809 . . . . 5  |-  ( x  =  C  ->  (
y  <Q  x  <->  y  <Q  C ) )
109anbi1d 453 . . . 4  |-  ( x  =  C  ->  (
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) )  <->  ( y  <Q  C  /\  ( *Q
`  y )  e.  ( 1st `  A
) ) ) )
1110exbidv 1748 . . 3  |-  ( x  =  C  ->  ( E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) )  <->  E. y ( y 
<Q  C  /\  ( *Q `  y )  e.  ( 1st `  A
) ) ) )
12 recexpr.1 . . . . 5  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
1312fveq2i 5232 . . . 4  |-  ( 2nd `  B )  =  ( 2nd `  <. { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) } ,  { x  |  E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >. )
14 nqex 6667 . . . . . 6  |-  Q.  e.  _V
152brel 4438 . . . . . . . . . 10  |-  ( x 
<Q  y  ->  ( x  e.  Q.  /\  y  e.  Q. ) )
1615simpld 110 . . . . . . . . 9  |-  ( x 
<Q  y  ->  x  e. 
Q. )
1716adantr 270 . . . . . . . 8  |-  ( ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  x  e.  Q. )
1817exlimiv 1530 . . . . . . 7  |-  ( E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
1918abssi 3078 . . . . . 6  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  C_  Q.
2014, 19ssexi 3936 . . . . 5  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  e.  _V
212brel 4438 . . . . . . . . . 10  |-  ( y 
<Q  x  ->  ( y  e.  Q.  /\  x  e.  Q. ) )
2221simprd 112 . . . . . . . . 9  |-  ( y 
<Q  x  ->  x  e. 
Q. )
2322adantr 270 . . . . . . . 8  |-  ( ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  x  e.  Q. )
2423exlimiv 1530 . . . . . . 7  |-  ( E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  x  e.  Q. )
2524abssi 3078 . . . . . 6  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  C_  Q.
2614, 25ssexi 3936 . . . . 5  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  e.  _V
2720, 26op2nd 5825 . . . 4  |-  ( 2nd `  <. { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) } ,  { x  |  E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >. )  =  { x  |  E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) }
2813, 27eqtri 2103 . . 3  |-  ( 2nd `  B )  =  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
2911, 28elab2g 2748 . 2  |-  ( C  e.  _V  ->  ( C  e.  ( 2nd `  B )  <->  E. y
( y  <Q  C  /\  ( *Q `  y )  e.  ( 1st `  A
) ) ) )
301, 8, 29pm5.21nii 653 1  |-  ( C  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  C  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   {cab 2069   _Vcvv 2610   <.cop 3419   class class class wbr 3805   ` cfv 4952   1stc1st 5816   2ndc2nd 5817   Q.cnq 6584   *Qcrq 6588    <Q cltq 6589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-2nd 5819  df-qs 6199  df-ni 6608  df-nqqs 6652  df-ltnqqs 6657
This theorem is referenced by:  recexprlemm  6928  recexprlemopu  6931  recexprlemupu  6932  recexprlemdisj  6934  recexprlemloc  6935  recexprlem1ssu  6938  recexprlemss1u  6940
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